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 and consequently

$$AC + CB' = AC + BC. \cos \omega = AB - BC. (1- \cos \omega) = AB' - BC. (1 - \cos \omega),$$

 i.e. , the path from $A$  to $B'$ through the point $C$  is shorter than the shortest line,  Q.E.A.

16.

With the theorem of the preceding article we associate another, which we state as follows: If on a curved surface we imagine any line whatever, from the different points of which are drawn at right angles and toward the same side an infinite number of shortest lines of the same length, the curve which joins their other extremities will cut each of the lines at right angles. For the demonstration of this theorem no change need be made in the preceding analysis, except that $\phi$  must denote the length of the given curve measured from an arbitrary point; or rather, a function of this length. Thus all of the reasoning will hold here also, with this modification, that $S = 0$ for $r = 0$  is now implied in the hypothesis itself. Moreover, this theorem is more general than the preceding one, for we can regard it as including the first one if we take for the given line the infinitely small circle described about the centre $A.$ Finally, we may say that here also geometric considerations may take the place of the analysis, which, however, we shall not take the time to consider here, since they are sufficiently obvious.

17. We return to the formula

$$\sqrt{E\, dp^{2} + 2F\, dp. dq + G\, dq^{2}},$$

which expresses generally the magnitude of a linear element on the curved surface, and investigate, first of all, the geometric meaning of the coefficients $E,$  $F,$  $G.$ We have already said in Art. 5 that two systems of lines may be supposed to lie on the curved surface, $p$  being variable, $q$  constant along each of the lines of the one system; and $q$  variable, $p$  constant along each of the lines of the other system. Any point whatever on the surface can be regarded as the intersection of a line of the first system with a line of the second; and then the element of the first line adjacent to this point and corresponding to a variation $dp$ will be equal to $\sqrt{E}. dp,$ and the element of the second line corresponding to the variation $dq$  will be equal to $\sqrt{G}. dq.$ Finally, denoting by $\omega$  the angle between these elements, it is easily seen that we shall have

$$\cos \omega = \frac{F}{\sqrt{EG}}.$$


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 Furthermore, the area of the surface element in the form of a parallelogram between the two lines of the first system, to which correspond $q,$  $q + dq,$ and the two lines of the second system, to which correspond $p,$  $p + dp,$  will be

$$\sqrt{EG - F^{2}}\, dp. dq.$$

Any line whatever on the curved surface belonging to neither of the two systems is determined when $p$  and $q$ are supposed to be functions of a new variable, or one of them is supposed to be a function of the other. Let $s$  be the length of such a curve, measured from an arbitrary initial point, and in either direction chosen as positive. Let $\theta$  denote the angle which the element

$$ds = \sqrt{E\, dp^{2} + 2F\, dp. dq + G\, dq^{2}}$$

makes with the line of the first system drawn through the initial point of the element, and, in order that no ambiguity may arise, let us suppose that this angle is measured from that branch of the first line on which the values of $p$ increase, and is taken as positive toward that side toward which the values of $q$  increase. These conventions being made, it is easily seen that

$$\begin{aligned} \cos \theta. ds &= \sqrt{E}. dp + \sqrt{G}. \cos \omega. dq = \frac{E\, dp + F\, dq}{\sqrt{E}} \\ \sin \theta. ds &= \sqrt{G}. \sin \omega. dq = \frac{\sqrt{(EG - F^{2})}. dq}{\sqrt{E}} \end{aligned}$$

18.

We shall now investigate the condition that this line be a shortest line. Since its length $s$ is expressed by the integral

$$s = \int \sqrt{E\, dp^{2} + 2F\, dp. dq + G\, dq^{2}}$$

the condition for a minimum requires that the variation of this integral arising from an infinitely small change in the position become equal to zero. The calculation, for our purpose, is more simply made in this case, if we regard $p$ as a function of $q.$  When this is done, if the variation is denoted by the characteristic $\delta,$  we have

$$\begin{aligned} \delta s &= \int \frac{\left(   \frac{\partial E}{\partial p} . dp^{2}  + 2\frac{\partial F}{\partial p} . dp . dq  + \frac{\partial G}{\partial p} . dq^{2}  \right) \delta p  + (2E\, dp + 2F\, dq)\, d\, \delta p}{2\, ds} \\ &= \frac{E\, dp + F\, dq}{ds}. \delta p\,+\end{aligned}$$

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


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 customs we are the best informed. The State required that every free citizen should fit himself for defending it in war, and should, upon that account, learn his military exercises. Bat it left him to learn them of such masters as he could find, and it seems to have advanced nothing for this purpose, but a public field or place of exercise, in which he should practice and perform them.

In the early ages both of the Greek and Roman republics, the other parts of education seem to have consisted in learning to read, write, and account according to the arithmetic of the times. These accomplishments the richer citizens seem frequently to have acquired at home, by the assistance of some domestic pedagogue, who was generally, either a slave, or a freedman; and the poorer citizens, in the schools of such masters as made a trade of teaching for hire. Such parts of education, however, were abandoned altogether to the care of the parents or guardians of each individual. It does not appear that the State ever assumed any inspection or direction of them. By a law of Solon, indeed, the children were acquitted from maintaining those parents in their old age, who had neglected to instruct them in some profitable trade or business.

In the progress of refinement, when philosophy and rhetoric came into fashion, the better sort of people used to send their children to the schools of philosophers and rhetoricians, in order to be instructed in these fashionable sciences. But those schools were not supported by the public. They were for a long time barely tolerated by it. The demand for philosophy and rhetoric was for a long time so small that the first professed teachers of either could not find constant employment in any one city, but were obliged to travel about from place to place. In this


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since I was a boy, and ah! long, long before that, I fancy, the one great anxiety of parents of the upper and middle classes blessed with large families has been, "What are we to do with our boys?" and the cry goes on increasing, being intensified by the depreciation in the value of land, and by our distant colonies getting a little overstocked with young gentlemen, who have been banished to them by thousands, to struggle and strive, sink or swim, as fate wills it. At home, all professions are full and everything has been tried; and, go where you will, even the children of the noble may be found B


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 $$\begin{aligned}+ \int \delta p \left(\frac{   \frac{\partial E}{\partial p} . dp^{2}  + 2\frac{\partial F}{\partial p} . dp . dq  + \frac{\partial G}{\partial p} . dq^{2}}{2\, ds}  - d . \frac{E\, dp + F\, dq}{ds}\right) \end{aligned}$$

and we know that what is included under the integral sign must vanish independently of $\delta p.$ Thus we have

$$\begin{aligned} \frac{\partial E}{\partial p}. dp^{2} &+ 2\frac{\partial F}{\partial p}. dp. dq + \frac{\partial G}{\partial p}. dq^{2} = 2\, ds. d. \frac{E\, dp + F\, dq}{ds} \\ &= 2\, ds. d. \sqrt{E}. \cos\theta \\ &= \frac{ds. dE. \cos\theta}{\sqrt{E}} - 2\, ds. d\theta. \sqrt{E}. \sin\theta \\ &= \frac{(E\, dp + F\, dq)\, dE}{E} - 2\sqrt{EG - F^{2}}. dq. d\theta \\ &= \left(\frac{E\, dp + F\, dq}{E}\right) . \left(\frac{\partial E}{\partial p} . dp + \frac{\partial E}{\partial q} . dq\right) - 2\sqrt{EG - F^{2}}. dq. d\theta \end{aligned}$$

This gives the following conditional equation for a shortest line:

$$\begin{array}{c} \sqrt{EG - F^{2}}. d\theta = \frac{1}{2}. \frac{F}{E}. \frac{\partial E}{\partial p}. dp + \frac{1}{2}. \frac{F}{E}. \frac{\partial E}{\partial q}. dq + \frac{1}{2}. \frac{\partial E}{\partial q}. dp \\ - \frac{\partial F}{\partial p}. dp - \frac{1}{2}. \frac{\partial G}{\partial p}. dq \end{array}$$

which can also be written

$$\sqrt{EG - F^{2}}. d\theta = \frac{1}{2}. \frac{F}{E}. dE + \frac{1}{2}. \frac{\partial E}{\partial q}. dp - \frac{\partial F}{\partial p}. dp - \frac{1}{2}. \frac{\partial G}{\partial p}. dq$$

From this equation, by means of the equation

$$\cot\theta = \frac{E}{\sqrt{EG - F^{2}}} \cdot \frac{dp}{dq} + \frac{F}{\sqrt{EG - F^{2}}}$$

it is also possible to eliminate the angle $\theta,$ and to derive a differential equation of the second order between $p$  and $q,$  which, however, would become more complicated and less useful for applications than the preceding.

19.

The general formulæ, which we have derived in Arts. 11, 18 for the measure of curvature and the variation in the direction of a shortest line, become much simpler if the quantities $p,$  $q$ are so chosen that the lines of the first system cut everywhere


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2 wrestling with those of the middle and working classes for every piece of bread that falls in the gutter. Nothing is infra dig. that brings in a shilling, and all has been and is being tried. The sons of the great are to be found shoulder to shoulder with "Tommy Atkins," up behind a hansom cab, keeping shops, selling wines, horses, cigars, coals, and generally endeavouring feebly to shoulder the son of the working man out of the race over the ropes. Fortunately Heaven tempers the wind to the shorn lamb, and I believe it has done so now. I believe kind Dame Nature during the last summer has stepped in and opened out an honourable path for many gentlemen's sons, that I think will be their salvation, and at all events, if it does not make them all rich, will, if they only follow it, make them most useful members of society and keep them out of mischief and out of their mammas' snug drawing-rooms.


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 orthogonally the lines of the second system;  i.e. , in such a way that we have generally $\omega = 90^{\circ},$ or $F = 0.$  Then the formula for the measure of curvature becomes

$$4E^{2}G^{2}k = E. \frac{\partial E}{\partial q}. \frac{\partial G}{\partial q} + E\left(\frac{\partial G}{\partial p}\right)^{2} + G. \frac{\partial E}{\partial p}. \frac{\partial G}{\partial p} + G\left(\frac{\partial E}{\partial q}\right)^{2} - 2EG\left(\frac{\partial^{2} E}{\partial q^{2}} + \frac{\partial^{2} G}{\partial p^{2}}\right),$$

and for the variation of the angle $\theta$

$$\sqrt{EG}. d\theta = \frac{1}{2}. \frac{\partial E}{\partial q}. dp - \frac{1}{2}. \frac{\partial G}{\partial p}. dq$$

Among the various cases in which we have this condition of orthogonality, the most important is that in which all the lines of one of the two systems,  e.g. , the first, are shortest lines. Here for a constant value of $q$ the angle $\theta$  becomes equal to zero, and therefore the equation for the variation of $\theta$  just given shows that we must have $\frac{\partial E}{\partial q} = 0,$  or that the coefficient $E$  must be independent of $q;$   i.e. , $E$  must be either a constant or a function of $p$  alone. It will be simplest to take for $p$ the length of each line of the first system, which length, when all the lines of the first system meet in a point, is to be measured from this point, or, if there is no common intersection, from any line whatever of the second system. Having made these conventions, it is evident that $p$  and $q$ denote now the same quantities that were expressed in Arts. 15, 16 by $r$  and $\phi,$ and that $E = 1.$  Thus the two preceding formulæ become:

$$\begin{aligned} 4G^{2}k &= \left(\frac{\partial G}{\partial p}\right)^{2} - 2G\, \frac{\partial^{2} G}{\partial p^{2}} \\ \sqrt{G}. d\theta &= -\frac{1}{2}. \frac{\partial G}{\partial p}. dq \end{aligned}$$

or, setting $\sqrt{G} = m,$

$$k = -\frac{1}{m}. \frac{\partial^{2} m}{\partial p^{2}},\quad d\theta = -\frac{\partial m}{\partial p}. dq$$

Generally speaking, $m$  will be a function of $p,$  $q,$ and $m\, dq$  the expression for the element of any line whatever of the second system. But in the particular case where all the lines $p$ go out from the same point, evidently we must have $m = 0$  for $p = 0.$  Furthermore, in the case under discussion we will take for $q$  the angle itself which the first element of any line whatever of the first system makes with the element of any one of the lines chosen arbitrarily. Then, since for an infinitely small value of $p$ the element of a line of the second system (which can be regarded as a circle described with radius $p$ ) is equal to $p\, dq,$  we shall have for an infinitely small value of $p,$  $m = p,$  and consequently, for $p = 0,$  $m = 0$  at the same time, and $\frac{\partial m}{\partial p} = 1.$


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20.

