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 it; 'Tis therefore to be presum'd that he is dead. After Twelve years expectation, Mr. Dryden gave it to the Players, having upon perusal of it, found that it deserv'd a better Fate than to be buried in obscurity: I have heard him say, that finding a Scene wanting, he supply'd it; and many have affirm'd, that the stile of it' (of the play, that is, in general; not by any means of the additional scene) 'is proper to the Subject, which is that the French call Basse Comedy (sic). The turns of it are natural,' (I should be loth to bet on the chance of any reader's agreement with the bookseller on this point) and the resemblance of one man to another, has not only been the foundation of this, but of many other Plays. Plautus his Amphitrion, was the Original of all, and Shakespear and Moliere have copied him with success. Nevertheless, if this Play in it self should be a trifle, which you have no reason to suspect, because that incomparable Person would not from his Ingenious labours lose so much time as to write a whole Scene in it, which in it self sufficiently makes you amends, for Poetry being like Painting, where, if a great Master have but touch'd upon an ordinary Piece, he makes it of Value to all understanding Men; as I doubt not but this will be by his Additions: As it is, I am resolv'd to detain you no longer from it, but subscribe myself,

After this somewhat Gampian example of publisher's English, the prologue naturally follows; and no reader who considers the date will be surprised to learn that neither prologue nor epilogue is presentable to eyes polite. Nor does either of these effusions-though certainly this is not an inevitable corollary to be inferred from the preceding proposition-show any trace whatever of the laureate's master-hand. Nor, again, will any reader who takes account of the subject and the model indicated by the admiring publisher be much amazed at the information that even under the regency of and  some passages of this 'Orphan Play'—as the pathetic prologue defines it—may have seemed almost exceptionally outrageous even to an audience not yet chaste enough—as in our own pure and happy period—to applaud the rankest ribaldry of foreign farces, while proscribing for


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 The general investigations developed in this treatise will, in the conclusion, be applied to the theory of triangles of shortest lines, of which we shall introduce only a couple of important theorems. If $a,$  $b,$  $c$ be the sides of such a triangle (they will be regarded as magnitudes of the first order); $A,$  $B,$  $C$  the angles opposite; $\alpha,$  $\beta,$  $\gamma$  the measures of curvature at the angular points; $\sigma$  the area of the triangle, then, to magnitudes of the fourth order, $\frac{1}{3}(\alpha + \beta + \gamma)\sigma$  is the excess of the sum $A + B + C$  over two right angles. Further, with the same degree of exactness, the angles of a plane rectilinear triangle whose sides are $a,$  $b,$  $c,$ are respectively

$$\begin{aligned} A &- \tfrac{1}{12}(2\alpha + \beta + \gamma)\sigma \\ B &- \tfrac{1}{12}(\alpha + 2\beta + \gamma)\sigma \\ C &- \tfrac{1}{12}(\alpha + \beta + 2\gamma)\sigma. \end{aligned}$$ We see immediately that this last theorem is a generalization of the familiar theorem first established by. By means of this theorem we obtain the angles of a plane triangle, correct to magnitudes of the fourth order, if we diminish each angle of the corresponding spherical triangle by one-third of the spherical excess. In the case of non-spherical surfaces, we must apply unequal reductions to the angles, and this inequality, generally speaking, is a magnitude of the third order. However, even if the whole surface differs only a little from the spherical form, it will still involve also a factor denoting the degree of the deviation from the spherical form. It is unquestionably important for the higher geodesy that we be able to calculate the inequalities of those reductions and thereby obtain the thorough conviction that, for all measurable triangles on the surface of the earth, they are to be regarded as quite insensible. So it is, for example, in the case of the greatest triangle of the triangulation carried out by the author. The greatest side of this triangle is almost fifteen geographical miles, and the excess of the sum of its three angles over two right angles amounts almost to fifteen seconds. The three reductions of the angles of the plane triangle are $4.95113,$ $4.95104,$  $4''.95131.$  Besides, the author also developed the missing terms of the fourth order in the above expressions. Those for the sphere possess a very simple form. However, in the case of measurable triangles upon the earth’s surface, they are quite insensible. And in the example here introduced they would have diminished the first reduction by only two units in the fifth decimal place and increased the third by the same amount.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


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 wages—which are barely twelve shillings a-week—five shillings a-week for more than two months to a poor labouring man, to prevent his going to the workhouse, and being parted from his wife and children.'

'Noble, indeed!' said Lancelot. 'I do not wonder now at the effect his conversation just now had on me.'

'Has he been talking to you?' said Honoria, eagerly. He seldom speaks to any one.'

'He has to me; and so well, that were I sure that the poor were as ill off as he says, and that I had the power of altering the system a hair, I could find it in my heart to excuse all political grievance-mongers, and turn one myself.'

Claude Mellot clapped his white woman-like hands.

'Bravo! bravo! Oh wonderful conversion! Lancelot has at last discovered that, beside the 'glorious Past,' there is a Present worthy of his sublime notice! We may now hope, in time, that he will discover the existence of a Future!'

'But, Mr. Mellot,' said Honoria, 'why have you been so unfaithful to your original? why have you, like all artists, been trying to soften and refine on your model?'

'Because, my dear lady, we are bound to see everything in its ideal,—not as it is, but as it ought to be, and will be, when the vices of this pitiful civilised world are exploded, and sanitary reform, and a variety of occupation, and harmonious education, let each


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Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/95

 NEW GENERAL INVESTIGATIONS

OF

CURVED SURFACES

[1825]

Although the real purpose of this work is the deduction of new theorems concerning its subject, nevertheless we shall first develop what is already known, partly for the sake of consistency and completeness, and partly because our method of treatment is different from that which has been used heretofore. We shall even begin by advancing certain properties concerning plane curves from the same principles.

1.

In order to compare in a convenient manner the different directions of straight lines in a plane with each other, we imagine a circle with unit radius described in the plane about an arbitrary centre. The position of the radius of this circle, drawn parallel to a straight line given in advance, represents then the position of that line. And the angle which two straight lines make with each other is measured by the angle between the two radii representing them, or by the arc included between their extremities. Of course, where precise definition is necessary, it is specified at the outset, for every straight line, in what sense it is regarded as drawn. Without such a distinction the direction of a straight line would always correspond to two opposite radii.

2.

In the auxiliary circle we take an arbitrary radius as the first, or its terminal point in the circumference as the origin, and determine the positive sense of measuring the arcs from this point (whether from left to right or the contrary); in the opposite direction the arcs are regarded then as negative. Thus every direction of a straight line is expressed in degrees, etc., or also by a number which expresses them in parts of the radius.


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 Such lines as differ in direction by $360^{\circ},$ or by a multiple of $360^{\circ},$  have, therefore, precisely the same direction, and may, generally speaking, be regarded as the same. However, in such cases where the manner of describing a variable angle is taken into consideration, it may be necessary to distinguish carefully angles differing by $360^{\circ}.$

If, for example, we have decided to measure the arcs from left to right, and if to two straight lines $l,$  $l'$ correspond the two directions $L,$  $L',$  then $L' - L$  is the angle between those two straight lines. And it is easily seen that, since $L' - L$ falls between $-180^{\circ}$  and $+180^{\circ},$  the positive or negative value indicates at once that $l'$  lies on the right or the left of $l,$  as seen from the point of intersection. This will be determined generally by the sign of $\sin(L' - L).$

If $aa'$  is a part of a curved line, and if to the tangents at $a,$  $a'$ correspond respectively the directions $\alpha,$  $\alpha',$  by which letters shall be denoted also the corresponding points on the auxiliary circles, and if $A,$  $A'$  be their distances along the arc from the origin, then the magnitude of the arc $\alpha\alpha'$  or $A' - A$  is called the amplitude of $aa'.$

The comparison of the amplitude of the arc $aa'$ with its length gives us the notion of curvature. Let $l$  be any point on the arc $aa',$ and let $\lambda,$  $\Lambda$  be the same with reference to it that $\alpha,$  $A$  and $\alpha',$  $A'$  are with reference to $a$  and $a'.$  If now $\alpha\lambda$  or $\Lambda - A$  be proportional to the part $al$  of the arc, then we shall say that $aa'$  is uniformly curved throughout its whole length, and we shall call

$$\frac{\Lambda - A}{al}$$

the measure of curvature, or simply the curvature. We easily see that this happens only when $aa'$  is actually the arc of a circle, and that then, according to our definition, its curvature will be $\pm\frac{1}{r},$ if $r$  denotes the radius. Since we always regard $r$ as positive, the upper or the lower sign will hold according as the centre lies to the right or to the left of the arc $aa'$  ($a$  being regarded as the initial point, $a'$  as the end point, and the directions on the auxiliary circle being measured from left to right). Changing one of these conditions changes the sign, changing two restores it again.

On the contrary, if $\Lambda - A$ be not proportional to $al,$  then we call the arc non-uniformly curved and the quotient

$$\frac{\Lambda - A}{al}$$


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 may then be called its mean curvature. Curvature, on the contrary, always presupposes that the point is determined, and is defined as the mean curvature of an element at this point; it is therefore equal to

$$\frac{d\Lambda}{d\,al}.$$

We see, therefore, that arc, amplitude, and curvature sustain a similar relation to each other as time, motion, and velocity, or as volume, mass, and density. The reciprocal of the curvature, namely,

$$\frac{d\,al}{d\Lambda},$$

is called the radius of curvature at the point $l.$ And, in keeping with the above conventions, the curve at this point is called concave toward the right and convex toward the left, if the value of the curvature or of the radius of curvature happens to be positive; but, if it happens to be negative, the contrary is true.

If we refer the position of a point in the plane to two perpendicular axes of coordinates to which correspond the directions $0$  and $90^{\circ},$ in such a manner that the first coordinate represents the distance of the point from the second axis, measured in the direction of the first axis; whereas the second coordinate represents the distance from the first axis, measured in the direction of the second axis; if, further, the indeterminates $x,$  $y$  represent the coordinates of a point on the curved line, $s$  the length of the line measured from an arbitrary origin to this point, $\phi$  the direction of the tangent at this point, and $r$  the radius of curvature; then we shall have

$$\begin{aligned} dx &= \cos\phi. ds, \\ dy &= \sin\phi. ds, \\ r &= \frac{ds}{d\phi}. \end{aligned}$$

If the nature of the curved line is defined by the equation $V = 0,$ where $V$  is a function of $x,$  $y,$  and if we set

$$dV = p\, dx + q\, dy,$$

then on the curved line

$$p\, dx + q\, dy = 0.$$

Hence

$$p\cos\phi + q\sin\phi = 0,$$


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Page:TheTreesOfGreatBritainAndIreland vol02B.djvu/84

242 Fountains Abbey, and close to two very tall spruce. This, though hard to measure correctly owing to its crowded position, which makes a satisfactory illustration impossible, is over 80 feet high and 11 feet in girth, but is forked at about 7 feet from the ground.

The next best is at Strathfieldsaye, a very spreading tree in damp soil, also forking near the ground. The two stems measure 9 feet 6 inches and 8 feet 3 inches, and the height in 1903 was about 75 feet, the branches weeping to the ground on all sides (Plate 71). At Althorp there is a fine old specimen on the lawn, of a more upright type, which in 1903 was 63 feet by 8 feet 10 inches. At Walcot, in Shropshire, the seat of the Earl of Powis, is one of the best grown trees I have seen, with a bole about 25 feet high, and measuring 60 feet by 8 feet 8 inches. At Mr. Heelas' residence, near Reading, part of the old White Knights estate, is a tree, probably planted 150 years ago, which Henry in 1904 found to be 67 feet by 8 feet. At Arley Castle there is a fine tree dividing into three stems, of which the largest is 6 feet 7 inches in girth and nearly 70 feet high.

At Hardwick, Bury St. Edmunds, there is a tree, forked at 30 feet up, 60 feet by 5 feet ro inches. At Beauport, Sussex, a tree measured in 1904, 65 feet by 7 feet. At Osberton, Notts, the seat of Mr. F. Savile Foljambe, there is a remarkably spreading old tree about 42 feet high, and dividing near the ground into three stems, each about 6 feet in girth. It has some layered branches which are over 20 feet high, and the total circumference is no less than 80 paces. Bunbury, Arboretum Notes, p. 140, mentions as the largest hemlock in the country one growing at Bowood, Wiltshire, the seat of the Marquess of Lansdowne, which, however, cannot now be found.

In Scotland, where the tree should succeed well, I have seen none of great size, except the tree at Dunkeld, which is growing in a thick wood of conifers mixed with beech on rocky ground, close to the Hermitage bridge. This is mentioned by Hunter as being 80 feet high by 1o feet in girth. Mr. D. Keir twenty years later made it 85 feet by 11 feet, and when he showed it to me in 1906 I found that, though the top is not easy to see, it is probably as much as 90 feet, and looks as if it would grow taller. It divides at about 12 feet into several stems, and is believed to be 140 to 150 years old.

At Dalkeith there was in 1891 a tree 42 feet high by 10 feet 6 inches in girth; and at Buchanan Castle, Stirlingshire, the seat of the Duke of Montrose, one measuring 45 feet by 6 feet 10 inches.

In Ireland the largest known to us is one at Carton, the seat of the Duke of Leinster, which in 1903 was 45 feet by 6½ feet.

Opinions as to the value of this wood differ a good deal, and I have no personal experience in the matter. Sargent says that it is light, soft, not strong, brittle coarse, crooked-grained, difficult to work, liable to wind-shake and splinter, and not


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 and therefore

$$\tan\phi = -\frac{p}{q}.$$

We have also

$$\cos\phi. dp + \sin\phi. dq - (p\sin\phi - q\cos\phi)\, d\phi = 0.$$

If, therefore, we set, according to a well known theorem,

$$\begin{aligned} dp &= P\, dx + Q\, dy, \\ dq &= Q\, dx + R\, dy, \end{aligned}$$

then we have

$$(P\cos^{2}\phi + 2Q\cos\phi\sin\phi + R\sin^{2}\phi)\, ds = (p\sin\phi - q\cos\phi)\, d\phi,$$

therefore

$$\frac{1}{r} = \frac{P\cos^{2}\phi + 2Q\cos\phi\sin\phi + R\sin^{2}\phi} {p\sin\phi - q\cos\phi},$$

or, since

$$ \cos\phi = \frac{\mp q}{\sqrt{p^{2} + q^{2}}},\qquad \sin\phi = \frac{\pm p}{\sqrt{p^{2} + q^{2}}}; $$

$$ \pm \frac{1}{r} = \frac{Pq^{2} - 2Qpq + Rp^{2}}{(p^{2} + q^{2})^{3/2}}. $$

4.

The ambiguous sign in the last formula might at first seem out of place, but upon closer consideration it is found to be quite in order. In fact, since this expression depends simply upon the partial differentials of $V,$ and since the function $V$  itself merely defines the nature of the curve without at the same time fixing the sense in which it is supposed to be described, the question, whether the curve is convex toward the right or left, must remain undetermined until the sense is determined by some other means. The case is similar in the determination of $\phi$ by means of the tangent, to single values of which correspond two angles differing by $180^{\circ}.$  The sense in which the curve is described can be specified in the following different ways.

I. By means of the sign of the change in $x.$ If $x$  increases, then $\cos\phi$  must be positive. Hence the upper signs will hold if $q$  has a negative value, and the lower signs if $q$  has a positive value. When $x$  decreases, the contrary is true.

II. By means of the sign of the change in $y.$ If $y$  increases, the upper signs must be taken when $p$  is positive, the lower when $p$  is negative. The contrary is true when $y$  decreases.

III. By means of the sign of the value which the function $V$ takes for points not on the curve. Let $\delta x,$  $\delta y$ be the variations of $x,$  $y$  when we go out from the


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Page:TheTreesOfGreatBritainAndIreland vol02B.djvu/88

244 diversifolia, and is remarkable for its limited distribution. It occurs at elevations of 2500 to 3000 feet, usually on dry rocky banks of mountain streams along the Blue Ridge, extending from south-western Virginia through South Carolina to northern Georgia. Sargent states that it occurs either in small groves or mingled with other species, and describes it as a beautiful tree of compact pyramidal habit, with dense dark-green lustrous foliage. Elwes saw it on the Blue Ridge in 1893, and brought home young plants, which, however, died in a year or two.

This tree was discovered in 1850 by Professor L.R. Gibbes. It was first raised in the Arnold Arboretum in 1881, and has proved there quite hardy. It was introduced from thence to England in 1886. There are two or three small specimens in the collection at Kew which are three or four feet in height and have a bushy, spreading habit. This species, judging from the slow rate of growth at Kew, is not likely to attain to timber size in England, and we know of no trees of any size living in this country.

A tree forming in the Himalayas, according to Hooker, a stately blunt pyramid, with branches spreading like the cedar, but not so stiff, and drooping gracefully on all sides, attaining 120 feet in height and 28 feet in girth. Incultivation in England it assumes a bushy habit, and never makes a clean stem, the trunk being concealed by the dense pendulous branches.

Bark thick and rough. Branchlets light brown in colour with a short and not very dense pubescence. Leaves long, 1 to 1¼ inch, narrow linear, gradually tapering towards the acute and recurved apex, serrulate in margin; upper surface dark green and deeply grooved; lower surface silvery white, the bands of stomata extending almost to the margins. Buds globose, flattened on the top; scales ovate, acute, pubescent.

Cones sessile, ovoid, an inch long, composed of about twenty-five woody scales, which are nearly orbicular, vertically striate, shining, showing externally a thickened ridge a little distance from and parallel to the thin entire margin; bract concealed. Seed two-thirds the length of the scale, with an oblong-ovate wing, which is decurrent on the outer side of the seed to its base.


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 curve toward the right, at right angles to the tangent, that is, in the direction $\phi + 90^{\circ};$ and let the length of this normal be $\delta\rho.$  Then, evidently, we have

$$\begin{aligned} \delta x &= \delta\rho. \cos(\phi + 90^{\circ}), \\ \delta y &= \delta\rho. \sin(\phi + 90^{\circ}), \end{aligned}$$

or

$$\begin{aligned} \delta x &= -\delta\rho. \sin\phi, \\ \delta y &= +\delta\rho. \cos\phi. \end{aligned}$$

Since now, when $\delta\rho$  is infinitely small,

$$\begin{aligned} \delta V &= p\, \delta x + q\, \delta y \\ &= (-p\sin\phi + q\cos\phi)\, \delta\rho \\ &= \mp\delta\rho\sqrt{p^{2} + q^{2}} \end{aligned}$$

and since on the curve itself $V$  vanishes, the upper signs will hold if $V,$ on passing through the curve from left to right, changes from positive to negative, and the contrary. If we combine this with what is said at the end of Art. 2, it follows that the curve is always convex toward that side on which $V$  receives the same sign as

$$Pq^{2} - 2Qpq + Rp^{2}.$$

For example, if the curve is a circle, and if we set

$$V = x^{2} + y^{2} - a^{2}$$

then we have

$$\begin{array}{c} p = 2x,\qquad q = 2y, \\ P = 2,\qquad Q = 0,\qquad R = 2, \\ Pq^{2} - 2Qpq + Rp^{2} = 8y^{2} + 8x^{2} = 8a^{2}, \\ (p^{2} + q^{2})^{3/2} = 8a^{3}, \\ r = \pm a \end{array}$$

and the curve will be convex toward that side for which

$$x^{2} + y^{2} > a^{2},$$

as it should be.

The side toward which the curve is convex, or, what is the same thing, the signs in the above formulæ, will remain unchanged by moving along the curve, so long as

$$\frac{\delta V}{\delta\rho}$$

does not change its sign. Since $V$  is a continuous function, such a change can take place only when this ratio passes through the value zero. But this necessarily presupposes that $p$  and $q$ become zero at the same time. At such a point the radius


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246 At Southampton, in the Red Lodge nursery belonging to Mr. W. H. Rogers, there was a tree twenty-five years old in 1884, about 20 feet high, which bore cones in profusion. At Kew a specimen planted in a sheltered position lived for many years, but ultimately succumbed. Sir Joseph Hooker knew of no good specimen nearer London than one on a south slope near Leith Hill in a very sheltered and well-watered valley.

At Fota, in the S.W. of Ireland, the seat of Lord Barrymore, Henry measured a tree about 4o feet by 4 feet 10 inches in 1904; and there are trees at Kilmacurragh and Powerscourt, in Co. Wicklow, which are about 30 feet high, all of very branching bushy habit, and with several main stems.

Sargent has never seen a specimen in the United States.

A tree attaining in Japan about 100 feet in height and 12 feet in girth, forming in England a small tree with a short bole and a dense crown of foliage, with numerous branches and pendulous branchlets.

Young shoots greyish in colour and quite glabrous. Leaves pectinately arranged, variable in size, the smaller on the upper side of the shoot, some of these being directed outwards at right angles to the general plane of the foliage. They are oblong, uniform in width, ¼ to 1 inch long, shining and dark green above with a median furrow continued to the rounded and emarginate apex; lower surface with green midrib and two narrow well-defined white bands of stomata; margin quite entire. Buds reddish, ovoid, slightly acute at the apex: scales glabrous on the surface, ciliate in margin.

Cones elongated ovoid, on a stalk about ¼ inch long, pendulous or deflected, composed of five series of orbicular scales, which are rounded at the apex and at the base and have a slightly bevelled margin. Bract included, very short and bifid. Seed with a long wing decurrent half-way along its outer side.

This tree has been much confused with the other Japanese species, from which it is very distinct in botanical characters. Koehne's proposed name, Tsuga Araragi, is not adopted by us, the name Sieboldii being the first one under the correct genus Tsuga.