We pause to investigate the case in which we suppose that $p$  denotes in a general manner the length of the shortest line drawn from a fixed point $A$ to any other point whatever of the surface, and $q$  the angle that the first element of this line makes with the first element of another given shortest line going out from $A.$  Let $B$  be a definite point in the latter line, for which $q = 0,$  and $C$  another definite point of the surface, at which we denote the value of $q$  simply by $A.$  Let us suppose the points $B,$  $C$  joined by a shortest line, the parts of which, measured from $B,$  we denote in a general way, as in Art. 18, by $s;$ and, as in the same article, let us denote by $\theta$  the angle which any element $ds$  makes with the element $dp;$  finally, let us denote by $\theta^{\circ},$  $\theta'$  the values of the angle $\theta$  at the points $B,$  $C.$  We have thus on the curved surface a triangle formed by shortest lines. The angles of this triangle at $B$  and $C$ we shall denote simply by the same letters, and $B$  will be equal to $180^{\circ} - \theta,$  $C$  to $\theta'$  itself. But, since it is easily seen from our analysis that all the angles are supposed to be expressed, not in degrees, but by numbers, in such a way that the angle $57^{\circ}\, 17'\, 45'',$ to which corresponds an arc equal to the radius, is taken for the unit, we must set

$$\theta^{\circ} = \pi - B,\quad \theta' = C$$

where $2\pi$  denotes the circumference of the sphere. Let us now examine the integral curvature of this triangle, which is equal to

$$\int k\, d\sigma,$$

$d\sigma$  denoting a surface element of the triangle. Wherefore, since this element is expressed by $m\, dp. dq,$ we must extend the integral

$$\iint km\, dp. dq$$

over the whole surface of the triangle. Let us begin by integration with respect to $p,$ which, because

$$k = -\frac{1}{m}. \frac{\partial^{2} m}{\partial p^{2}},$$

gives

$$dq. \left(\text{const.} - \frac{\partial m}{\partial p}\right),$$

for the integral curvature of the area lying between the lines of the first system, to which correspond the values $q,$  $q + dq$ of the second indeterminate. Since this inte-


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 orchards, whose trees were laden with noble fruit, gleamed forth the thatched roofs of pretty cottages, the friendly smoke from whose chimnies rose curling into the clear air. At intervals on the mountain slopes, frowned turreted fortresses, at once dominating and defending the vallies beneath. In the blooming meadows which varied the scene, pastured herds of sheep and of cattle; the while, amid pleasant groves, re-echoed the soothing notes of shepherds’ pipes.

The novelty of the thing, and the agreeable aspect of the new region, so astonished and delighted the subterranean sovereign, that he never once thought of being angry with those who had, without his permission, taken possession of his territory, or of disturbing them in their operations. Like the good-natured farmer who suffers the sociable swallow, or even the troublesome sparrow, to retain unmolested the nest it has built beneath his eaves, Rubezahl benevolently resolved that the busy mortals should receive no interruption from him. On the contrary, he conceived a fancy to make acquaintance with these creatures, intermediate between spirits and animals, and see what they were made of. ’Twould serve to amuse him awhile. Forthwith, he assumed the form of a lusty hind, and offering his services to the first farmer he met with, was accepted. All that he undertook throve in the hands of Rips, for so he called himself, who soon became noted as


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Rh I have followed the path myself, and, after fifty years' tramp down it, have been forced to abandon it owing to gout and rheumatism. I have not picked up a big fortune at it, or become celebrated, except quite locally; but I have had a good time and helped the world in general, and am content with my past life.

I was the son of a worthy country parson, who in my youth proposed to me in turn to become a judge, a bishop, a general, a Gladstone, a Nelson, a Sir James Paget, and a ritualistic curate; but when talking to me on the subject the good old man always said, "Mind, my boy, though I propose these various positions for you, yet, if you have any decided preference yourself, I will not thwart you, I will not fly in the face of nature."

For some time I thought I should rather like to be a bishop, and to this day I think I should have made a good one; but the B 2


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 gral curvature must vanish for $p = 0,$ the constant introduced by integration must be equal to the value of $\frac{\partial m}{\partial q}$  for $p = 0,$   i.e. , equal to unity. Thus we have

$$dq\left(1 - \frac{\partial m}{\partial p}\right){,}$$

where for $\frac{\partial m}{\partial p}$ must be taken the value corresponding to the end of this area on the line $CB.$  But on this line we have, by the preceding article,

$$\frac{\partial m}{\partial q}. dq = -d\theta,$$

whence our expression is changed into $dq + d\theta.$ Now by a second integration, taken from $q = 0$  to $q = A,$  we obtain for the integral curvature

$$A + \theta'- \theta^{\circ},$$

or

$$A + B + C - \pi.$$

The integral curvature is equal to the area of that part of the sphere which corresponds to the triangle, taken with the positive or negative sign according as the curved surface on which the triangle lies is concavo-concave or concavo-convex. For unit area will be taken the square whose side is equal to unity (the radius of the sphere), and then the whole surface of the sphere becomes equal to $4\pi.$ Thus the part of the surface of the sphere corresponding to the triangle is to the whole surface of the sphere as $\pm(A + B + C - \pi)$  is to $4\pi.$  This theorem, which, if we mistake not, ought to be counted among the most elegant in the theory of curved surfaces, may also be stated as follows:

The excess over $180^{\circ}$ of the sum of the angles of a triangle formed by shortest lines on a concavo-concave curved surface, or the deficit from $180^{\circ}$  of the sum of the angles of a triangle formed by shortest lines on a concavo-convex curved surface, is measured by the area of the part of the sphere which corresponds, through the directions of the normals, to that triangle, if the whole surface of the sphere is set equal to $720$  degrees.

More generally, in any polygon whatever of $n$  sides, each formed by a shortest line, the excess of the sum of the angles over $(2n - 4)$  right angles, or the deficit from $(2n - 4)$  right angles (according to the nature of the curved surface), is equal to the area of the corresponding polygon on the sphere, if the whole surface of the sphere is set equal to $720$  degrees. This follows at once from the preceding theorem by dividing the polygon into triangles.


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26  Thy Nashua meadows lay green and unshorn, And the hills of Pentucket were tasselled with corn.

But thy Pennacook valley was fairer than these, And greener its grasses and taller its trees, Ere the sound of an axe in the forest had rung, Or the mower his scythe in the meadows had swung.

In their sheltered repose looking out from the wood The bark-builded wigwams of Pennacook stood; There glided the corn-dance, the council-fire shone, And against the red war-post the hatchet was thrown.

There the old smoked in silence their pipes, and the young To the pike and the white-perch their baited lines flung; There the boy shaped his arrows, and there the shy maid Wove her many-hued baskets and bright wampum braid.

O Stream of the Mountains! if answers of thine Could rise from thy waters to question of mine, Methinks through the din of thy thronged banks a moan Of sorrow would swell for the days which have gone.

Nor for thee the dull jar of the loom and the wheel, The gliding of shuttles, the ringing of steel; But that old voice of waters, of bird and of breeze, The dip of the wild-fowl, the rustling of trees!

Lift we the twilight curtains of the Past,
 * And, turning from familiar sight and sound,

Sadly and full of reverence let us cast
 * A glance upon Tradition’s shadowy ground,

Led by the few pale lights which, glimmering round
 * That dim, strange land of Eld, seem dying fast;

And that which history gives not to the eye, The faded coloring of Time’s tapestry, Let Fancy, with her dream-dipped brush, supply.

Roof of bark and walls of pine, Through whose chinks the sunbeams shine, Tracing many a golden line
 * On the ample floor within;

Where, upon the earth-floor stark, Lay the gaudy mats of bark, With the bear’s hide, rough and dark,
 * And the red-deer’s skin.

Window-tracery, small and slight, Woven of the willow white, Lent a dimly checkered light;
 * And the night-stars glimmered down,

Where the lodge-fire’s heavy smoke, Slowly through an opening broke, In the low roof, ribbed with oak,
 * Sheathed with hemlock brown.

Gloomed behind the changeless shade By the solemn pine-wood made; Through the rugged palisade,
 * In the open foreground planted,

Glimpses came of rowers rowing, Stir of leaves and wild-flowers blowing, Steel-like gleams of water flowing,
 * In the sunlight slanted.

Here the mighty Bashaba Held his long-unquestioned sway, From the White Hills, far away,
 * To the great sea’s sounding shore;

Chief of chiefs, his regal word All the river Sachems heard, At his call the war-dance stirred,
 * Or was still once more.

There his spoils of chase and war, Jaw of wolf and black bear’s paw, Panther’s skin and eagle’s claw,
 * Lay beside his axe and bow;

And, adown the roof-pole hung, Loosely on a snake-skin strung,


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21.