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 of curvature becomes infinite or the curvature vanishes. Then, generally speaking, since here

$$-p\sin\phi + q\cos\phi$$

will change its sign, we have here a point of inflexion.

5.

The case where the nature of the curve is expressed by setting $y$  equal to a given function of $x,$ namely, $y = X,$  is included in the foregoing, if we set

$$V = X - y.$$

If we put

$$dX = X'\, dx,\qquad dX' = X''\, dx,$$

then we have

$$\begin{array}{c} p = X',\qquad q = -1, \\ P = X'', \qquad Q = 0,\qquad R = 0, \end{array}$$

therefore

$$\pm\frac{1}{r} = \frac{X''}{(1 + X'^{2})^{3/2}}.$$

Since $q$  is negative here, the upper sign holds for increasing values of $x.$ We can therefore say, briefly, that for a positive $X$  the curve is concave toward the same side toward which the $y$ -axis lies with reference to the $x$ -axis; while for a negative $X$  the curve is convex toward this side.

6.

If we regard $x,$  $y$ as functions of $s,$  these formulæ become still more elegant. Let us set

$$\begin{alignedat}{2} \frac{dx}{ds} &= x',\qquad& \frac{dx'}{ds} &= x'', \\ \frac{dy}{ds} &= y',\qquad& \frac{dy'}{ds} &= y''. \end{alignedat}$$

Then we shall have

$$\begin{alignedat}{2} x' &= \cos\phi,\qquad & y' &= \sin\phi, \\ x &= -\frac{\sin\phi}{r},\qquad & y &= \frac{\cos\phi}{r}; \end{alignedat} $$

or

$$ \begin{alignedat}{2} y' &= -rx,\qquad& x' &= ry, \end{alignedat}$$


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Rh

, the father of Lemminkainen.

, or, the Evil Power.

, the mother of Ilmarinen.

, the mistress of Pohjola.

, one of the daughters of Tuoni, and the mother of the Plagues.

, Daughter of Creation, a name applied to Ilmatar, and other celestial goddesses.

, name of a bay.

, a name of Kauppi.

, name of a cow.

, or, the God of Hades.

, or, Hades.

= Mana.

, Daughter of Mana.

name of a cow.

, ''the mother of Väinämöinen’s supplanter. She is usually identified with the Virgin Mary''.

, name or epithet of a cowherd who has been exposed to the rain.

, the goddess of the Rudder.

, the Woodlands, from metsa, a forest.

, the Mistress of the Forests, the spouse of Tapio.

, a name of Mielikki.

, a dog’s name.

, name of a cow.

, the son of Tapio.

, a name of Kaleva.

= Kalevala.

, an epithet of Väinämöinen.

, the daughter of Osmo.

, the constellation of the Great Bear.

, pet name for the bear.

, a name of the Evil Power.

, the Daughter of the Sun.

, the personified Frost.

, apparently the same as Tuuri.

, the son of the Sun.

, vide Sampsa.

, the nymph of the Mountain-Ash tree.

, the handmaid of Marjatta.

, a name of Pohjola.

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Page:TheTreesOfGreatBritainAndIreland vol02B.djvu/94

248 per 100 cubic feet. The growth of the tree from seed is very slow at first as in the allied species.

Tsuga Sieboldii was introduced into Europe by Siebold in 1850. Cones both of this species and of Tsuga diversifolia were brought from Japan by John Gould Vetch in 1861, and the latter species was sent out under the name Abies Tsuga, var. nana. Specimens cultivated at Kew as Tsuga Sieboldii, var. nana, belong to Tsuga diversifolia.

Though both species have been introduced long enough to prove their hardiness in favoured parts of the South of England, we have never seen even a moderately large tree, and doubt much if either species will attain timber size in this country. The Japanese hemlocks seem to prefer a light moist rich soil, free from lime, with shade and shelter from cold winds. They will not grow at all on the limestone soil of Colesborne. The best specimen we know is in the garden of Mr. W. H. Griffiths at Campden, Gloucestershire, and is about 15 feet high. It bore cones in 1905.

Sargent says that Tsuga Sieboldii is one of the most graceful and satisfactory of the exotic conifers cultivated in American gardens, where it promises to grow to a large size; but in the garden of Mr. Hunnewell at Wellesley, Massachusetts, which I visited in May 1904, I noted that it had been almost killed to the snow line by the exceptionally severe winter of 1903–1904, though it had produced cones in the preceding year.


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 or also

$$1 = r(x'y - y'x),$$

so that

$$x'y - y'x$$

represents the curvature, and

$$\frac{1}{x'y - y'x}$$

the radius of curvature.

7.

We shall now proceed to the consideration of curved surfaces. In order to represent the directions of straight lines in space considered in its three dimensions, we imagine a sphere of unit radius described about an arbitrary centre. Accordingly, a point on this sphere will represent the direction of all straight lines parallel to the radius whose extremity is at this point. As the positions of all points in space are determined by the perpendicular distances $x,$  $y,$  $z$ from three mutually perpendicular planes, the directions of the three principal axes, which are normal to these principal planes, shall be represented on the auxiliary sphere by the three points $(1),$  $(2),$  $(3).$  These points are, therefore, always $90^{\circ}$  apart, and at once indicate the sense in which the coordinates are supposed to increase. We shall here state several well known theorems, of which constant use will be made.

1) The angle between two intersecting straight lines is measured by the arc [of the great circle] between the points on the sphere which represent their directions.

2) The orientation of every plane can be represented on the sphere by means of the great circle in which the sphere is cut by the plane through the centre parallel to the first plane.

3) The angle between two planes is equal to the angle between the great circles which represent their orientations, and is therefore also measured by the angle between the poles of the great circles.

4) If $x,$  $y,$  $z;$ $x',$  $y',$  $z'$  are the coordinates of two points, $r$  the distance between them, and $L$  the point on the sphere which represents the direction of the straight line drawn from the first point to the second, then

$$\begin{alignedat}{2} x' &= x &&+ r\cos(1)L, \\ y' &= y &&+ r\cos(2)L, \\ z' &= z &&+ r\cos(3)L. \end{alignedat}$$

5) It follows immediately from this that we always have

$$\cos^{2}(1)L + \cos^{2}(2)L + \cos^{2}(3)L = 1$$


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 [and] also, if $L'$  is any other point on the sphere,

$$\cos(1)L. \cos(1)L' + \cos(2)L. \cos(2)L' + \cos(3)L. \cos(3)L' = \cos LL'.$$

We shall add here another theorem, which has appeared nowhere else, as far as we know, and which can often be used with advantage.

Let $L,$ $L',$  $L,$  $L$  be four points on the sphere, and $A$  the angle which $LL$  and $L'L$  make at their point of intersection. [Then we have]

$$\cos LL'. \cos LL' - \cos LL''. \cos L'L = \sin LL. \sin L'L''. \cos A.$$

The proof is easily obtained in the following way. Let

$$AL = t,\qquad AL' = t',\qquad AL = t,\qquad AL = t;$$

we have then

$$\begin{alignedat}{6} &\cos L L'   &&= \cos t  &&\cos t'   &&+ \sin t  &&\sin t'  &&\cos A, \\ &\cos LL &&= \cos t&&\cos t &&+ \sin t&&\sin t'&&\cos A, \\ &\cos L L  &&= \cos t  &&\cos t  &&+ \sin t  &&\sin t'' &&\cos A, \\ &\cos L' L &&= \cos t' &&\cos t &&+ \sin t' &&\sin t'''&&\cos A. \end{alignedat}$$

Therefore

$$\begin{alignedat}{1} \cos LL' &\cos LL' - \cos LL \cos L'L \\ &= \cos A \{\cos t \cos t'  \sin t\sin t' + \cos t\cos t'\sin t \sin t' \\ &\qquad\qquad - \cos t\cos t\sin t'\sin t - \cos t'\cos t\sin t\sin t\} \\ &= \cos A (\cos t \sin t - \cos t\sin t)         (\cos t'\sin t  - \cos t \sin t') \\ &= \cos A \sin (t' - t) \sin(t - t') \\ &= \cos A \sin LL' \sin L'L. \end{alignedat}$$

Since each of the two great circles goes out from $A$ in two opposite directions, two supplementary angles are formed at this point. But it is seen from our analysis that those branches must be chosen, which go in the same sense from $L$ toward $L'$  and from $L'$  toward $L.$

Instead of the angle $A,$ we can take also the distance of the pole of the great circle $LL$  from the pole of the great circle $L'L.$  However, since every great circle has two poles, we see that we must join those about which the great circles run in the same sense from $L$  toward $L$  and from $L'$  toward $L,$  respectively.

The development of the special case, where one or both of the arcs $LL'$  and $L'L$ are $90^{\circ},$  we leave to the reader.

6) Another useful theorem is obtained from the following analysis. Let $L,$  $L',$  $L''$ be three points upon the sphere and put


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 1200, was Chan-chêng, a district included in Cochin China. It might be conjectured that specimens of Tu-Ting-yao, included among these imports, found their way to Japan from Chan-chêng, and were identified with their port of shipment rather than with their original place of manufacture, as was the case with the well known "Old Japan" ware of Western collectors, which, though produced at Arita, was everywhere called Imari-yaki, after its place of export (Imari). Against such a theory, however, has to be set the well established fact that there existed, during the Sung dynasty, a much brisker direct trade between China and Japan than between the former and Cochin China. Indeed, the very author (Chao Ju-kua) quoted by Dr. Hirth in this context, says that large Chinese junks were engaged, in his time, transporting timber—notably planks of the Cryptomeria Japonica—from Japan to the Chinese port of Ch'üan-chou, and it is scarcely credible that these junks, on their return journey, would have failed to bring Chinese keramic wares to Japan, where such objects of art had been held in high esteem ever since the sev- enth or eighth century of the Christian era. Of course a general inference of this sort does not necessarily include a special product like the Tu-Ting-yao. Specimens of that particular ware may have reached Japan viâ Cochin China, and thus been erroneously attributed to the factories of the latter. Unfortunately the evidence available does not suffice to elucidate this matter.

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 art during that long interval. But in 1127 the Sung emperors changed their capital from Peking to Nankin, and that event, generally spoken of in Chinese history as "the passage of the Sung to the south," resulted in the division of the empire into two Kingdoms, the invading Tartars holding sway in the north, and the Sung sovereigns reigning in the south. Tradition says that the change of capital seriously influenced the potter's trade. The factory at Ting-chou in Pechili was transferred to Nan-chang in Kiang-si, and the manufacture of Ting-yao, ceasing at the former place, was re-commenced at the latter. This Nan-chang was in the immediate neighbourhood of Ching-tê-chên, the great centre of Chinese keramic industry. But in the early part of the twelfth century the resources of the place do not appear to have been developed, for the Ting-yao produced there did not enjoy as high a reputation as its northern predecessor. Connoisseurs distinguished the two wares, that of the northern factory of Ting-chou and that of the southern factory of Nan-chang, as Pai-ting, or white Ting, and Nan-ting, or southern Ting. The latter was also called Fan-ting, or "rice-flour" Ting. The manufacture of the former lasted from 960 to 1126; that of the latter from 1127 to 1279. These dates refer, of course, to Sung Ting-yao only. The reader will readily understand that as Nan-chang, or Chingtê-chên, continued in after years to be the very metropolis of Chinese pottery, its experts did not lose but rather developed, their ability to produce Ting-yao. Under the Sung emperors, however, it is asserted that the material employed for the biscuit of the Ting-yao made at Nan-chang was not so fine or close-grained as that used by the Ting-chou potters. The


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 $$\begin{alignedat}{6} &\cos L &&(1) = x,       &&\cos L   &&(2) = y,       &&\cos L   &&(3) = z, \\ &\cos L' &&(1) = x',     &&\cos L'  &&(2) = y',      &&\cos L'  &&(3) = z', \\ &\cos L&&(1) = x,\quad&&\cos L &&(2) = y,\quad&&\cos L &&(3) = z. \end{alignedat}$$

We assume that the points are so arranged that they run around the triangle included by them in the same sense as the points $(1),$  $(2),$  $(3).$ Further, let $\lambda$  be that pole of the great circle $L'L''$  which lies on the same side as $L.$  We then have, from the above lemma,

$$\begin{alignedat}{3} &y'z &&- z'y &&= \sin L'L''. \cos\lambda(1), \\ &z'x &&- x'z &&= \sin L'L''. \cos\lambda(2), \\ &x'y &&- y'x &&= \sin L'L''. \cos\lambda(3). \end{alignedat}$$

Therefore, if we multiply these equations by $x,$  $y,$  $z$ respectively, and add the products, we obtain

$$xy'z + x'yz + xyz' - xyz' - x'yz - xy'z = \sin L'L''. \cos\lambda L,$$

wherefore, we can write also, according to well known principles of spherical trigonometry,

$$\begin{alignedat}{2} \sin L'L''. &\sin L L''&&. \sin L' \\ = \sin L'L''. &\sin L L' &&. \sin L'' \\ = \sin L'L''. &\sin L'L''&&. \sin L, \end{alignedat}$$

if $L,$  $L',$  $L''$ denote the three angles of the spherical triangle. At the same time we easily see that this value is one-sixth of the pyramid whose angular points are the centre of the sphere and the three points $L,$  $L',$  $L$ (and indeed positive'', if etc.).

8.

The nature of a curved surface is defined by an equation between the coordinates of its points, which we represent by

$$f(x, y, z) = 0.$$

Let the total differential of $f(x, y, z)$ be

$$P\, dx + Q\, dy + R\, dz,$$

where $P,$  $Q,$  $R$ are functions of $x,$  $y,$  $z.$  We shall always distinguish two sides of the surface, one of which we shall call the upper, and the other the lower. Generally speaking, on passing through the surface the value of $f$ changes its sign, so that, as long as the continuity is not interrupted, the values are positive on one side and negative on the other.


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 The direction of the normal to the surface toward that side which we regard as the upper side is represented upon the auxiliary sphere by the point $L.$ Let

$$\cos L(1) = X,\qquad \cos L(2) = Y,\qquad \cos L(3) = Z.$$

Also let $ds$  denote an infinitely small line upon the surface; and, as its direction is denoted by the point $\lambda$ on the sphere, let

$$\cos \lambda(1) = \xi,\qquad \cos \lambda(2) = \eta,\qquad \cos \lambda(3) = \zeta.$$

We then have

$$dx = \xi\, ds,\qquad dy = \eta\, ds,\qquad dz = \zeta\, ds,$$

therefore

$$P\xi + Q\eta + R\zeta = 0,$$

and, since $\lambda L$ must be equal to $90^{\circ},$  we have also

$$X\xi + Y\eta + Z\zeta = 0.$$

Since $P,$  $Q,$  $R,$ $X,$  $Y,$  $Z$  depend only on the position of the surface on which we take the element, and since these equations hold for every direction of the element on the surface, it is easily seen that $P,$  $Q,$  $R$  must be proportional to $X,$  $Y,$  $Z.$  Therefore

$$P = X\mu,\qquad Q = Y\mu,\qquad R = Z\mu.$$

Therefore, since

$$\begin{array}{c} X^{2} + Y^{2} + Z^{2} = 1; \\ \mu = PX + QY + RZ \end{array} $$

and

$$ \mu^{2} = P^{2} + Q^{2} + R^{2}, $$

or

$$ \mu = \pm\sqrt{P^{2} + Q^{2} + R^{2}}. $$

If we go out from the surface, in the direction of the normal, a distance equal to the element $\delta\rho,$ then we shall have

$$\delta x = X\, \delta\rho,\qquad \delta y = Y\, \delta\rho,\qquad \delta z = Z\, \delta\rho$$

and

$$\delta f = P\, \delta x + Q\, \delta y + R\, \delta z = \mu\, \delta\rho.$$

We see, therefore, how the sign of $\mu$ depends on the change of sign of the value of $f$  in passing from the lower to the upper side.

9.

Let us cut the curved surface by a plane through the point to which our notation refers; then we obtain a plane curve of which $ds$  is an element, in connection with which we shall retain the above notation. We shall regard as the upper side of the plane that one on which the normal to the curved surface lies. Upon this plane


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250

I. Leaflets not serrate; usually entire or sinuate (Plate 73).

II. ''Leaflets serrate. Leaf-scars without a pubescent band on their upper edge.''

 * Leaflets glabrous beneath, except for the axil tufts.

 * Leaflets pubescent beneath.

III. ''Leaflets serrate. Leaf-scars with a transverse pubescent band on their upper edge.''


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284

, name of a mountain.

, the North.

, the North Country; (a) A dark and dismal country to the north of Lapland, but sometimes identified with Lapland itself; (b) The castle or homestead of Louhi, to which the name of the country itself was applied.

, the North Wind.

, name of a cow.

, the headman of a village. (Herod, according to the commentators.)

, a cataract, said to be the same as Turja.

, an epithet of Lemminkainen.

, an island, especially the island now called Kronstadt.

, a magic corn, salt and coin-mill.

, the genius of agriculture (from pellon or pelto, a field), the servant or agent of Väinämöinen.

a province of Finland.

, a Sound in Pohjola.

, a nymph who colours flowers blue.

 * the protecting nymphs of ducks.

, Finland.

, the nymph of the veins.

, Death, or the God of Death.

, name of an old woman.

(the land of still waters], a name of Väinölä.

, an epithet of Väinämöinen.

, the goddess of Summer.

, an ogress, the mother of the serpents.

, name of a cow.

, the goddess of the oak tree.

, name of the builder of a castle.

, the God of the Forests.

, the dominions of Tapio.

, the daughter of Tapio, but in some passages apparently identified with Mielikki.

, the goddess of the Clouds.

, Lemminkainen’s comrade in arms.

, the goddess of the Bird Cherry.

, name of a cow.

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 superiority in respect of delicacy of pâte and purity of colour thus rested with the Pai-ting. On the other hand, it is certain that technical processes were continually progressing through this long interval, and that the ware of the thirteenth century appreciably excelled that of the tenth in many important features. Chinese authors themselves state that the most beautiful pieces of northern Ting-yao were manufactured in the interval between 1111 and 1125; that is to say, just before political troubles compelled the transfer of the factory from Ting-chou to Nan-chang. It is reasonable to suppose that the potters did not leave their technical skill behind them at Ting-chou, and that their work in the south continued to improve as it had done in the north. But there is little to guide in this matter. Practical experience of the Ting-yao of the Sung dynasty leaves the student completely in the dark in respect of such fine distinctions as Pai-ting and Nan-ting.

It will be well to pass from the Ting-yao to the Ju-yao because the latter is said to have had its origin in technical defects of the former. The Tao-lu says that the glaze of the Ting-yao was often disfigured by fissures and other faults, due to imperfectly prepared materials or unskilled stoving. These blemishes proved so embarrassing and unavoidable that, in 1130, imperial orders were issued for the establishment of a special factory at Juchou, in the province of Kiang-su. Here the Ju-yao was produced. A Chinese writer, whose work was published at the close of the sixteenth century, says that the Chai and Ju porcelains, though the best of all, had ceased to exist in his time. The same writer's father, however, mentions that in his day specimens of Ju-yao were


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 constantly met with. Its chief variety, indeed its only variety so far as is known, was céladon. The primary purpose of establishing the factory at Juchou seems to have been to produce this highly esteemed monochrome. Julien, in his translation of the Tao-lu, falls into the error of calling the Ju-yao blue. He repeats this same error more than once, and his translation has thus been the means of deceiving connoisseurs with respect to the nature of several of the early Chinese wares. Julien's misconception—originally pointed out by the writer of these pages in 1881, in the Chrysanthemum—is alluded to at length in a recent brochure by Dr. Hirth. The fact is that the ideograph Ching employed by the author of the Tao-lu, may be properly rendered by "blue" in the great majority of cases. Thus the decoration on blue-and-white porcelain of later generations is called Chinghwa, and the same ideograph (ching) is used to describe the colour of the blue cotton coats worn by the lower orders in China. Julien was naturally deceived, not having the aid of research in loco and practical knowledge to verify his opinion. And of course when, in one instance, he had committed himself to the rendering "blue," he adhered to it consistently throughout, although it involved him in such anomalies as "onion-blue." The Ju-yao was unquestionably céladon. As for the nature of the ware, it is on record that the clay employed at Juchou was red; though there is nothing to indicate whether it was red originally or whether it became red in the furnace—a peculiar property, as will be seen by and by, representative of early Chinese céladon. In the Tao-lu the pâte of the Ju-yao is said to have been of fine quality and to have shone like copper, from


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 which and other evidences it may safely be inferred that there is here no question of translucid porcelain. Indeed, the statement may be made at once that all the choice céladons of the Sung, the Yuan, and even the Ming dynasties were stone-ware, showing considerable variation in respect to fineness of pâte and thinness of biscuit, but never becoming true translucid porcelain. Of course the conclusion is not to be drawn that to manufacture translucid porcelain was beyond the keramic competence of the time. On the contrary, an opaque pâte seems to have been deliberately preferred as a suitable basis for Ching-tsu (green-coloured) glaze. It is exceedingly probable that, like all early-period céladons, that Ju-yao had pâte which was white except at places directly exposed to the heat of the kiln; that, in short, its clay, when not protected by the glaze, assumed a red, or red-brown tinge in the oven. Paucity of authenticated specimens precludes absolute certainty about these points. Japanese connoisseurs maintain, however, that this so-called "iron base" is not necessarily found in the best examples of Ju-yao, though it does constitute a mark of authenticity in the case of early céladons generally. Reference will be made to the point hereafter. The glaze of the Ju-yao presented great merits. It was so soft and lustrous that connoisseurs compared it to congealed fat. Its colour varied from a green almost verging upon a blue to white barely tinged with green. Very frequently the surface was crackled; sometimes it was entirely without crackle, specimens of the latter character being most highly prized. In the Tao-lu it is stated that the crackle of the Ju-yao was of two varieties. In the first case the surface was covered with a network of close, cir-