Let us again give to the symbols $p,$  $q,$ $E,$  $F,$  $G,$  $\omega$  the general meanings which were given to them above, and let us further suppose that the nature of the curved surface is defined in a similar way by two other variables, $p',$  $q',$  in which case the general linear element is expressed by

$$\sqrt{E'\, dp'^{2} + 2F'\, dp'. dq' + G'\, dq'^{2}}$$

Thus to any point whatever lying on the surface and defined by definite values of the variables $p,$  $q$ will correspond definite values of the variables $p',$  $q',$  which will therefore be functions of $p,$  $q.$  Let us suppose we obtain by differentiating them

$$\begin{alignedat}{2} dp' &= \alpha\, dp &&+ \beta\, dq \\ dq' &= \gamma\, dp &&+ \delta\, dq \end{alignedat}$$

We shall now investigate the geometric meaning of the coefficients $\alpha,$  $\beta,$  $\gamma,$  $\delta.$

Now four systems of lines may thus be supposed to lie upon the curved surface, for which $p,$  $q,$ $p',$  $q'$  respectively are constants. If through the definite point to which correspond the values $p,$  $q,$ $p',$  $q'$  of the variables we suppose the four lines belonging to these different systems to be drawn, the elements of these lines, corresponding to the positive increments $dp,$  $dq,$  $dp',$  $dq',$  will be

$$\sqrt{E}. dp,\quad \sqrt{G}. dq,\quad \sqrt{E'}. dp',\quad \sqrt{G'}. dq'.$$

The angles which the directions of these elements make with an arbitrary fixed direction we shall denote by $M,$  $N,$ $M',$  $N',$  measuring them in the sense in which the second is placed with respect to the first, so that $\sin(N - M)$  is positive. Let us suppose (which is permissible) that the fourth is placed in the same sense with respect to the third, so that $\sin(N' - M')$ also is positive. Having made these conventions, if we consider another point at an infinitely small distance from the first point, and to which correspond the values $p + dp,$ $q + dq,$  $p' + dp',$  $q' + dq'$  of the variables, we see without much difficulty that we shall have generally,  i.e. , independently of the values of the increments $dp,$  $dq,$  $dp',$  $dq',$

$$\sqrt{E}. dp. \sin M + \sqrt{G}. dq. \sin N = \sqrt{E'}. dp'. \sin M' + \sqrt{G'}. dq'. \sin N'$$

since each of these expressions is merely the distance of the new point from the line from which the angles of the directions begin. But we have, by the notation introduced above,

$$N - M = \omega.$$

In like manner we set

$$N' - M' = \omega',$$


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 and also

$$N - M' = \psi.$$ Then the equation just found can be thrown into the following form:

$$\begin{alignedat}{1} \sqrt{E}. dp. \sin(M' &- \omega + \psi) + \sqrt{G}. dq. \sin(M' + \psi) \\ &= \sqrt{E'}. dp'. \sin M' + \sqrt{G'}. dq'. \sin(M' + \omega') \end{alignedat}$$

or

$$\begin{alignedat}{1} \sqrt{E}. dp. \sin(N' &- \omega - \omega' + \psi) - \sqrt{G}. dq. \sin(N' - \omega' + \psi) \\ &= \sqrt{E'}. dp'. \sin(N' - \omega') + \sqrt{G'}. dq'. \sin N' \end{alignedat}$$

And since the equation evidently must be independent of the initial direction, this direction can be chosen arbitrarily. Then, setting in the second formula $N' = 0,$ or in the first $M' = 0,$  we obtain the following equations:

$$\begin{aligned} \sqrt{E'}. \sin \omega'. dp' &= \sqrt{E}. \sin(\omega + \omega' - \psi). dp  + \sqrt{G}. \sin(\omega' - \psi). dq \\ \sqrt{G'}. \sin \omega'. dq' &= \sqrt{E}. \sin(\psi - \omega). dp + \sqrt{G}. \sin\psi. dq \end{aligned}$$

and these equations, since they must be identical with

$$\begin{alignedat}{2} dp' &= \alpha\, dp &&+ \beta\, dq \\ dq' &= \gamma\, dp &&+ \delta\, dq \end{alignedat}$$

determine the coefficients $\alpha,$  $\beta,$  $\gamma,$  $\delta.$ We shall have

$$\begin{aligned} \alpha &= \sqrt{\frac{E}{E'}}. \frac{\sin(\omega + \omega' - \psi)}{\sin\omega'}, & \beta &= \sqrt{\frac{G}{E'}}. \frac{\sin(\omega' - \psi)}{\sin\omega'} \\ \gamma &= \sqrt{\frac{E}{G'}}. \frac{\sin(\psi - \omega)}{\sin\omega'}, & \delta &= \sqrt{\frac{G}{G'}}. \frac{\sin\psi}{\sin\omega'} \end{aligned}$$

These four equations, taken in connection with the equations

$$\begin{aligned} \cos\omega &= \frac{F}{\sqrt{EG}}, & \cos\omega' &= \frac{F'}{\sqrt{E'G'}}, \\ \sin\omega &= \sqrt{\frac{EG - F^{2}}{EG}}, & \sin\omega' &= \sqrt{\frac{E'G' - F'^{2}}{E'G'}}, \end{aligned}$$

may be written

$$\begin{aligned} \alpha\sqrt{E'G' - F'^{2}} &= \sqrt{EG'}. \sin(\omega + \omega' - \psi) \\ \beta \sqrt{E'G' - F'^{2}} &= \sqrt{GG'}. \sin(\omega' - \psi) \\ \gamma\sqrt{E'G' - F'^{2}} &= \sqrt{EE'}. \sin(\psi - \omega) \\ \delta\sqrt{E'G' - F'^{2}} &= \sqrt{GE'}. \sin \psi \end{aligned}$$

Since by the substitutions

$$\begin{alignedat}{2} dp' &= \alpha\, dp &&+ \beta\, dq, \\ dq' &= \gamma\, dp &&+ \delta\, dq \end{alignedat}$$


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 the trinomial

$$E'\, dp'^{2} + 2F'\, dp'. dq' + G'\, dq'^{2}$$

is transformed into

$$E\, dp^{2} + 2F\, dp. dq + G\, dq^{2},$$

we easily obtain

$$EG - F^{2} = (E'G' - F'^{2})(\alpha\delta - \beta\gamma)^{2}$$

and since, vice versa, the latter trinomial must be transformed into the former by the substitution

$$(\alpha\delta - \beta\gamma)\, dp = \delta\, dp' - \beta\, dq',\quad (\alpha\delta - \beta\gamma)\, dq = -\gamma\, dp' + \alpha\, dq',$$

we find

$$\begin{alignedat}{1} &E\delta^{2} - 2F\gamma\delta + G\gamma^{2} = \frac{EG - F^{2}}{E'G' - F'^{2}}. E' \\ -&E\beta\delta + F(\alpha\delta + \beta\gamma) - G\alpha\gamma = \frac{EG - F^{2}}{E'G' - F'^{2}}. F' \\ &E\beta^{2} - 2F\alpha\beta + G\alpha^{2} = \frac{EG - F^{2}}{E'G' - F'^{2}}. G' \end{alignedat}$$

22.

From the general discussion of the preceding article we proceed to the very extended application in which, while keeping for $p,$  $q$ their most general meaning, we take for $p',$  $q'$  the quantities denoted in Art. 15 by $r,$  $\phi.$ We shall use $r,$  $\phi$  here also in such a way that, for any point whatever on the surface, $r$  will be the shortest distance from a fixed point, and $\phi$  the angle at this point between the first element of $r$  and a fixed direction. We have thus

$$E' = 1,\quad F' = 0,\quad \omega' = 90^{\circ}.$$

Let us set also

$$\sqrt{G'} = m,$$

so that any linear element whatever becomes equal to

$$\sqrt{dr^{2} + m^{2}\, d\phi^{2}}.$$

Consequently, the four equations deduced in the preceding article for $\alpha,$  $\beta,$  $\gamma,$  $\delta$ give

$$\begin{alignedat}{2} &\sqrt{E}. \cos(\omega - \psi) = \frac{\partial r}{\partial p} \quad.&\quad.\quad.\quad.\quad.\quad.\quad.\quad{(1)} \\ &\sqrt{G}. \cos \psi = \frac{\partial r}{\partial q} \qquad.\quad.&\quad.\quad.\quad.\quad.\quad.\quad.\quad{(2)} \end{alignedat}$$


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 $$\begin{alignedat}{2} &\sqrt{E}. \sin(\psi - \omega) = m. \frac{\partial \phi}{\partial p} &\quad.\quad.\quad.\quad.\quad.\quad.\quad{(3)} \\ &\sqrt{G}. \sin\psi = m. \frac{\partial \phi}{\partial q} \quad.\quad.&\quad.\quad.\quad.\quad.\quad.\quad.\quad{(4)} \end{alignedat}$$

But the last and the next to the last equations of the preceding article give

$$\begin{aligned} EG - F^{2} = E\left(\frac{\partial r}{\partial q}\right)^{2} - 2F. \frac{\partial r}{\partial p}. \frac{\partial r}{\partial q} + G\left(\frac{\partial r}{\partial p}\right)^{2} .&\quad.\quad{(5)}\;\quad\qquad \\ \left(E . \frac{\partial r}{\partial q} - F . \frac{\partial r}{\partial p}\right). \frac{\partial \phi}{\partial q} = \left(F . \frac{\partial r}{\partial q} - G . \frac{\partial r}{\partial p}\right). \frac{\partial \phi}{\partial p} &\quad.\quad{(6)}\;\quad\qquad \end{aligned}$$

From these equations must be determined the quantities $r,$  $\phi,$  $\psi$ and (if need be) $m,$  as functions of $p$  and $q.$  Indeed, integration of equation (5) will give $r;$  $r$  being found, integration of equation (6) will give $\phi;$  and one or other of equations (1), (2) will give $\psi$  itself. Finally, $m$  is obtained from one or other of equations (3), (4).

The general integration of equations (5), (6) must necessarily introduce two arbitrary functions. We shall easily understand what their meaning is, if we remember that these equations are not limited to the case we are here considering, but are equally valid if $r$  and $\phi$ are taken in the more general sense of Art. 16, so that $r$  is the length of the shortest line drawn normal to a fixed but arbitrary line, and $\phi$  is an arbitrary function of the length of that part of the fixed line which is intercepted between any shortest line and an arbitrary fixed point. The general solution must embrace all this in a general way, and the arbitrary functions must go over into definite functions only when the arbitrary line and the arbitrary functions of its parts, which $\phi$  must represent, are themselves defined. In our case an infinitely small circle may be taken, having its centre at the point from which the distances $r$ are measured, and $\phi$  will denote the parts themselves of this circle, divided by the radius. Whence it is easily seen that the equations (5), (6) are quite sufficient for our case, provided that the functions which they leave undefined satisfy the condition which $r$  and $\phi$ satisfy for the initial point and for points at an infinitely small distance from this point.

Moreover, in regard to the integration itself of the equations (5), (6), we know that it can be reduced to the integration of ordinary differential equations, which, however, often happen to be so complicated that there is little to be gained by the reduction. On the contrary, the development in series, which are abundantly sufficient for practical requirements, when only a finite portion of the surface is under consideration, presents no difficulty; and the formulæ thus derived open a fruitful source for


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 red is the standard red mixed with a smaller quantity of orange. With the disks, pure hues are secured only by mixing two standards adjacent in the spectrum circuit.

For convenience in speaking and writing about colors in this system of color instruction, all the spectrum colors other than the six standard spectrum colors are designated as intermediate spectrum hues, and often for convenience in speaking of them they are called simply spectrum hues. To these are also added the colors between red and violet which are not in the spectrum. When so used the term must be considered as purely technical in this particular relation, because a color between the standard blue and the standard green is in the abstract no more a hue than either of these colors. If two standards not adjacent in the spectrum circuit are combined the result is not a pure spectrum hue but always some broken spectrum color.

Local Color.—A term applied to the natural color of an object when seen in ordinarily good daylight and at a convenient distance, as a sheet of paper at arms length, a tree at twice its height, etc.

Tint.—Any pure or full color mixed with white, or reduced by strong sunlight. In the disk combinations a spectrum color combined with white.

Shade.—A full color in shadow, i. e., with a low degree of illumination. In disk combinations a spectrum color combined with a black disk produces by rotation a shade of -that color. In pigments the admixture of black does not usually produce as satisfactory shades of a color as may be secured with some other pigments, and each artist has his own preferences in making shades of the various colors on his palette.