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 cular meshes, not inaptly likened by connoisseurs to the roe of fishes: in the second, the crackle assumed an appearance distinguished as the "crab's claw" fashion. Julien's words are:—"Suivant l'ouvrage intitulé khe-kou-yao-lun, ceux qui offraient dans le vernis des yeux (boutons) de Tsong (Aralia) imitant les raies des pattes de crabes, étaient encore plus beaux:" Concerning this perplexing passage M. Salvétat makes only the following remark:—"Les pièces de cette sorte sont extrêmement rares." What is to be understood by crackle—if indeed there is question of crackle at all—compared, at the same time, to crab's claws and Aralia leaves? Dr. Bushell, than whom no higher authority is to be found among Western connoisseurs, says that these "crab's-claw" marks, so far from being an embellishment, were imperfections that they were simply little holes in the glaze, as though a crab had walked over it before firing. How pieces showing such blemishes can have been highly esteemed, one is at a loss to understand. The simile of "boutons de Aralia" has reference to chagrined glaze, something quite different from the so-called "crab's-claw" marking. In the glaze of these antique wares—especially in choice specimens of Sing Chün-yao, to be presently described—serrated, or V-shaped, lines are sometimes found that have been erroneously supposed to belong to the "crab's-claw" type. These lines are not crackle: they may be better described as dappling. They are not continuous, and it is almost inconceivable that they can have been produced at will. Resulting, probably, from more or less accidental conditions of temperature in the kiln, their effect, nevertheless, is to impart to the surface of the piece an appearance


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 The revenue of a Tartar or Arabian chief consists in profit. It arises principally from the milk and increase of his own herds and flocks, of which he himself superintends the management, and is the principal shepherd or herdsman of his own horde or tribe. It is, however, in this earliest and rudest state of civil government only that profit has ever made the principal part of the public revenue of a monarchical State.

Small republics have sometimes derived a considerable revenue from the profit of mercantile projects. The republic of Hamburg is said to do so from the profits of a public wine cellar and apothecary's shop. The State cannot be very great of which the sovereign has leisure to carry on the trade of a wine merchant or apothecary. The profit of a public bank has been a source of revenue to more considerable States. It has been so not only to Hamburg, but to Venice and Amsterdam. A revenue of this kind has even by some people been thought not below the attention of so great an empire as that of Great Britain. Reckoning the ordinary dividend of the Bank of England at five and a half per cent and its capital at ten millions seven hundred and eighty thousand pounds, the net annual profit, after paying the expense of management, must amount, it is said, to five hundred and ninety-two thousand nine hundred pounds. Government, it is pretended, could borrow this capital at three per cent interest, and by taking the


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 of softness and richness greatly prized by Chinese virtuosi.

In the "Illustrated Catalogue" of H'siang, translated by Dr. Bushell, three specimens of Ju-yao are depicted and described. They are céladons, having a glaze of "bluish-green" tint. The shapes and decoration are copied from ancient bronzes, designs incised and in relief—scroll pattern, spirals, and so forth—being employed to relieve the uniformity of the surface. To produce the peculiar delicate-green of this ware, the potters are said to have powdered red calcedony and added it to the glazing material. Experience had evidently taught them that this highly siliceous mineral turns white under the action of heat. It is seen from H'siang's pictures that the potters of the Ju-yao exercised admirable expertness in modelling, moulding, and graving. Chinese keramists have always been remarkable for such work. The great test of skill at the Juchou factory was the quality of the céladon glaze. Its delicate greenish blue colour, lustre, and softness combined with solidity belong to a very high range of achievement. The best specimens are said to have been without crackle, but H'siang's catalogue does not support that dictum.

In the early days of its manufacture certain pieces of Ju-yao were stoved in an inverted position, and in order to hide their unglazed rims slender rings of copper or silver were fitted to them. This device, originally dictated by the presence of a defect, subsequently came to be regarded as a distinctive mark, and continued to be employed long after its real purpose had ceased to exist. Allusion is made to the practice in the Tao-lu, but the translator, not unnatur-


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 management of the bank into its own hands, might make a clear profit of two hundred and sixty-nine thousand five hundred pounds a year. The orderly, vigilant and parsimonious administration of such aristocracies as those of Venice and Amsterdam, is extremely proper, it appears from experience, for the management of a mercantile project of this kind. But whether such a government as that of England, which, whatever may be its virtues, has never been famous for good economy; which, in time of peace, has generally conducted itself with the slothful and negligent profusion that is perhaps natural to monarchies; and in time of war has constantly acted with all the thoughtless extravagance that democracies are apt to fall into, could be safely trusted with the management of such a project, must at least be a good deal more doubtful.

The post-office is properly a mercantile project. The government advances the expense of establishing the different offices, and of buying or hiring the necessary horses or carriages, and is repaid with a large profit by the duties upon what is carried. It is, perhaps, the only mercantile project which has been successfully managed by, I believe, every sort of government. The capital to be advanced is not very considerable. There is no mystery in the business. The returns are not only certain, but immediate.

Princes, however, have frequently engaged in many other mercantile projects, and have been willing, like private persons, to mend their fortunes by becoming adventurers in the common branches of trade. They have scarce ever succeeded. The profusion with which the affairs of princes are always managed, renders it almost impossible that they should. The agents of a prince regard the wealth of their master as inexhaustible; are care-


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 ally perplexed by so strange a device, misinterpreted it. Most Western collectors have doubtless observed that bowls and cups of early-period Chinese wares generally have their rims protected, or, to speak more correctly, concealed by strips of metal, but few are likely to place much reliance on such a feature as a means of identification.

It is stated in the Tao-lu that on the bottom of Ju-yao vases flowers of the sesame were "painted." The same criticism applies to this as to the so-called "painted" designs of the Ting-yao: the Sung potters did not paint their wares by way either of decoration or of mark. The sesame flowers referred to here were either engraved in the paste, or moulded in slight relief under the glaze. Whether they were invariably employed to mark choice examples of the ware it is impossible to tell, but the specimens figured in H'siang's catalogue do not appear to be thus distinguished.

Not less important than the Ting-yao and the Ju-yao among wares of the Sung dynasty was the Kuan-yao, or "Imperial Ware." The quality of the ware did not procure for it its distinguished title. It was called "Imperial" simply because the Emperor himself (1107 ) established the factory where it was produced, at Peng-liang or Kai-fêng-fu, in the province of Honan. The clay is said to have been fine, but that it was not a porcelain stone may be gathered from the fact that the rims of the pieces, after stoving, sometimes had a purple-brown tint, and that the pâte at the base showed an iron-red colour. It was, in short, stone-ware. Wherever the thickness of the glaze did not suffice to conceal the paste completely, the dark colour of the latter became more or less


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 apparent. In the early days of the manufacture three varieties were produced; namely, clair-de-lune, called by the Chinese Yueh-pai, i.e., moon-white; light green, and dark green. The first variety did not win public esteem, and after a time the typical Kuan-yao became a céladon, of various shades of green. The great aim of the potters was to produce that peculiar delicate greenish blue compared to the tint of the firmament between rain-clouds. It will be seen, therefore, that as to colour the Ju-yao and Kuan-yao potters worked on the same lines. If a distinction is to be drawn between the results they achieved, the verdict will be that the Ju-yao showed a more delicate tinge and verged more closely upon the ideal cerulean than the Kuan-yao. Moreover, in richness and lustre of glaze the advantage is said to have been slightly on the side of the Ju-yao. As a general rule the Kuan-yao was crackled. The crackle was large and regular. It is compared by Chinese connoisseurs to the markings of starred ice. In their treatment of this crackle, the Kuan-yao potters struck out a new line. For while the piece was still hot, after emerging from the kiln—that is to say, before the cracks had entirely contracted in the process of cooling—vermilion was strongly rubbed over the surface. The pigment thus became permanently fixed in the main crackle, as well as in the fringe of subsidiary and almost imperceptible fissures that radiated from its edges; and in the result the surface of the piece appeared covered with a vermilion network, bordered here and there by little clouds of red. The effect was novel and pretty.

The potters of the Kuan-yao adopted the same models and decorative designs as the potters of the


text: Page:Poems Osgood.djvu/63

title: Page:Poems Osgood.djvu/63


text:
Page:The wealth of nations, volume 3.djvu/224

 less at what price they buy; are careless at what price they sell; are careless at what expense they transport his goods from one place to another. Those agents frequently live with the profusion of princes, and sometimes, too, in spite of that profusion, and by a proper method of making up their accounts, acquire the fortunes of princes. It was thus, as we are told by Machiavel, that the agents of Lorenzo of Medicis, not a prince of mean abilities, carried on his trade. The republic of Florence was several times obliged to pay the debt into which their extravagance had involved him. He found it convenient, accordingly, to give up the business of merchant, the business to which his family had originally owed their fortune, and in the latter part of his life to employ both what remained of that fortune, and the revenue of the State of which he had the disposal, in projects and expenses more suitable to his station.

No two characters seem more inconsistent than those of trader and sovereign. If the trading spirit of the English East India Company renders them very bad sovereigns, the spirit of sovereignty seems to have rendered them equally bad traders. While they were traders only they managed their trade successfully, and were able to pay from their profits a moderate dividend to the proprietors of their stock. Since they became sovereigns, with a revenue which, it is said, was originally more than three millions sterling, they have been obliged to beg the ordinary assistance of government in order to avoid immediate bankruptcy. In their former situation, their servants in India considered themselves as the clerks of merchants; in their present situation, those servants consider themselves as the ministers of sovereigns.


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title: Page:Poems Osgood.djvu/64


text: Page:Poems Osgood.djvu/65

title: Page:Poems Osgood.djvu/65


text: Oxford Book of English Verse 1250-1900/Ode to Psyche


text: Kalevala (Kirby 1907)/Glossary

title: Kalevala (Kirby 1907)/Glossary


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 A State may sometimes derive some part of its public revenue from the interest of money, as well as from the profits of stock. If it has amassed a treasure, it may lend a part of that treasure, either to foreign States, or to its own subjects.

The canton of Berne derives a considerable revenue by lending a part of its treasure to foreign States; that is, by placing it in the public funds of the different indebted nations of Europe, chiefly in those of France and England. The security of this revenue must depend, first, upon the security of the funds in which it is placed, or upon the good faith of the government which has the management of them; and, secondly, upon the certainty or probability of the continuance of peace with the debtor nation. In the case of a war, the very first act of hostility, on the part of the debtor nation, might be the forfeiture of the funds of its creditor. This policy of lending money to foreign States is, so far as I know, peculiar to the canton of Berne.