Scale.—A scale of color is a series of colors consisting of a pure or full color at the center and graduated by a succession of steps to a light tint on one side and a deep shade on the other.

Tone.—Each step in a color scale is a tone of that color, and the full color may be called the normal tone in that scale. In art this word has had such a variety of meaning as to ren-


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 the solution of many important problems. But here we shall develop only a single example in order to show the nature of the method.

23.

We shall now consider the case where all the lines for which $p$  is constant are shortest lines cutting orthogonally the line for which $\phi = 0,$ which line we can regard as the axis of abscissas. Let $A$  be the point for which $r = 0,$ $D$  any point whatever on the axis of abscissas, $AD = p,$  $B$  any point whatever on the shortest line normal to $AD$  at $D,$  and $BD = q,$  so that $p$  can be regarded as the abscissa, $q$  the ordinate of the point $B.$  The abscissas we assume positive on the branch of the axis of abscissas to which $\phi = 0$  corresponds, while we always regard $r$  as positive. We take the ordinates positive in the region in which $\phi$  is measured between $0$  and $180^{\circ}.$

By the theorem of Art. 16 we shall have

$$\omega = 90^{\circ},\quad F = 0,\quad G = 1,$$

and we shall set also

$$\sqrt{E} = n.$$

Thus $n$  will be a function of $p,$  $q,$ such that for $q = 0$  it must become equal to unity. The application of the formula of Art. 18 to our case shows that on any shortest line whatever we must have

$$d\theta = \frac{\partial n}{\partial q}. dp,$$

where $\theta$  denotes the angle between the element of this line and the element of the line for which $q$  is constant. Now since the axis of abscissas is itself a shortest line, and since, for it, we have everywhere $\theta = 0,$ we see that for $q = 0$  we must have everywhere

$$\frac{\partial n}{\partial q} = 0.$$

Therefore we conclude that, if $n$  is developed into a series in ascending powers of $q,$ this series must have the following form:

$$n = 1 + fq^{2} + gq^{3} + hq^{4} + \text{etc.}$$

where $f,$  $g,$  $h,$  etc., will be functions of $p,$ and we set

$$\begin{alignedat}{4} f &= f^{\circ} &&+ f'p &&+ f''p^{2} &&+ \text{etc.} \\ g &= g^{\circ} &&+ g'p &&+ g''p^{2} &&+ \text{etc.} \\ h &= h^{\circ} &&+ h'p &&+ h''p^{2} &&+ \text{etc.} \end{alignedat}$$


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1888, Mr. Whittier supervised the preparation of a collective edition of his writings which was published in seven volumes, under the title of the Riverside Edition, uniform in general plan with the Riverside Edition of Longfellow’s writings. For this edition the poet furnished introductions and head-notes, and in many cases revised the text. He decided which of his earlier poems to discard altogether, which to insert in an appendix, and which to include in the body of his poetry. He also determined on a classification of his poems, and divided the four volumes containing them into definite subdivisions, nine in all besides a small group of his sister’s poems which he wished preserved with his own. Thus, very near the end of his life, he formed what was a definitive edition of his writings. He continued, however, to send out poems occasionally in the remaining four years, and these were gathered after his death into a small volume entitled “At Sundown.” This little book was indeed the extension of one which he had issued privately in the last year of his life.

The present Cambridge Edition is based upon the original Riverside Edition. It contains the same text in the same topical arrangement, together with “At Sundown” and a few poems which were gleaned after Mr. Whittier’s death and included in the authorized biography. The head-notes and the notes at the end of the volume are for the most part copies or abridgments of those used in the Riverside Edition, but a few have been added containing facts brought to light after Mr. Whittier’s death. These are distinguished by being inclosed in brackets [ ]. As in the Cambridge Edition of Longfellow’s Complete Works, a biographical sketch has been provided. The introduction which follows the sketch is that prepared by Mr. Whittier for the Riverside Edition.


text: Quarterly Journal of the Geological Society of London/Volume 33/On the Strata and their Fossil Contents between the Borrowdale Series of the North of England and the Coniston Flags


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 extract, as it shows both her conception of what fictitious writing ought to be, and her always kindly interest in what I was doing.

1em

"Thank you for your letter; it was as pleasant as a quiet chat, as welcome as spring showers, as reviving as a friend's visit; in short, it was very like a page of 'Cranford.' A thought strikes me. Do you, who have so many friends,—so large a circle of acquaintance,—find it easy, when you sit down to write, to isolate yourself from all those ties, and their sweet associations, so as to be your own woman, uninfluenced or swayed by the consciousness of how your work may affect other minds; what blame or what sympathy it may call forth? Does no luminous cloud ever come between you and the severe Truth, as you know it in your own secret and clear-seeing soul? In a word, are you never tempted to make your characters more amiable than the Life, by the inclination to assimilate your thoughts to the thoughts of those who always feel kindly, but sometimes fail to see justly? Don't answer the question; it is not intended to be answered Your account of Mrs. Stowe was stimulatingly interesting. I long to see you, to get you to say it, and many other things, all over again. My father continues better. I am better too; but to-day I have a headache again, which will hardly let me write coherently. Give


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8

Lendeth to its bow; And the soft breeze from the west Scarcely broke their dreamy rest.

Far behind was Ocean striving With his chains of sand; Southward, sunny glimpses giving, ’Twixt the swells of land, Of its calm and silvery track, Rolled the tranquil Merrimac.

Over village, wood, and meadow Gazed that stranger man. Sadly, till the twilight shadow Over all things ran, Save where spire and westward pane Flashed the sunset back again.

Gazing thus upon the dwelling Of his warrior sires, Where no lingering trace was telling Of their wigwam fires, Who the gloomy thoughts might know Of that wandering child of woe?

Naked lay, in sunshine glowing, Hills that once had stood Down their sides the shadows throwing Of a mighty wood, Where the deer his covert kept, And the eagle’s pinion swept!

Where the birch canoe had glided Down the swift Powow, Dark and gloomy bridges strided Those clear waters now; And where once the beaver swam, Jarred the wheel and frowned the dam.

For the wood-bird’s merry singing, And the hunter’s cheer, Iron clang and hammer’s ringing Smote upon his ear; And the thick and sullen smoke From the blackened forges broke.

Could it be his fathers ever Loved to linger here? These bare hills, this conquered river,— Could they hold them dear, With their native loveliness Tamed and tortured into this?

Sadly, as the shades of even Gathered o’er the hill, While the western half of heaven Blushed with sunset still, From the fountain’s mossy seat Turned the Indian’s weary feet.

Year on year hath flown forever, But he came no more To the hillside on the river Where he came before. But the villager can tell Of that strange man’s visit well.

And the merry children, laden With their fruits or flowers,— Roving boy and laughing maiden, In their school-day hours, Love the simple tale to tell Of the Indian and his well.

How sweetly on the wood-girt town The mellow light of sunset shone! Each small, bright lake, whose waters still Mirror the forest and the hill, Reflected from its waveless breast The beauty of a cloudless west, Glorious as if a glimpse were given Within the western gates of heaven, Left, by the spirit of the star Of sunset’s holy hour, ajar!

Beside the river’s tranquil flood The dark and low-walled dwellings stood, Where many a rood of open land Stretched up and down on either hand,


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 or

$$\begin{alignedat}{2} n = 1 + f^{\circ}q^{2} &+ f'pq^{2}  &&+ f''p^{2}q^{2} + \text{etc.} \\ &+ g^{\circ} q^{3} &&+ g'pq^{3} + \text{etc.} \\ &            &&+ h^{\circ}q^{4} + \text{etc. etc.} \end{alignedat}$$

24.

The equations of Art. 22 give, in our case,

$$\begin{aligned} &n\sin\psi = \frac{\partial r}{\partial p},\quad \cos\psi = \frac{\partial r}{\partial q},\quad -n\cos\psi = m. \frac{\partial \phi}{\partial r},\quad \sin\psi = m. \frac{\partial \phi}{\partial q}, \\ &n^{2} = n^{2}\left(\frac{\partial r}{\partial q}\right)^{2} + \left(\frac{\partial r}{\partial p}\right)^{2},\quad n^{2}. \frac{\partial r}{\partial q}. \frac{\partial \phi}{\partial q}     + \frac{\partial r}{\partial p}. \frac{\partial \phi}{\partial p} = 0 \end{aligned}$$

By the aid of these equations, the fifth and sixth of which are contained in the others, series can be developed for $r,$  $\phi,$  $\psi,$  $m,$ or for any functions whatever of these quantities. We are going to establish here those series that are especially worthy of attention.

Since for infinitely small values of $p,$  $q$ we must have

$$r^{2} = p^{2} + q^{2},$$

the series for $r^{2}$ will begin with the terms $p^{2} + q^{2}.$  We obtain the terms of higher order by the method of undetermined coefficients, by means of the equation

$$\left(\frac{1}{n} . \frac{\partial(r^{2})}{\partial p}\right)^{2} + \left(\frac{\partial(r^{2})}{\partial q}\right)^{2} = 4r^{2}$$

Thus we have [1] $\displaystyle\begin{alignedat}{3} r^{2} &= p^{2} + \tfrac{2}{3}f^{\circ}p^{2}q^{2} &&+ \tfrac{1}{2}f'p^{3}q^{2} &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f^{\circ}}^{2})p^{4}q^{2}\quad\text{etc.} \\ &+ q^{2} &&+ \tfrac{1}{2}g^{\circ}p^{2}q^{3} &&+ \tfrac{2}{5}g'p^{3}q^{3} \\ &&& &&+(\tfrac{2}{5}h^{\circ} - \tfrac{7}{45}{f^{\circ}}^{2})p^{2}q^{4} \end{alignedat}$

Then we have, from the formula

$$r\sin\psi = \frac{1}{2n}. \frac{\partial(r^{2})}{\partial p},$$

[2] $\displaystyle\begin{alignedat}{2} r\sin\psi = p - \tfrac{1}{3}f^{\circ}pq^{2} & -\tfrac{1}{4}f'p^{2}q^{2} &&- (\tfrac{1}{5}f'' + \tfrac{8}{45}{f^{\circ}}^{2})p^{3}q^{2}\quad\text{etc.} \\ & -\tfrac{1}{2}g^{\circ}pq^{3} &&-\tfrac{2}{5}g'p^{2}q^{3} \\ &&& -(\tfrac{3}{5}h^{\circ} - \tfrac{8}{45}{f^{\circ}}^{2})pq^{4} \end{alignedat}$


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Rh

a paper read before the Society in January 1875, I discussed the relations of the Gault and Chalk Marl in Bedfordshire and South Cambridgeshire, and endeavoured to show that the fossil contents of the so-called Cambridge Greensand are for the most part remaniés and have been derived from the upper beds of the Gault.

The conclusions arrived at were briefly these:—that the Upper Greensand does not extend further in a north-westerly direction than West-end Hill near Cheddington in Bucks; that stratigraphically the Cambridge Greensand is of no importance, being a mere nodule-bed at the base of the Chalk Marl; that it rests unconformably upon denuded Gault; that a greater portion of the fauna belonged originally to the Upper Gault; that the remainder, really belonging to the bed itself, are species proper to the Chalk Marl rather than to the Upper Greensand.

A full list of the species then known as existing in the formation was appended; and the derived forms were therein separated from those presumed to be indigenous to the Greensand itself. This list was mainly prepared from an examination of the large collection of fossils in the Woodwardian Museum; and my investigations resulted in the determination of many species that were previously unnamed in that collection, and in the identification of others which had been wrongly named. All these species were separately noticed, with remarks upon their determination and synonymy.

These observations, however, were by no means intended as an exhaustive study of the fauna of the Cambridge Greensand. Many specimens remained which it was not then possible to identify; and some of them appeared to be altogether new and undescribed; the consideration of these was purposely deferred until I should be able to obtain more information regarding them.

Two years have now elapsed, during which I have had the opportunity of inspecting many more specimens both from this and from other Cretaceous beds; and I am now better able, therefore, to offer some remarks upon these forms.

At the same time it is with considerable hesitation that I venture to name and place on record four species which do not appear to have been hitherto described; for I consider the establishment of such new forms as a very serious matter, and one upon which too much care cannot be spent.

It was, I think, Professor Forbes who remarked that few are aware how many species have been named from single and often broken specimens. Now it is obvious that such species cannot be


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486 of the same intrinsic value as those which are founded upon a number of perfect examples; neither can they be of the same use or importance to the comparative palæontologist. We are led, therefore, directly to the conclusion that some species are of more palæontological value than others ; and it may be worth while considering a little more fully the other circumstances by which this value is modified or increased, especially as I am not aware that any one has recently discussed the conditions under which new species may be constituted, or given any directions to those intending to name and describe such forms as may appear to have been previously unnoticed.

The importance of a proposed new species must greatly depend, I think, upon the following circumstances:—

1. The number of specimens examined (whether many or few).

2. The number of localities where it has been found.

3. The extent to which the original shell or test has been preserved.

4. The amount of difference between it and nearly allied species of the same genus.

In the first place it is evident that a species founded on a single specimen, however well preserved, has not the same value as one the description of which has been drawn from an examination of many individuals. Much, however, will depend upon the character of the fossil; and supposing all the other conditions to be very favourable, if it is in a good state of preservation, retaining all its parts, and if it be markedly different from any other species, so that it is unlikely to be merely a monstrosity or local variety, then its intrinsic value is much enhanced; but at the same time it remains of little use when considered as a member of the fauna to which it is added; for, being found only at one locality, it cannot be utilized in a comparison of one fauna with another, but must always be omitted from such calculations.

Again, new species which have been described, from single imperfect specimens in the state of casts, and even those founded on casts which are abundant and perfect so far as they go, are, as a rule, very unsatisfactory; for it is only in exceptional cases that the cast indicates the form and ornamentation of the shell with which it was covered. In cases where the cast does give such evidence its description is perhaps of some value; but where the casts are nearly smooth while the test was probably ornamented, as in Bivalve shells, Echinoderms, and many Gasteropods, it is worse than useless to describe them; for it is often impossible to compare them properly with specimens from other deposits which retain the shell, and the result is an unnecessary multiplication of specific terms: it is far easier to give a specific name to such casts than it is to disprove their title to the same.

Again, it is exceedingly difficult, even where specimens are abundant and their state of preservation good, to be quite sure that the form has never been elsewhere described. If not known in the British Isles, it may have been figured and described on the Con-


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 230 and God is the Beloved One (Mʾashuq). This Divine love is the theme of most of the Persian and Pushtu poems, which abound in Sufistic expressions which are difficult of interpretation to an ordinary English reader. For instance, Sharáb, "wine," expresses the domination of Divine love in the heart. Gísú, "a ringlet," the details of the mysteries of Divinity. Mai Khána, "a tavern," a stage of the journey. "Mirth," "wantonness," and "inebriation," signify religious enthusiasm and abstraction from worldly things.

The eight stages which we have given are those usually taught by Súfí teachers in their published works; but in North India we have frequently met with persons of this sect, who have learnt only the four following stages:—

The first, Násút, "humanity," for which there is the Shariʾat, or law. The second Malakút, "the nature of angels," for which there is Taríqat, or the pathway of purity. The third is Jabarút, "the possession of power," for which there is Mʾarifat, or knowledge. And the fourth is Láhút, "extinction," for which there is Haqíqat, or truth.

The Súfí mystic seeks, by concentration of


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488 48b A. J. jukes-browne's supplementary

Nautilus I have described, but have not named, since the characters of its original shelly covering are hardly indicated on the casts, although these form a very common fossil in the Cambridge Green- sand.

In conclusion, I may state that I regard the description of any of these species as of far less importance than the recognition among the fauna of other previously known species, and the demon- stration of the identity of the separately named forms.

A few errors and misapprehensions which had crept into my former notes are now corrected.

Regarding the general conclusions arrived at in my former paper, I will only say that my opinions have been confirmed by subse- quent experience during the progress of the Geological Survey in Cambridgeshire ; the only point in which I am disposed to alter the views previously expressed is with regard to those fossils which are preserved in light phosphates. Similar nodules and fossils are found in the uppermost beds of the Gault both at Folkestone and in Bedfordshire ; and I now consider them to have been derived from this horizon, very few, if any, of the phosphate nodules having originated in the bed where they are now found. I regard, there- fore, a still larger proportion of the Cambridge fossils as derived, the invertebrate fauna of the Greensand itself being very small and only numbering 36 species.

CEPHALOPODA.

Belemnites plexus, Blainville.

This name was admitted into my former list in consequence of three ot four specimens being so named in the Woodwardian Museum. A subsequent examination, however, in company with M. Ch. Barrois, of Lille, has convinced me that they do not belong to this species, since none of them exhibit the peculiar truncation of the alveolar end, which is its great characteristic. They bear most resemblance to large specimens of B. ultimus or attenuatus.

Nautilus arcuatus, Desh.

Nautilus arcuatus, Desh. Mem. Soc. Geol. France, vol. v. pi. xviii. ,£g. 1.

This species is described as inflated towards the middle, with rounded and subangular back, without any umbilicus ; chambers as wide as high and very sinuate (tres-arquees) ; siphon situated a little below the middle ; test thin and nearly smooth.

It somewhat resembles JV. Fleuriausianus, but has still more enveloping whorls, different chambers, and a differently situated siphon, that of N. Fleuriausianus lying close upon the inner whorl.

Several casts agreeing in every respect with the figure and description of N. arcuatus have been found in the " Nodule-bed," though they are by no means common. They are easily distin- guished from all others by the absence of any umbilicus, and by


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490 490 a. .t. jukes-browne's supplementary

In size, number of chambers, and position of siphuncle they resemble iV. Bouchardianus ; but the septa of the latter are described as straight.

Ammonites rhamnonotus, Seeley.

A. rhamnonotus, Seeley, Ann. & Mag. Nat. Hist. ser. 3, xvi. p. 233, pi. xi. fig. 7.

A. navicularis, var. nothus, Seeley, torn. cit. p. 232.

A. gardonicus, Hebert & Munier-Chalmas (Bassin d'Uchaux), Ann. des Sc. Geol. tome vi. pi. 4. figs. 1, 2.

Tn 1875 (Quart. Journ. Geol. Soc. vol. xxxi. p. 28G) I pointed out that the fossils indicated by the second of the above names had a closer relationship with A. Mantelli than Avith A. navicularis, but had also some characters in common with A. dispar, D'Orb., as figured and described by MM. Pictet and Campiche.

In the same year Prof. Hebert published his description of the Bassin d'Uchaux, in the appendix to which he described and figured the young and old forms of an Ammonite from the Gault of Yal- bonne under the name of A. gardonicus. The resemblance between these and some of the Cambridge specimens is very great, the young form being very like the A. rhamnonotus of Mr. Seeley, while the older individual approaches nearly to that called by him A. navicularis, var. nothus. A comparison of the descriptions given by Prof. Hebert and Prof. Seeley respectively will show how closely A. rhamnonotus and A. gardonicus are allied. After describing the young form, M. Hebert observes, "in older individuals the ribs pass over the back, and only present three slight elevations, which sometimes finish by disappearing altogether."

Mr. Seeley says : — " On the back the ribs are rather less distinct, and each bears in its centre a small sharp tubercle. In a younger state there are also tubercles at the extreme edge of the back, which seem to disappear with a diameter of twelve lines."

Another common characteristic is that only some of the ribs spring from the umbilicus, the intermediate ones arising from near the middle of the whorl ; this, however is also the case with the ribs ornamenting the so-called A. navicidaris, var. nothus ; and there are specimens now in the Woodwardian Museum which plainly con- nect this with the former. In one of these the earlier ribs are fre~ quent and rounded, while the later ribs are wider apart and become rudely nodulated ; in others the tubercles appear to be lost with age, as in Prof. Hebert 's specimen ; another form resembles the fragment named A. Wiestii by Sharpe. In view, therefore, of the great variation exhibited in this series of specimens, I propose that they be all considered as varieties of one species, for which Mr. Seeley 's name A. r*hamnonotus has the prioritv.

[Since the above was written I find that Mr. Seeley has obtained casts of A. gardonicus and placed them in the Museum, identi- fying them at the same time with his A. rhamnonotus; this identity may therefore be regarded as established.]


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ward was right in considering this supposed new species as simply a monstrosity, although he was probably wrong in referring it to A. lautus. It appears to me to be a malformed individual of the species A. auritus, the dorsal tubercles of one side being atrophied and thus allowing those of the other side to take a more median posi- tion. In every other detail of form and ornamentation it agrees with A. auritus; and there are now in the Woodwardian Museum other partially atrophied forms which clearly show the way in which this particular specimen came to possess such deceptive characters.

The other individual which shared the tablet of A. acaniho- notus is a very different form, being large, strong, and inflated ; it is, in fact, a similarly distorted specimen of A. Studeri, Pict. & Camp. The collection also possesses some interesting deformities belonging apparently to A. vraconnensis or to other varieties of A. Studeri. It is clear, therefore, that A. acanthonotus must be struck out of future lists of Cambridge fossils.

I may also call attention to a similarly malformed species, A. Ramsay anus, figured in Sharpe's Chalk Mollusca, and described at page 51. One side of it bears a very great resemblance to that named A. Woodwardi by Mr. Seeley, torn. cit. p. 236, pi. xi. fig. 3.

SCAPHITES HuGABDIANTJS, D'Orb.

Scajjhites Hugardianus, D'Orb., Heb. &Mun.-Ch., he. cit. p. 117, pi. v. fig. 7.

Scajphites Meriani, Pict. &Camp., see Quart. Journ. Geol. Soc. xxxi. p. 287, pi. xiv. fig. 1-3.

I was glad to observe that MM. Hebert and Munier-Chalmas also regard these two forms as varieties of one species, distinguishing them at the same time from S. cequalis as I had done in my former note on the Cambridge Scaphites. I ought to have given the pre- ference to D'Orbigny's name.

Tuerilites Wiestii, Sharpe.

The casts which in 1874 (see Quart. Journ. Geol. Soc. xxxi. p. 289) I identified as belonging to this species, appear in reality to be some- what intermediate between T. Wiestii and T. costatus, or rather to hold such a relation to both forms as we might expect were it the common progenitor of both. It resembles T. Wiestii in most particulars, especially in the position and number of the tubercles ; these, how- ever, are developed from indistinct costae, which are continued under the base of the whorls ; and their impress is seen on broken casts. In the typical T. Wiestii the tubercles are not connected by such elevations or ribs.