The city of Hamburg has established a sort of public pawnshop, which lends money to the subjects of the State upon pledges at six per cent interest. This pawnshop or Lombard, as it is called, affords a revenue, it is pretended, to the State of a hundred and fifty thousand crowns, which, at four-and-sixpence the crown, amounts to £33,750 sterling.

The government of Pennsylvania, without amassing any treasure, invented a method of lending, not money indeed, but what is equivalent to money, to its subjects. By advancing to private people, at interest, and upon land security to double the value, paper bills of credit to be redeemed fifteen years after their date, and in the meantime made


text: Index:Tree Crops (1953).pdf

title: Index:Tree Crops (1953).pdf


text: Template:Facsimile page

title: Template:Facsimile page


text:
Stanzas (Keats)

 * "Stanzas", in The Poetical Works of John Keats (1884)
 * "Stanzas", in The complete poetical works and letters of John Keats (1899), edited by Horace Elisha Scudder

title: Stanzas (Keats)


text: Category:Facsimile editions of works

title: Category:Facsimile editions of works


text: Category:Facsimile templates

title: Category:Facsimile templates


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Page:Messages of the President of the United States on the Relations of the United States to Spain (1898).djvu/38

 Rh Havana. The sufferings of the reconcentrado class have been terrible beyond description, but in Havana less than in other places on the island; yet Dr. Brunner, acting United States sanitary inspector here, informed me this morning that the death rate of the reconcentrados in this city was about 50 per cent in other places of the island, and when it is remembered that there have been several hundred thousands of these noncombatants, or pacificos, mainly women and children, who are concentrated under General Weyler's order, some idea can be formed of the mortality among them.

In this city matters are assuming better shape. Under charitable committees large numbers of them have been gathered together in houses, and are now fed and cared for by private subscriptions. I visited them yesterday and found their condition comparatively good, and there will be a daily improvement among them, though the lives of all can not be saved. I witnessed many terrible scenes and saw some die while I was present. I am told General Blanco will give $100,000 to the relief fund.

Consul-General.

Havana, November 27, 1897.

One of two gentlemen who visited the reconcentrados after they were concentrated in los fosos (the ditches) in this city handed me to-day the inclosed paper. The names of these two gentlemen are not signed to it for obvious reasons.

I personally know the gentleman who brought the communication, and know that he stands high in this community as a man of integrity and character.

The number of reconcentrados here, as I had the honor to report already, have always been less than elsewhere. I am able to say now that they will be taken care of and fed by committees of charitably disposed persons.

The ayuntaminto (city government) of Havana has ordered an additional tax of 5 per cent to be levied upon real estate in this city. I am informed that this sum has already reached the amount of $80,000, and that it is to be devoted exclusively to the relief of the reconcentrados.

The $100,000 reported in a former dispatch as being given by the Governor-General, is in Spanish silver, and is to be applied to the reconcentrados over the whole island.

Consul-General.


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Page:Messages of the President of the United States on the Relations of the United States to Spain (1898).djvu/39

 10

The public rumor of the horrible state in which the reconcentrados of the municipal council of Havana were found in the fosos having reached us, we resolved to pay a visit there, and we will relate to you what we saw with our own eyes:

Four hundred and sixty women and children thrown on the ground, heaped pell-mell as animals, some in a dying condition, others sick and others dead, without the slightest cleanliness, nor the least help, not even to give water to the thirsty, with neither religious or social help, each one dying wherever chance laid them, and for this limited number of reconcentrados the deaths ranged between forty and fifty daily, giving relatively ten days of life for each person, with great joy to the authorities who seconded fatidically the polities of General Weyler to exterminate the Cuban people, for these unhappy creatures received food only after having been for eight days in the Fosos, if during this time they could feed themselves with the bad food that the dying refused.

On this first visit we were present at the death of an old man who died through thirst. When we arrived he begged us, for God's sake, to give him a drink. We looked for it and gave it to him, and fifteen minutes afterwards he breathed his last, not having had even a drink of water for three days before. Among the many deaths we witnessed there was one scene impossible to forget. There is still alive the only living witness, a young girl of 18 years, whom we found seemingly lifeless on the ground; on her right-hand side was the body of a young mother, cold and rigid, but with her young child still alive clinging to her dead breast; on her left-hand side was also the corpse of a dead woman holding her son in a dead embrace; a little farther on a poor, dying woman having in her arms a daughter of fourteen, crazy with pain, who after five or six days also died in spite of the care she received.

In one corner a poor woman was dying, surrounded by her children, who contemplated her in silence, without a lament or shedding a tear, they themselves being real specters of hunger, emaciated in a horrible manner. This poor woman augments the catalogue already large of the victims of the reconcentration in the fosos.

The relation of the pictures of misery and horror which we have witnessed would be never ending were we to narrate them all.

It is difficult and almost impossible to express by writing the general aspect of the inmates of the fosos, because it is entirely beyond the line of what civilized humanity is accustomed to see; therefore no language can describe it.

The circumstances which the municipal authorities could reunite there are the following: Complete accumulation of bodies dead and alive, so that it was impossible to take one step without walking over them; the greatest want of cleanliness, want of light, air, and water; the food lacking in quality and quantity what was necessary to sustain life, thus sooner putting an end to these already broken-down systems; complete absence of medical assistance; and what is more terrible than all, no consolation whatever, religious or moral.

If any young girl came in any way nice looking, she was infallibly condemned to the most abominable of traffics.

At the sight of such horrible pictures the two gentlemen who went there resolved in spite of the ferocious Weyler, who was still Captain-General of the island, to omit nothing to remedy a deed so dishonorable


text:
Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/105

 we erect a normal whose direction is expressed by the point $\mathfrak{L}$ of the auxiliary sphere. By moving along the curved line, $\lambda$  and $L$ will therefore change their positions, while $\mathfrak{L}$  remains constant, and $\lambda L$  and $\lambda\mathfrak{L}$  are always equal to $90^{\circ}.$  Therefore $\lambda$  describes the great circle one of whose poles is $\mathfrak{L}.$  The element of this great circle will be equal to $\frac{ds}{r},$  if $r$  denotes the radius of curvature of the curve. And again, if we denote the direction of this element upon the sphere by $\lambda',$ then $\lambda'$  will evidently lie in the same great circle and be $90^{\circ}$  from $\lambda$  as well as from $\mathfrak{L}.$  If we now set

$$\cos \lambda'(1) = \xi',\qquad \cos \lambda'(2) = \eta',\qquad \cos \lambda'(3) = \zeta',$$

then we shall have

$$d\xi = \xi'\, \frac{ds}{r},\qquad d\eta = \eta'\, \frac{ds}{r},\qquad d\zeta = \zeta'\, \frac{ds}{r},$$

since, in fact, $\xi,$  $\eta,$  $\zeta$ are merely the coordinates of the point $\lambda$  referred to the centre of the sphere.

Since by the solution of the equation $f(x, y, z) = 0$ the coordinate $z$  may be expressed in the form of a function of $x,$  $y,$  we shall, for greater simplicity, assume that this has been done and that we have found

$$z = F(x, y).$$

We can then write as the equation of the surface

$$z - F(x, y) = 0,$$

or

$$f(x, y, z) = z - F(x, y).$$

From this follows, if we set

$$\begin{array}{c} dF(x, y) = t\, dx + u\, dy, \\ P = -t,\qquad Q = -u,\qquad R = 1, \end{array}$$

where $t,$  $u$ are merely functions of $x$  and $y.$  We set also

$$dt = T\, dx + U\, dy,\qquad du = U\, dx + V\, dy.$$

Therefore upon the whole surface we have

$$dz = t\, dx + u\, dy$$

and therefore, on the curve,

$$\zeta = t\xi + u\eta.$$

Hence differentiation gives, on substituting the above values for $d\xi,$  $d\eta,$  $d\zeta,$

$$\begin{aligned} (\zeta' - t\xi' - u\eta') \frac{ds}{r} &= \xi\, dt + \eta\, du \\ &= (\xi^{2}T + 2\xi\eta U + \eta^{2}V)\, ds, \end{aligned}$$


text:
On —— 'Think not of it, sweet one, so'

 * "On ****", in The Poetical Works of John Keats (1884)
 * "On —— 'Think not of it, sweet one, so'", in The complete poetical works and letters of John Keats (1899), edited by Horace Elisha Scudder

title: On —— 'Think not of it, sweet one, so'


text:
Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/106

 or

$$\begin{aligned} \frac{1}{r} &= \frac{\xi^{2}T + 2\xi\eta U + \eta^{2}V}{-\xi' t - \eta'SIC{\mu}{u} + \zeta'} \\ &= \frac{Z(\xi^{2}T + 2\xi\eta U + \eta^{2}V)}{X\xi' - Y\eta' + Z\zeta'} \\ &= \frac{Z(\xi^{2}T + 2\xi\eta U + \eta^{2}V)}{\cos L\lambda'}. \end{aligned}$$

10.

Before we further transform the expression just found, we will make a few remarks about it.

A normal to a curve in its plane corresponds to two directions upon the sphere, according as we draw it on the one or the other side of the curve. The one direction, toward which the curve is concave, is denoted by $\lambda',$ the other by the opposite point on the sphere. Both these points, like $L$  and $\mathfrak{L},$ are $90^{\circ}$  from $\lambda,$  and therefore lie in a great circle. And since $\mathfrak{L}$  is also $90^{\circ}$  from $\lambda,$ $\mathfrak{L} L = 90^{\circ} - L\lambda',$  or $= L\lambda' - 90^{\circ}.$  Therefore

$$\cos L\lambda' = \pm\sin \mathfrak{L} L,$$

where $\sin \mathfrak{L} L$ is necessarily positive. Since $r$  is regarded as positive in our analysis, the sign of $\cos L\lambda'$ will be the same as that of

$$Z(\xi^{2}T + 2\xi\eta U + \eta^{2}V).$$

And therefore a positive value of this last expression means that $L\lambda'$  is less than $90^{\circ},$ or that the curve is concave toward the side on which lies the projection of the normal to the surface upon the plane. A negative value, on the contrary, shows that the curve is convex toward this side. Therefore, in general, we may set also

$$\frac{1}{r} = \frac{Z(\xi^{2}T + 2\xi\eta U + \eta^{2}V)}{\sin \mathfrak{L} L},$$

if we regard the radius of curvature as positive in the first case, and negative in the second. $\mathfrak{L} L$  is here the angle which our cutting plane makes with the plane tangent to the curved surface, and we see that in the different cutting planes passed through the same point and the same tangent the radii of curvature are proportional to the sine of the inclination. Because of this simple relation, we shall limit ourselves hereafter to the case where this angle is a right angle, and where the cutting


text:
Page:Messages of the President of the United States on the Relations of the United States to Spain (1898).djvu/40

 Rh to humanity, and so contrary to all Christianity. They did not fail to find persons animated with like sentiments, who, putting aside all fear of the present situation, organized a private committee with the exclusive end of aiding materially and morally the reconcentrados. This neither has been nor is at present an easy task. The great number of the poor and scarcity of means make us encounter constant conflicts. This conflict is more terrible with the official elements and in a special manner with the mayor of the city and the civil authorities, who try by all means to annihilate this good work. The result of the collections are very insignificant if we bear in mind the thousands of people who suffer from the reconcentrations; but it serves for some consolation to see that in Havana some 159 children and 84 women are well cared for in the asylum erected in Cadiz street, No. 82, and 93 women and children are equally well located in a large saloon erected for them in the second story of the fosos, with good food and proper medical assistance, as also everything indispensable to civilized life.