Prom T. costatus it differs by its narrower whorls and their greater spiral angle, as well as in the position of the tubercles, though, as I have elsewhere remarked, the ribs and tubercles of T. costatus appear to be very variable in their position.

Although it is sufficiently distinguished from T. Bergeri by only


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GASTEKOPODA.

Apoeehais, Da Costa.

An excellent account of this genus and its fossil forms, hitherto included under the heads of Rostellaria and Pteroceras, is given by Mr. J. S. Gardner in the ' Geological Magazine ' for 1875, Dec. ii. vol. ii. p. 49. He concludes by dividing them into four generic or subgeneric groups, viz. Aporrhais, Omithepus, Tridac- tylus and Dimorpliosoma (torn. cit. p. 394).

Apoeehais maeoinata, Sow.

Rostellaria marginata, Sow. Geol. Trans, vol. iv. pi. xi. fig. 18. Rostellaria Orbignyana, Pictet & Koux, pi. xxiv. fig 4. Pterodonta marginata, Seeley, Ann. & Mag. Nat. Hist. 1861 T vol. vii. pi. xi. fig. 2.

Pterodonta longispira, Seeley, torn. cit. pi. xi. fig. 3.

In the papers above referred to, Mr. Gardner has carefully inves- tigated the history of A. marginata, and has demonstrated its identity with A. Orbignyana, the cause of their separation having been that the lower keel was not clearly shown in the original engraving of the former. This being so, when Mr. Seeley (at p. 283) says of his supposed Pterodonta longispira that it almost as closely resembles A. Orbignyana as P. marginata does A. marginata, it is evident that there cannot be much difference between the two species referred to Pterodonta.

Further, it is to be remembered that both were described from imperfect casts, without any trace of the original shell, so that the characters were necessarily indefinite.

Moreover, from an examination of Mr. Carter's specimens, I am inclined to think that in the case of P. marginata the supposed notch is only an accidentally exaggerated inflection of the concavity below the keel, and that it is not the impression of any definite internal tooth-like ridge.

The figure of P. longispira is stated to be slightly restored from specimens in the same collection ; these, however, appear to have been mislaid, and are not now to be found : I cannot, therefore, say more than that the circular hollow represented in the figure is unlike that caused by the vertically elongated ridge shown in D'Orbigny's figures of Pterodonta, and that the other characters of the figure appear to be those of A. marginata (Orbignyana).

Further evidence must therefore be forthcoming before the pre- sence of Pterodonta in the bed can be regarded as ascertained.

Oenithoptjs, Gardner.

Mr. Gardner has proposed the above name (Geol. Mag. dec. 2 r vol. ii. p. 394) to include the small shells which have hitherto been grouped under the recent genus Pteroceras.

There can, I think, be no doubt that they merited a new generic


text: Index:Poetical works (IA poeticalworks00grayrich).pdf


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Mesochilotoma striata, Seeley.

Mesochilotoma striata, Seeley Ann. & Mag. Nat. Hist. ser. 3, vol. vii. p. 284.

The single specimen to receive which the above genns and species have been proposed, is a small cast, not half an inch long, and imbedded in phosphate, so that only one side of it is exposed to view.

It is, moreover, imperfect, both the apical and anterior portions being broken off, so that only parts of four whorls are visible. The onter lip is therefore not present, although Mr. Seeley describes it as having " a notch which forms a keel round the middle of the whorl ; " this keel he states to be crossed by the lines of growth ; and perhaps it was from the evidence these would afford that he presumed the outer lip to have been notched.

These lines are not very distinct ; there is no doubt, however, about the presence of the keel and of the faint striae above and below it ; but in my opinion the keel bears a greater resemblance to that on some forms of Aporrhais than to that of Pleuroiomaria. Mr. Seeley believes it to have had a short canal ; but he adds, " I have not yet seen the canal, and only predict its being short from an examination of the broken uppermost whorl."

Now, so far as such a small and badly preserved specimen enables me to judge, I should be inclined to consider it the cast of a small Aporrhais carinella, a species which has just such a median keel with striae above and below. At any rate Mr. Seeley's description would apply equally well to the shell above mentioned ; and it is certainly much more likely to belong to this than to an entirely new genus and species, for the existence of which no other evidence is forthcoming.

Brachtstoma angtjlare, Seeley, sp.

Scalaria angularis, Seeley, Ann. & Mag. Nat. Hist. ser. 3, vii. p. 286, pi. xi. fig. 9.

Brachystoma angularis, Gardner, Geol. Mag. dec. 2, vol. iii. p. 160, pi. iv. fig. 11.

In 1861 Mr. Seeley described an imperfect cast in the collection of Mr. J. Carter under the name of Scalaria angularis, one side only of two whorls being visible on this specimen ; another frag- ment in the "Woodwardian Museum exhibited the three uppermost whorls of the spire.

There was also in the Museum a fine and nearly perfect cast, which, from the slight keel on the body-whorl, I supposed to be a new species of Aporrhais ; and last year I obtained a second specimen, but never suspected their identity with 8. angulans. Having, however, sent one of these to Mr. J. S. Gardner, he forwarded me a specimen from the Gault of Folkestone in which the shell was preserved, asking me to compare it with the Cambridge casts, as he suspected they all belonged to the same species. A careful inspec-


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doubt of this identity ; those from the Perte du Rhone which retain the shell agree entirely with Mr. Seeley's type, which has the shell converted into phosphate of lime, and is preserved in Mr. Carter's collection.

D'Orbigny's figure does not express the rugose lamellae under- neath, which coexist with the striae said to be characteristic ; the nodules also are made too regular and too distinctly double, whereas they are only elongated or bilobed.

Mr. Seeley's figure of the under surface is very good ; but the side view gives too high an elevation, and the base is badly drawn ; the nodulations, again, are too distinctly single ; on the specimen itself they are more irregular, more elongate, and may have been bilobate. The side view of this and also of a Rhone specimen are given in PI. XXII. figs. 3 & 5.

Those from the Perte du Rhone vary much in height, some having the elevation of T. nodosa, others being low and resembling that called T. eoopansa by Mr. Seeley. One individual of the Cambridge T. expansa shows protuberances on the cast caused by the tuber- culations of the shell : they are probably only varieties of the same species.

Trochus cancellatus, Seeley.

Comp. Turbo indecisus, D'Orb. Pal. Fr. p. 230.

I think MM. Pictet and Campiche must have been mistaken in referring any Cambridge specimens to T. Tollotianus. I have recently had an opportunity of comparing them with specimens from the Perte du Rhone ; and though these are somewhat similar to T. cancelled us in shape, yet they are quite smooth, showing neither spiral lines nor cancellated markings ; neither is there any trace of the occasional varices.

I am inclined to think therefore that Trochus cancellatus is a distinct species, and that T. ToHoticums does not exist in the Cambridge Greensand. I may mention that it was by a mistake that the latter name was inserted in the Explanation of plate xiv. appended to my former paper (Quart. Journ. Geol. Soc. vol. xxxi. p. 313).

[Since writing the above I have seen the collection of D'Orbigny's types at the Jardin des Plantes in Paris : and among these I found specimens named Turbo indecisus, which appear identical with T. cancellatus ; one of them shows the longitudinal markings, and another the full cancellations, in the same variable way as the Cam- bridge casts exhibit them. D'Orbigny, p. 230, says that it occurs at Eocragnoles, Perte du Rhone, and Clansaye, but that he only knew the internal cast.]

Natica levistriata, new sp. PI. XXI. fig. 6.

Shell globose ; spire moderately high : whorls flattened above, so as to produce a slight angulation which runs round the spire : below this on the body- whorl there are 10 or 11 slightly elevated spiral striae. The angulation may be otherwise described as the highest


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 and from the formula

$$r\cos\psi = \tfrac{1}{2}\, \frac{\partial(r^{2})}{\partial q}$$

[3] $$\begin{alignedat}{2} r\cos\psi = q + \tfrac{2}{3}f^{\circ}p^{2}q & +\tfrac{1}{2}f'p^{3}q &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f^{\circ}}^{2})p^{4}q\quad\text{etc.} \\ & +\tfrac{3}{4}g^{\circ}p^{2}q^{2} &&+\tfrac{3}{5}g'p^{3}q^{2} \\ &&& +(\tfrac{4}{5}h^{\circ} - \tfrac{14}{45}{f^{\circ}}^{2})p^{2}q^{3} \end{alignedat}$$

These formulæ give the angle $\psi.$ In like manner, for the calculation of the angle $\phi,$  series for $r\cos\phi$  and $r\sin\phi$  are very elegantly developed by means of the partial differential equations

$$\begin{aligned} &\frac{\partial. r\cos\phi}{\partial p} = n\cos\phi. \sin\psi - r\sin\phi. \frac{\partial \phi}{\partial p} \\ &\frac{\partial. r\cos\phi}{\partial q} = \phantom{0}\cos\phi. \cos\psi - r\sin\phi. \frac{\partial \phi}{\partial q} \\ &\frac{\partial. r\sin\phi}{\partial p} = n\sin\phi. \sin\psi + r\cos\phi. \frac{\partial \phi}{\partial p} \\ &\frac{\partial. r\sin\phi}{\partial q} = \phantom{0}\sin\phi. \cos\psi + r\cos\phi. \frac{\partial \phi}{\partial q} \\ &n\cos\psi. \frac{\partial \phi}{\partial q} + \sin\psi. \frac{\partial \phi}{\partial p} = 0 \end{aligned}$$ A combination of these equations gives

$$\begin{alignedat}{3} &\frac{r\sin\psi}{n}. \frac{\partial. r\cos\phi}{\partial p} &&+ r\cos\psi. \frac{\partial. r\cos\phi}{\partial q} &&= r\cos\phi \\ &\frac{r\sin\psi}{n}. \frac{\partial. r\sin\phi}{\partial p} &&+ r\cos\psi. \frac{\partial. r\sin\phi}{\partial q} &&= r\sin\phi \end{alignedat}$$

From these two equations series for $r\cos\phi,$ $r\sin\phi$  are easily developed, whose first terms must evidently be $p,$  $q$  respectively. The series are

[4] $$\begin{alignedat}{3} r\cos\phi &= p + \tfrac{2}{3}f^{\circ}pq^{2} &&+ \tfrac{5}{12}f'p^{2}q^{2} &&+ (\tfrac{3}{10}f'' - \tfrac{8}{45}{f^{\circ}}^{2})p^{3}q^{2}\quad\text{etc.} \\ &&&+ \tfrac{1}{2}g^{\circ}pq^{3} &&+ \tfrac{7}{20}g'p^{2}q^{3} \\ &&&&&+ (\tfrac{2}{5}h^{\circ} - \tfrac{7}{45}{f^{\circ}}^{2})pq^{4} \end{alignedat}$$

[5] $ \begin{alignedat}{3} r\sin\phi &= q - \tfrac{1}{3}f^{\circ}p^{2}q &&- \tfrac{1}{6}f'p^{3}q &&- (\tfrac{1}{10}f'' - \tfrac{7}{90}{f^{\circ}}^{2})p^{4}q\quad\text{etc.} \\ &&&- \tfrac{1}{4}g^{\circ}p^{2}q^{2} &&- \tfrac{3}{20}g'p^{3}q^{2} \\ &&&&&- (\tfrac{1}{5}h^{\circ} + \tfrac{13}{90}{f^{\circ}}^{2})p^{2}q^{3} \end{alignedat}$

From a combination of equations [2], [3], [4], [5] a series for $r^{2}\cos(\psi + \phi),$ may be derived, and from this, dividing by the series [1], a series for $\cos(\psi + \phi),$  from


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forms which are not uncommon among the Cambridge Pleuroto- marice. They are easily distinguishable from casts of P. Rouxi or P. Rhodani by their flat or slightly concave base.