According to the information which we have been able to acquire since August until the present day, 1,700 persons have entered the Fosos proceeding from Jaruco, Campo Florido, Guanabo, and Tapaste, in the province of Havana. Of these, only 243 are living now and are to be found in Cadiz street—82 in the saloon already mentioned and 61 in the Quinta del Rey and the Hospital Mercedes, the whole amounting to about 397, and of these a great many will die on account of the great sufferings and hunger they have gone through.

From all this we deduct that the number of deaths among the reconcentrados has amounted to 77 per cent.

Havana, December 3, 1897.

Referring to my cipher telegram of the 1st instant, which I beg to confirm, reading as follows:

I have the honor to state that rumors have been more or less frequent regarding the riotous intentions of some of the dissatisfied elements toward citizens of the United States dwelling here and in other parts of the island. Any riotous demonstrations here must come from the Spanish noncombatants or from the volunteer forces. I do not think there is any danger from the former, many of whom seem to be in favor of annexation, rather than for real autonomy or for an independent Cuban Republic. And I am inclined to think if General Blanco can manage the volunteers as yesterday he said he could, the trouble from that source is diminishing. The origin of the mobs in this city in the past has always been located in the ranks of the volunteers, who alone have organization and arms.

The Governor and Captain-General is now investigating the Matanzas rumors and will, I am sure, deal promptly with any conspirators found there.


text:
Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/107

 plane, therefore, is passed through the normal of the curved surface. Hence we have for the radius of curvature the simple formula

$$\frac{1}{r} = Z(\xi^{2}T + 2\xi\eta U + \eta^{2}V).$$

11.

Since an infinite number of planes may be passed through this normal, it follows that there may be infinitely many different values of the radius of curvature. In this case $T,$  $U,$  $V,$  $Z$ are regarded as constant, $\xi,$  $\eta,$  $\zeta$  as variable. In order to make the latter depend upon a single variable, we take two fixed points $M,$  $M'$ $90^{\circ}$  apart on the great circle whose pole is $L.$  Let their coordinates referred to the centre of the sphere be $\alpha,$  $\beta,$  $\gamma;$  $\alpha',$  $\beta',$  $\gamma'.$  We have then

$$\cos\lambda(1) = \cos\lambda M. \cos M(1) + \cos\lambda M'. \cos M'(1) + \cos\lambda L. \cos L(1).$$

If we set

$$\lambda M = \phi,$$

then we have

$$\cos\lambda M' = \sin\phi,$$

and the formula becomes

$$\begin{aligned} \xi  &= \alpha\cos\phi + \alpha'\sin\phi, \end{aligned}$$

and likewise

$$\begin{aligned} \eta &= \beta \cos\phi + \beta' \sin\phi, \\ \zeta &= \gamma\cos\phi + \gamma'\sin\phi. \end{aligned}$$

Therefore, if we set

$$\begin{aligned} A &= (\alpha^{2}T + 2\alpha\beta U + \beta^{2}V)Z, \\ B &= (\alpha\alpha'T + (\alpha'\beta + \alpha\beta')U + \beta\beta'V)Z, \\ C &= (\alpha'^{2}T + 2\alpha'\beta' U + \beta'^{2}V)Z, \end{aligned}$$

we shall have

$$\begin{aligned} \frac{1}{r} &= A\cos^{2}\phi + 2B\cos\phi \sin\phi + C\sin^{2}\phi \\ &= \frac{A + C}{2} + \frac{A - C}{2}\cos 2\phi + B\sin 2\phi. \end{aligned}$$

If we put

$$\begin{aligned} \frac{A - C}{2} &= E\cos 2\theta, \\ B &= E\sin 2\theta, \end{aligned}$$


text:
Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/108

 where we may assume that $E$  has the same sign as $\frac{A - C}{2},$ then we have

$$\frac{1}{r} = \tfrac{1}{2}(A + C) + E\cos 2(\phi - \theta).$$

It is evident that $\phi$  denotes the angle between the cutting plane and another plane through this normal and that tangent which corresponds to the direction $M.$ Evidently, therefore, $\frac{1}{r}$  takes its greatest (absolute) value, or $r$  its smallest, when $\phi = \theta;$  and $\frac{1}{r}$  its smallest absolute value, when $\phi = \theta + 90^{\circ}.$  Therefore the greatest and the least curvatures occur in two planes perpendicular to each other. Hence these extreme values for $\frac{1}{r}$ are

$$\tfrac{1}{2}(A + C) \pm \sqrt{\left(\frac{A - C}{2}\right)^{2} + B^{2}}.$$

Their sum is $A + C$ and their product $AC - B^{2},$  or the product of the two extreme radii of curvature is

$$= \frac{1}{AC - B^{2}}.$$

This product, which is of great importance, merits a more rigorous development. In fact, from formulæ above we find

$$AC - B^{2} = (\alpha\beta' -\beta\alpha')^{2}(TV - U^{2})Z^{2}.$$

But from the third formula in [Theorem] 6, Art. 7, we easily infer that

$$\alpha\beta' - \beta\alpha' = \pm Z,$$

therefore

$$AC - B^{2} = Z^{4}(TV - U^{2}).$$

Besides, from Art. 8 ,

$$\begin{aligned} Z &= \pm\frac{R}{\sqrt{P^{2} + Q^{2} + R^{2}}} \\ &= \pm\frac{1}{\sqrt{1 + t^{2} + u^{2}}}, \end{aligned}$$

therefore

$$AC - B^{2} = \frac{TV - U^{2}}{(1 + t^{2} + u^{2})^{2}}.$$

Just as to each point on the curved surface corresponds a particular point $L$ on the auxiliary sphere, by means of the normal erected at this point and the radius of


text:
La Belle Dame sans Merci (Keats)

 * "La Belle Dame sans Merci", in The Poetical Works of John Keats (1884)
 * "La Belle Dame sans Merci", in The complete poetical works and letters of John Keats (1899), edited by Horace Elisha Scudder
 * "La Belle Dame sans Merci" (revised, unsourced)

title: La Belle Dame sans Merci (Keats)


text:
Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/109

 the auxiliary sphere parallel to the normal, so the aggregate of the points on the auxiliary sphere, which correspond to all the points of a line on the curved surface, forms a line which will correspond to the line on the curved surface. And, likewise, to every finite figure on the curved surface will correspond a finite figure on the auxiliary sphere, the area of which upon the latter shall be regarded as the measure of the amplitude of the former. We shall either regard this area as a number, in which case the square of the radius of the auxiliary sphere is the unit, or else express it in degrees, etc., setting the area of the hemisphere equal to $360^{\circ}.$

The comparison of the area upon the curved surface with the corresponding amplitude leads to the idea of what we call the measure of curvature of the surface. If the former is proportional to the latter, the curvature is called uniform; and the quotient, when we divide the amplitude by the surface, is called the measure of curvature. This is the case when the curved surface is a sphere, and the measure of curvature is then a fraction whose numerator is unity and whose denominator is the square of the radius.

We shall regard the measure of curvature as positive, if the boundaries of the figures upon the curved surface and upon the auxiliary sphere run in the same sense; as negative, if the boundaries enclose the figures in contrary senses. If they are not proportional, the surface is non-uniformily curved. And at each point there exists a particular measure of curvature, which is obtained from the comparison of corresponding infinitesimal parts upon the curved surface and the auxiliary sphere. Let $d\sigma$  be a surface element on the former, and $d\Sigma$  the corresponding element upon the auxiliary sphere, then

$$\frac{d\Sigma}{d\sigma}$$

will be the measure of curvature at this point.

In order to determine their boundaries, we first project both upon the $xy$ -plane. The magnitudes of these projections are $Z\, d\sigma,$  $Z\, d\Sigma.$ The sign of $Z$  will show whether the boundaries run in the same sense or in contrary senses around the surfaces and their projections. We will suppose that the figure is a triangle; the projection upon the $xy$ -plane has the coordinates

$$x,\ y;\qquad x + dx,\ y + dy;\qquad x + \delta x,\ y + \delta y.$$

Hence its double area will be

$$2Z\, d\sigma = dx. \delta y - dy. \delta x.$$

To the projection of the corresponding element upon the sphere will correspond the coordinates:


text:
Index:Color standards and color nomenclature (Ridgway, 1912).djvu/styles.css

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Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/110

 $$\begin{array}{c} X, \\ X + \frac{\partial X}{\partial x}. dx + \frac{\partial X}{\partial y}. dy, \\ X + \frac{\partial X}{\partial x}. \delta x + \frac{\partial X}{\partial y}. \delta y, \end{array} \qquad \begin{array}{c} Y, \\ Y + \frac{\partial Y}{\partial x}. dx + \frac{\partial Y}{\partial y}. dy, \\ Y + \frac{\partial Y}{\partial x}. \delta x + \frac{\partial Y}{\partial y}. \delta y, \end{array}$$

From this the double area of the element is found to be

$$\begin{aligned} 2Z\, d\Sigma &= \phantom{-} \left(\frac{\partial X}{\partial x} . dx     + \frac{\partial X}{\partial y} . dy\right) \left(\frac{\partial Y}{\partial x} . \delta x     + \frac{\partial Y}{\partial y} . \delta y\right) \\ &\phantom{={}} -\left(\frac{\partial X}{\partial x} . \delta x     + \frac{\partial X}{\partial y} . \delta y\right) \left(\frac{\partial Y}{\partial x} . dx     + \frac{\partial Y}{\partial y} . dy\right) \\ &= \phantom{-} \left(\frac{\partial X}{\partial x} . \frac{\partial Y}{\partial y}     - \frac{\partial X}{\partial y} . \frac{\partial Y}{\partial x}\right) (dx . \delta y - dy . \delta x). \end{aligned}$$

The measure of curvature is, therefore,

$$= \frac{\partial X}{\partial x}. \frac{\partial Y}{\partial y} - \frac{\partial X}{\partial y}. \frac{\partial Y}{\partial x} = \omega.$$

Since

$$\begin{array}{c} X = -tZ,\qquad Y = -uZ, \\ (1 + t^{2} + u^{2})Z^{2} = 1, \end{array}$$

we have

$$\begin{aligned} dX &= -Z^{3}(1 + u^{2})\, dt + Z^{3}tu. du, \\ dY &= +Z^{3}tu. dt - Z^{3}(1 + t^{2})\, du, \end{aligned}$$

therefore

$$\begin{alignedat}{2} \frac{\partial X}{\partial x} &= Z^{3}\bigl\{-(1 + u^{2})T + tuU\bigr\},\qquad& \frac{\partial Y}{\partial x} &= Z^{3}\bigl\{tuT - (1 + t^{2})U\bigr\}, \\ \frac{\partial X}{\partial y} &= Z^{3}\bigl\{-(1 + u^{2})U + tuV\bigr\},\qquad& \frac{\partial Y}{\partial y} &= Z^{3}\bigl\{tuU - (1 + t^{2})V\bigr\}, \end{alignedat}$$

and

$$\begin{aligned} \omega &= Z^{6}(TV - U^{2})\bigl((1 + t^{2})(1 + u^{2}) - t^{2}u^{2}\bigr) \\ &= Z^{6}(TV - U^{2})(1 + t^{2} + u^{2}) \\ &= Z^{4}(TV - U^{2}) \\ &= \frac{TV - U^{2}}{(1 + t^{2} + u^{2})^{2}}, \end{aligned}$$

the very same expression which we have found at the end of the preceding article. Therefore we see that


text:
Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/111

 ::"The measure of curvature is always expressed by means of a fraction whose numerator is unity and whose denominator is the product of the maximum and minimum radii of curvature in the planes passing through the normal."