In the abundance of its Pleurotomariai the Cambridge Greensand resembles the continental Gault Superieur ; and it is remarkable also that most of the species appear to be identical.

Pleurotomaria Itieriana ? Pict. and Ex. PI. XXI. fig. 2.

Casts of a more elevated species are often found, having 3 or 4 whorls ornamented with a spiral rib corresponding to the sinus- band ; the base is flattened ; and the general form resembles that of P. Itieriana, Pict. and Ex. pi. 22. fig 3. The same fossil also occurs in the Gault of Folkestone ; but I have never seen it named. I also possess what I believe to be a unique specimen of a reversed variety, apparently belonging to this species ; the whorls are more separated, as is frequently the case in sinistral forms, see PI. XXI. fig. 2.

LAMELLIBEANCHIATA.

Pecten aptiensis, D'Orb.

i Pecten inter striatus, D'Orb. Pal. Pr. vol. iii. pi. ccccxxxiii. figs.

1-5 ; and Pict. & Roux, Gres Verts, p. 516, pi. xlvi. fig. 4 ; subse- quently named

Pecten aptiensis, D'Orb.

Pecten Dutemplei, D'Orb. Pal. Fr. pi. ccccxxxiii. figs. 10-14 ; and Pictet & Eoux, Gres Verts, p. 516, pi. 46. fig. 4.

Pecten Barretti, Seeley, Ann. & Mag. N. H. ser. 3, vol. vii. p. 118, pi. vi. fig. 1.

I cannot help thinking that these three forms are much more nearly related to one another than has hitherto been supposed.

All three are about the same size, have nearly the same number of ribs, and are characterized by similar interstitial striations.

Of P. Barretti Mr. Seeley says, "the left valve is ornamented with some 48 radiating, rather depressed ribs, of which about a dozen are subordinate, the remainder nearly equal ; all are imbri- cated. The interstitial spaces are very finely striated, those in the centre longitudinally, and those of the sides obliquely, reminding one of P. aptiensis. On the flat valve the ribs appear less promi- nent and more numerous." I cannot see that it has any resemblance to Pecten Espaillaci ; nor do I think the latter existed with it, the specimen so named in the Woodwardian Museum being only a large individual of P. Barretti preserved in a lighter-coloured phosphate. There is no doubt, however, of its close affinity with P. aptiensis ; indeed it becomes a matter of doubt whether the differences are sufficiently great to justify its being considered a separate species.

The upper and lower valves of P. aptiensis differ similarly in the number of ribs : D'Orbigny gives 20 to 23 as the number on the upper (convex) valve, and 46 on the lower; Pictet and Eoux,


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502 502 A. J. JUKES-BROWNE'S SUPPLEMENTARY

specimen in the Woodwardian Museum presents the characters above mentioned. The ribs are strongly marked on the casts ; the shell varies in length from 13 to 17 lines. D'Orbigny gives it as a Turonian form ; but it appears also to occur in the Gault Superieur of Cheville (' Renevier,' p. 163).

Lima Rauliniana, D'Orb.

Lima Rauliniana, D'Orb. Pal. Fr. iii. pi. ccccxvii. figs. 5-8.

This species bears a considerable resemblance to Lima aspera, Mant. : and the Cambridge specimens have hitherto been confounded with that form.

Great differences, however, are discovered on a close examination. L. aspera has wide, flattened ribs, separated by very narrow inter- spaces, mere grooved lines, in fact, which are pitted at intervals.

L. Rauliniana has low rounded ribs with peculiar imbrications, and the interspaces are nearly as wide as the ribs. M. d'Orbigny's description is correct so far as it goes ; and fig. 7 shows the peculiar ribs " sur lesquelles sont comme des tuiles obliques en dehors, espacees ot alternes ;" he does not, however, mention the ornamen- tation of the interspaces ; these are not pitted, but crossed with irregular transverse grooves or striations, very minute and closely set.

The main characters of the shell may be thus described : — valves nearly equal and equiconvex, each bearing between 40 and 50 low rounded ribs, interrupted occasionally by the lines of growth ; the ribs are ornamented with elongate imbrications, projecting slightly over their inner side, viz. over the left side of one half of the ribs, and over the right side of the other half, on each side of a median line, where the ribs divaricate ; when worn, the ribs appear to be simply notched ; the interspaces are grooved in the way above mentioned.

The specimens are generally preserved in the lighter-coloured phosphate, which at one time I thought had been formed in situ : I have since, however, been led to think that these phosphates have also been derived, but probably from rather higher beds of the Gault than those from which most of the black nodules came.

Lima Rauliniana is found in the Gaize or Gault Superieur of the Paris basin ; and casts of similar shape are quoted from Cheville by Prof. Renevier (Taune de Cheville,' p. 163) ; but I am not aware that it has been previously discovered in England.

Lima interlineata, new sp. PI. XXI. fig. 10.

Shell transversely oval, or obliquely quadrate : valves equiconvex, and each ornamented with 10 or 11 strong rounded ribs radiating from the umbo ; interspaces wide and flat, marked with a few distinct longitudinal lines, which vary in number from 1 to 5, but are most numerous on the more central interspaces. There appear to have been several stronger striae or small ribs on the anterior side or buccal region ; but the shell is here worn off.


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surn-r.MKM'AKY xotks on nir. Cambridge greens and.

Although I have never seen a T. angutatus with distinct epitheca and expanded base, yet, judging from the analogy of T. conulus, it

is reasonable to suppose that they were originally present : indeed it is exceedingly rare to find even a trace of the costnlate wall, their usual condition being that of dark phosphate casts.

Range of the species mentioned in the foregoing notes as not previously identified in the Cambridge Greensand : —

Oault. . English or Foreign.

Upper Gault of

England.

Vraoonien i

or G*ult

Superieur.

Turbo Pietetianus, D' 0>'h

Pleurotomaria regina, P'. >v R.r. ...

? Itierinna, Pt. # R.r.

Pootou Raulinianus, D Orb

subacutus. D Orh

Lima Rauliniana. D Orh

■*■

1

Names of those to be struck out of former list

Belenmites plenus, Blainv. Ammonites dispar. D'Orb.

« aeanthonotus. Sahy.

Pterodonta marginata. Seetey,

longispira. Stela/.

Pteroceras Fittoni. Meeochilotoma striata. Seeley. Troelms Tollotianus. Ft. § fcr. Turboidea nodosa. SMSfly.

Turboidea expansa, Seetey. Spondylus Dutempleanus, D'Orb. trnnoatus. JfOrb,

Lima aspera, Ma/it. Hemiaster asterias (=£piaster).

Sareimda favosa (= Isastr:va Cidaris elavigora. Konig. Chenendopora = Phaivtrospongia. So! las.

Fig.l.

9.

10.

EXPLANATION OF PLATE XXI.

Turrilites uobilis, now >p.. Geological-Survey collection, phosphate cast.

A reversed Pleurotomaria, probably PL Ifieriana. Survey collection.

Turbo Picftfiauus. D'Orb. {Turboidea nodosa, Seeley), a unique specimen in the cabinet of Mr. James Carter. F.G.S.

Turbo Pietetianus, D'Orb., a phosphate cast.

Turbo Pictetianus, D'Orb.. a Perte-du-Rhone specimen retaining the shell. from the Woodwardian Museum.

Xatica Icvistriata. new sp. From a phosphate cast in the Woodwardian Museum, Circumstances have prevented the delineation of a speci- men showing the shell.

8. Nerita nodulosa, new sp. From a unique specimen in the cabinet of Mr. J. Carter. F.G.S.

Lima Rauliniaua. D'Orb. Magnified view of a portion of the shell.

Lima intcrlineata. new sp. From a specimen in the Woodwardian Museum. Fig. 10(7 shows a portion of the shell on the other side.

For tht Discussion on this paper see p. 446.


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The species, indeed, is very rare ; and only a few portions of the shell remain on that figured, which measured inch by about 1^ inch. The shell appears to have been moderately thick ; and the ribs stand out in strong relief on the cast.

It somewhat resembles L. Galliennei of D'Orbigny ; but this species is wider, is much more compressed, and is strongly striated over both ribs and interspaces — characters which sufficiently distin- guish it.

BRACHIOPODA.

Rhtnchonella sulcata, Park.

Rhynchonella sulcata, Park, Geol. Trans, ser. 1. vol. v. p. 59 ; Dav. Cret. Brach. pi. x. figs. 18-36.

The varieties of this species are very numerous, all gradations being found between the large and sparsely ribbed shells, and the smaller forms which bear numerous closely set ribs. The specimen noticed in my previous paper as R. comjpressa (really dimicliata) may be only another extreme variety. Mr. Davidson, to whom it has since been sent, remarks that it is much more transverse than any R. sulcata which has come under his notice, and that it is very like some forms of R. dimicliata ; but for all that he would not be surprised if it were only a compressed and abnormal form of R. sulcata, so difficult is it to state positively to which species a single and isolated individual belongs.

Those shells previously known as R. latissima are now considered to be R. dimicliata, var. convexa ; but the Cambridge specimens may after all be only an extreme variation of R. sulcata.

ACTINOZOA.

Trochocyathus angtjlattjs, Duncan, sp.

Smilotrochus angulatus, Duncan, Suppl. Cret. Corals, pi. vii. figs. 7 and 8.

In 1867 Prof. Duncan described as new forms two species of corals from the Cambridge Greensand, referring them to the genus Smilotrochus ; one of these, S. elongatus, I have previously shown to be really a Trochocyathus, by discovering that it possessed epitheca, exhibited a columella, and was clearly pediculate — that it was, in fact, identical with Trochocyathus conulus of Edw. and Haime.

I had not then any proof that the second species, S. angulatus, belonged to the same genus, though I regarded it as exceedingly likely. Having recently, however, split open many specimens of this coral, I have ascertained that it possesses as distinct a colu- mella as its larger congener, though, of course, shorter in proportion to the squat growth of the species.

Edwards and Haime's description, viz. " columella well deve- loped, composed of prismatic or twisted processes disposed fascicu- larly or in a single row," applies equally to both species.

Q.J.G. S. No. 131. 2l


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however, reckon the number on the upper valve as 35 to 38, but have not seen the lower valve.

Pecten Dutemplei is said to have about 40 or 50 ribs on the upper valve ; " and a small simple linear rib occurs between these ribs, but not regularly." This arrangement is better shown in Pictet and Roux's figure than in that of D'Orbigny ; and the number of ribs given by the latter, viz. 70 to 80, is probably an exaggeration.

Both P. aptiensis and P. Dutemplei have the same curious oblique striation across the interspaces, and are in all other respects very like one another, the chief difference mentioned by D'Orbigny being the presence of the occasional intermediate simple rib on the latter species.