12.

We will now investigate the nature of shortest lines upon curved surfaces. The nature of a curved line in space is determined, in general, in such a way that the coordinates $x,$  $y,$  $z$ of each point are regarded as functions of a single variable, which we shall call $w.$  The length of the curve, measured from an arbitrary origin to this point, is then equal to

$$\int \sqrt{\left(\frac{dx}{dw}\right)^{2} + \left(\frac{dy}{dw}\right)^{2} + \left(\frac{dz}{dw}\right)^{2}}. dw.$$

If we allow the curve to change its position by an infinitely small variation, the variation of the whole length will then be

$$\begin{aligned} &\qquad\qquad = \int \frac{\frac{dx}{dw}. d\, \delta x          + \frac{dy}{dw}. d\, \delta y          + \frac{dz}{dw}. d\, \delta z}           {\sqrt{\left(\frac{dx}{dw}\right)^{2} + \left(\frac{dy}{dw}\right)^{2} + \left(\frac{dz}{dw}\right)^{2}}} = \frac{\frac{dx}{dw}. \delta x     + \frac{dy}{dw}. \delta y     + \frac{dz}{dw}. \delta z}      {\sqrt{\left(\frac{dx}{dw}\right)^{2} + \left(\frac{dy}{dw}\right)^{2} + \left(\frac{dz}{dw}\right)^{2}}} \\ &- \int\left\{ \delta x. d\frac{\frac{dx}{dw}} {\sqrt{\left(\frac{dx}{dw}\right)^{2} + \left(\frac{dy}{dw}\right)^{2} + \left(\frac{dz}{dw}\right)^{2}}}\right. + \delta y. d\frac{\frac{dy}{dw}} {\sqrt{\left(\frac{dx}{dw}\right)^{2} + \left(\frac{dy}{dw}\right)^{2} + \left(\frac{dz}{dw}\right)^{2}}} \\ &\qquad\qquad+ \left.\delta z. d\frac{\frac{dz}{dw}} {\sqrt{\left(\frac{dx}{dw}\right)^{2} + \left(\frac{dy}{dw}\right)^{2} + \left(\frac{dz}{dw}\right)^{2}}}\right\}. \end{aligned}$$

The expression under the integral sign must vanish in the case of a minimum, as we know. Since the curved line lies upon a given curved surface whose equation is

$$P\, dx + Q\, dy + R\, dz = 0,$$

the equation between the variations $\delta x,$  $\delta y,$  $\delta z$

$$P\, \delta z + Q\, \delta y + R\, \delta z = 0$$

must also hold. From this, by means of well known principles, we easily conclude that the differentials


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Page:Messages of the President of the United States on the Relations of the United States to Spain (1898).djvu/41

 12 The Weyler police have all been changed and the officers of the volunteers, too, when the Government here has reason to doubt their loyalty.

In consequence of all this, and the assurances of the governmental authorities that American life and property will, if necessary, be protected by them at a moment's notice, I have declined to make an application for the presence of one or more war ships in this harbor, and have advised those of our people who have wives and children here not to send them away, at least for the present, because such proceedings would not, in my opinion, be justifiable at this time, from the standpoint of personal security.

I still think that two war ships at least should be at Key West, prepared to move here at short notice, and that more of them should be sent to Dry Tortugas, and a coal station be established there. Such proceedings would seem to be in line with that prudence and foresight necessary to afford safety to the Americans residing on the island, and to their properties, both of which, I have every reason to know, are objects of the greatest concern to our Government.

Consul-General.

Havana, December 3, 1897.

I have the honor to state that a representative of a Madrid paper here says that:

The Lucha to-day publishes that Canalejas has said "that the economic condition of the Pinar del Rio province is deplorable, there being 40,000 reconcentrados absolutely destitute, 15,000 of which are children, most of whom are orphans; that they are unequally distributed throughout the different towns in the province, there being only 460 at the capital, city of Pinar del Rio, while in small towns like Consolacioa and Candelaria there are over 4,000. The municipalities can not incur any expense, because the taxes can not be collected, because most of the taxpayers, if not all, have been ruined by the war."

I am, etc.,

Consul-General.


text:
Author:Henry Marvin Wharton

Works

 * War Songs and Poems of the Southern Confederacy, 1861 - 1865 (1904)

title: Author:Henry Marvin Wharton


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 Rh

Havana, December 7, 1897.

(The consul-general informs the Assistant Secretary of State that measures for the relief of the reconcentrados are not sufficiently energetic to be effective, and that he is advised by the Governor-General that authority to admit articles of food and clothing from the United States to Cuban ports free of duty rested with the authorities at Madrid.)

I see no effects of the governmental distribution to the reconcentrados. I am informed that only $12,500, in Spanish silver, had been dedicated to the Havana province out of the $100,000 said to have been set aside for the purpose of relieving them on the island, and that reports from all parts of the province show that 50 per cent have already died and that many of those left will die. Most of these are women and children, I do not believe the Government here is really able to relieve the distress and sufferings of these people.

I am informed an order has been issued in some parts of the island suspending the distribution of rations to reconcentrados. The condition of these people is simply terrible.

I inclose herewith an official copy of the comparative mortality in Havana for the six months ending November 30. It will be perceived that there has been a great increase in the death rate, and without adequate means in the future to prevent it the mortality will increase. I hear of much suffering in the Spanish hospitals for want of food and among the Spanish soldiers. I hear, also, that the Spanish merchants in some parts of the island are placing their establishments in the names of foreigners in order to avoid their provisions being purchased on credit by the military administration, and that the Spanish army is suffering much from sickness and famine, and that a great deal of money is needed at once to relieve their condition. In some parts of the island, I am told, there is scarcely any food for soldiers or citizens, and that even cats are used for food purposes, selling at 30 cents apiece.

It is a fair inference therefore to draw from the existing conditions, that it is not possible for the Governor-General of this island to relieve the present situation with the means at his disposal.

I am, etc.,

Consul-General.

Havana, December 13, 1897. (Received December 18.)

I have the honor to make the following report:

The contest for and against autonomy is most unequal. For it, there are five or six of the head officers at the palace, and twenty or thirty other persons here in the city.


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Page:The London Magazine, volume 7 (January–June 1823).djvu/671

 with the waiter of the Double-barreled Gun and that young female attendant of the Princess, whose kitten had been persecuted by Juno, had framed the whole plot, and had written the letters which Mr. Schnackenberger had ascribed to her Highness. He had scarce patience to hear out the remainder. In some way or other, Von Pilsen had so far mistaken our hero, as to pronounce him “chicken-hearted:” and upon this ground, he invited his whole audience to an evening party at the public rooms of the Double-barreled Gun—where he promised to play off Mr. Schnackenberger as a glorious exhibition for this night only.

Furious with wrath, and moreover anxious to escape before Von Pilsen and his party should see him, and know that this last forgery no less than the others had succeeded in duping him into a punctual observance of the appointment, Mr. Schnackenberger rushed out of the room, seized his horse’s bridle—and was just on the point of mounting, when up came his female tormentor, Mrs. Sweetbread.

“Come, come, now,” said she, smiling in her most amiable manner; “we were both under a mistake yesterday morning: and both of us were too hasty. The booby of a lad took you to the Gun, when you wanted nothing but the Sow: you were a little ‘fresh,’ and didn’t know it; and I thought you did it on purpose. But I know better now. And here I am to fetch you back to the Sow: so come along: and we’ll forget and forgive on both sides.”

So saying, she would have taken his arm most lovingly: but Mr. Schnackenberger stoutly refused. He had nothing to do with her but to pay his bill; he wanted nothing of her but his back-sword, which he had left at the Sow; and he made a motion towards his stirrup. But Mrs. Sweetbread laid her hand upon his arm, and asked him tenderly—if her person were then so utterly disgusting to him that, upon thus meeting him again by his own appointment, he had at once forgotten all his proposals?

“Proposals! what proposals?” shrieked the persecuted student; “Appointment! what appointment?”

“Oh you base, low-lived villain! don’t you go for to deny it, now: didn’t you offer to be reconciled? didn’t you bid me to come here, that we might settle all quietly in the forest? Aye, and we will settle it: and nothing shall ever part us more; nothing in the world; for what God has joined

“Drunken old witch!” interrupted Mr. Jeremiah, now sufficiently admonished by the brandy fumes which assailed him as to the proximate cause of Mrs. Sweetbread’s boldness; “seek lovers elsewhere.” And hastily turning round to shake her off, he perceived to his horror that an immense crowd had by this time assembled behind them. In the rear, and standing upon the steps of the Forester’s house, stood Von Pilsen and his party, convulsed with laughter; immediately below them was the whole body of the hunters, who had called here for refreshment—upon whose faces struggled a mixed expression of merriment and wonder: and at the head of the whole company stood a party of butchers and butchers’ boys returning from the hunt, whose fierce looks and gestures made it evident that they sympathized with the wrongs of Mrs. Sweetbread, the relict of a man who had done honour to their body—and were prepared to avenge them in any way she might choose. She, mean time, whose whole mighty love was converted into mighty hatred by the opprobrious words and fierce repulse of Mr. Schnackenberger, called heaven and earth, and all present, to witness her wrongs; protested that he had himself appointed the meeting at the Forest-house; and in confirmation drew forth a letter.

At sight of the letter, a rattling peal of laughter from Mr. Von Pilsen left no room to doubt, in our student’s mind, from whose witty manufactory it issued; and a rattling peal of wrath from the butchers’ boys left no room to doubt in any body’s mind what would be its consequences. The letter was, in fact, pretty much what Mrs. Sweetbread alleged: it contained a large and unlimited offer of Mr. Schnackenberger’s large and unlimited person; professed an ardour of passion which could brook no delay; and entreated her to grant him an interview for the, 1823.