Now occasional simple ribs are also distinctly visible on some specimens of P. Barretti, especially on that previously referred to as having been mistaken for P. Espaillaci. P. Barretti, therefore, would appear to be most nearly allied to the Grault form of the species, viz. P. Dutemplei.

Mr. Seeley finds some differences in the ornamentation of the ears ; but his remarks regarding them (p. 119) are not very clear : neither does the variation seem very important ; for the combination of transverse and radiating lines is very frequent on the ears of allied species, and sometimes one set of markings, sometimes the other, stands out more prominently, according to growth and state of preservation.

All the differences above noted are merely slight variations from the earlier type of P. aptiensis, under which head I think both P. Dutemplei and P. Barretti should be ranged as varieties. The latter has been found by Mr. H. G. Fordham in the Greensand at Swanage.

Pecten Rauliniantjs ? D'Orb.

Pecten Raulinianus, D'Orb. Pal. Fr. iii. pi. ccccxxxiii. figs. 6-9.

A specimen recently placed in the Woodwardian Museum is about the same size as P. Barretti, but possesses very different characters. No interstitial striations are visible ; the ribs are nume- rous, and ornamented with closely set, narrow, lamellated imbrica- tions. It appears to agree more closely with P. Raulinianus than with any other known to me ; indeed, as far as can be seen, in ornamentation it agrees exactly with that species.

Pecten stjbactjtus, D'Orb.

Pecten subacutus, D'Orb. Pal. Fr. p. 603, pi. 435. figs. 5-10.

This species has about 25 simple ribs, ornamented with squamose elevations set at short intervals ; the ribs are rounded on the upper valve, but more angular and acute on the lower ; slight striations run down each side of the ribs into the interspaces, which are nearly as broad as the ribs.

It is a very rare Pecten at Cambridge ; and most of those I have seen were much worn, so that the ribs appeared quite plain ; but a


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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/52

 which may be found a series for the angle $\psi + \phi$ itself. However, the same series can be obtained more elegantly in the following manner. By differentiating the first and second of the equations introduced at the beginning of this article, we obtain

$$\sin\psi. \frac{\partial n}{\partial q} + n\cos\psi. \frac{\partial \psi}{\partial q} +  \sin\psi. \frac{\partial \psi}{\partial p} = 0$$

and this combined with the equation

$$n\cos\psi. \frac{\partial \phi}{\partial q} + \sin\psi. \frac{\partial \phi}{\partial p} = 0$$

gives

$$\frac{r\sin\psi}{n}. \frac{\partial n}{\partial q} + \frac{r\sin\psi}{n}. \frac{\partial(\psi + \phi)}{\partial p} + r\cos\psi. \frac{\partial(\psi + \phi)}{\partial q} = 0$$

From this equation, by aid of the method of undetermined coefficients, we can easily derive the series for $\psi + \phi,$ if we observe that its first term must be $\frac{1}{2}\pi,$  the radius being taken equal to unity and $2\pi$  denoting the circumference of the circle,

[6] $\displaystyle\begin{alignedat}{2} \psi + \phi = \tfrac{1}{2}\pi - f^{\circ}pq &- \tfrac{2}{3}f'p^{2}q &&- (\tfrac{1}{2}f'' - \tfrac{1}{6}{f^{\circ}}^{2})p^{3}q\quad\text{etc.} \\ &- g^{\circ}pq^{2} &&- \tfrac{3}{4}g'p^{2}q^{2} \\ &&&- (h^{\circ} - \tfrac{1}{3}{f^{\circ}}^{2})pq^{3} \end{alignedat}$

It seems worth while also to develop the area of the triangle $ABD$ into a series. For this development we may use the following conditional equation, which is easily derived from sufficiently obvious geometric considerations, and in which $S$  denotes the required area:

$$\frac{r\sin\psi}{n}. \frac{\partial S}{\partial p} + r\cos\psi. \frac{\partial S}{\partial q} = \frac{r\sin\psi}{n}. \int n\, dq$$

the integration beginning with $q = 0.$ From this equation we obtain, by the method of undetermined coefficients,

[7] $\displaystyle\begin{alignedat}{4} S = \tfrac{1}{2}pq &- \tfrac{1}{12}f^{\circ}p^{3}q &&- \tfrac{1}{20}f'p^{4}q &&- (\tfrac{1}{30}f'' - \tfrac{1}{60}{f^{\circ}}^{2})p^{5}q\quad\text{etc.} \\ &- \tfrac{1}{12}f^{\circ}pq^{3} &&- \tfrac{3}{40}g^{\circ}p^{3}q^{2} &&- \tfrac{1}{20}g'p^{4}q^{2} \\ &&&- \tfrac{7}{120}f'p^{2}q^{3} &&- (\tfrac{1}{15}h^{\circ} + \tfrac{2}{45}f'' + \tfrac{1}{60}{f^{\circ}}^{2})p^{3}q^{3} \\ &&&- \tfrac{1}{10}g^{\circ}pq^{4} &&- \tfrac{3}{40}g'p^{2}q^{4} \\ &&&&&- (\tfrac{1}{10}h^{\circ} - \tfrac{1}{30}{f^{\circ}}^{2})pq^{5} \end{alignedat}$


text:
Page:Thomas Patrick Hughes - Notes on Muhammadanism - 2ed. (1877).djvu/252

 Rh his thoughts and affections on God, to lose his own identity; and the following fable, related by Jalál-ud-dín, the author of the Masnawí, illustrates their views on the subject. It represents Human Love seeking admission into the Sanctuary of Divinity:—

"One knocked at the door of the Beloved, and a voice from within inquired 'Who is there?' Then he answered, It is I.' And the voice said, 'This house will not hold me and thee.' So the door remained shut. Then the Lover sped away into the wilderness, and fasted and prayed in solitude. And after a year he returned, and knocked again at the door, and the voice again demanded, 'Who is there?' And the Lover said, It is Thou.' Then the door was opened."

In Professor Max Müller's address to the Aryan section of the International Congress of Orientalists assembled in London, in September, 1874, he said:—"We have learnt already one lesson, that behind the helpless expressions


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Page:Complete Poetical Works of John Greenleaf Whittier (1895).djvu/39

Rh From lips as the lips of Hylas sweet, And moved like twin roses which zephyrs meet!

I cross my floor with a nervous tread,
 * I whistle and laugh and sing and shout,

I flourish my cane above his head,
 * And stir up the fire to roast him out;

I topple the chairs, and drum on the pane, And press my hands on my ears, in vain!

I ’ve studied Glanville and James the wise,
 * And wizard black-letter tomes which treat

Of demons of every name and size
 * Which a Christian man is presumed to meet,

But never a hint and never a line Can I find of a reading fiend like mine.

I ’ve crossed the Psalter with Brady and Tate,
 * And laid the Primer above them all,

I ’ve nailed a horseshoe over the grate,
 * And hung a wig to my parlor wall

Once worn by a learned Judge, they say, At Salem court in the witchcraft day!

“Conjuro te, sceleratissime,
 * Abire ad tuum locum!”—still

Like a visible nightmare he sits by me,—
 * The exorcism has lost its skill;

And I hear again in my haunted room The husky wheeze and the dolorous hum!

Ah! commend me to Mary Magdalen
 * With her sevenfold plagues, to the wandering Jew,

To the terrors which haunted Orestes when
 * The furies his midnight curtains drew,

But charm him off, ye who charm him can, That reading demon, that fat old man!

on thy journey toiling By the swift Powow, With the summer sunshine falling On thy heated brow, Listen, while all else is still. To the brooklet from the hill.

Wild and sweet the flowers are blowing By that streamlet’s side, And a greener verdure showing Where its waters glide, Down the hill-slope murmuring on. Over root and mossy stone.

Where yon oak his broad arms flingeth O'er the sloping hill, Beautiful and freshly springeth That soft-flowing rill, Through its dark roots wreathed and bare, Gushing up to sun and air.

Brighter waters sparkled never In that magic well, Of whose gift of life forever Ancient legends tell, In the lonely desert wasted, And by mortal lip untasted.

Waters which the proud Castilian Sought with longing eyes, Underneath the bright pavilion Of the Indian skies, Where his forest pathway lay Through the blooms of Florida.

Years ago a lonely stranger, With the dusky brow Of the outcast forest-ranger, Crossed the swift Powow, And betook him to the rill And the oak upon the hill.

O’er his face of moody sadness For an instant shone Something like a gleam of gladness, As he stooped him down To the fountain’s grassy side, And his eager thirst supplied.

With the oak its shadow throwing O’er his mossy seat, And the cool, sweet waters flowing Softly at his feet. Closely by the fountain’s rim That lone Indian seated him.

Autumn’s earliest frost had given To the woods below Hues of beauty, such as heaven


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Page:Quarterly Journal of the Geological Society of London, vol. 33.djvu/585

Rh NOTES ON THE CAMBRIDGE GREENLAND. 499

of these ridges. There are very few striated Naticce known from Cretaceous beds ; and this appears to differ from all of them. The above description is from a specimen in the national collection, from the Gault of Folkestone. It has the shell partially preserved ; but where this is absent a cast is disclosed of the same shape and dimen- sions as those from the Cambridge bed (fig. 6), which in my previous paper (p. 292) I provisionally referred to N. Rauliniana, D'Orb.

Nerita nodulosa, new sp. PI. XXI. figs. 7, 8.

Shell very thick ; spire depressed ; whorls angulated and flattened above, ornamented when young with numerous small ribs, which proceed from the spire and, passing over the head, run vertically for a short way down the side of the whorls ; they are then broken up into small nodules or tubercles. As the shell increases in growth, however, the upper parts of the ribs also become nodulose, so that near the mouth the ornamentation is entirely changed ; large, irregular, nodulose tubercles are developed on the keel, apparently by the union of two or three ribs ; and above each, on the flat portion of the whorl, are two smaller nodules. At the same time the outer lip is enlarged, flattened, and applied against the inner whorls, somewhat as in recent Neritce.

A single specimen only is known, in the cabinet of J. Carter, Esq. Length about 13 lines, breadth 9 lines, height unknown ; and the lower part of this is broken away, so that the characters of the base cannot be ascertained.

Although the fossil above described is unique and imperfect, yet so much of the original thick shell is retained ; and this is so remarkable in its shape and ornamentation that I have thought some account of it might be published.

At the same time I have had great difficulty in deciding to what genus it probably belongs. It might almost be referred to Strapa- rollus, or Euomphalus ; but the angle which the outer lip makes with the inner whorl is greater than in these genera, and leads to the conclusion that the last whorl was considerably produced below. This is the case with a shell named Nerita rugosa by MM. Briart and Cornet (Meule de Bracquegnies, pi. iii. figs. 50-52), which the Cambridge fossil appears to much resemble in shape and general features.

N. rugosa has thick, radiating ribs of a uniform character, each ending in a nodular projection at the keel, below which the shell is smooth ; the base is not umbilicated ; it is thus specifically different, though it appears to be generically identical. Whether, however, they are true Neritce some doubt may be entertained.

Pleurotomaria regina, Pict. & Rx.

Pleurotomaria regina, Pict. & Roux, Gres Verts, pi. xxiv. fig. 2.

Casts of this species are among the fossils from the Perte du Rhone recently placed in the Woodwardian Museum ; and from them I have been able to identify certain low, closely whorled