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Translation:Allgemeine Auflosung der Aufgabe die Theile einer gegebenen Flache auf einer andern gegebnen Flache so abzubilden, dass die Abbildung dem Abgebildeten in den kleinsten Theilen ahnlich wird.

1.
The nature of a curved surface is determined by an equation involving the coordinates $$x, \ y, \ z$$ relative to each of its points. By virtue of this equation, each of these three variables can be considered as a function of the other two. The generality is even greater if we introduce two new variables $$t, \ u$$ and represent each of the variables $$x, y, z$$ as a function of $$t $$ and $$ u,$$ so that, at least in general, specific values of $$t $$ and $$u$$ always correspond to a specific point on the surface and vice versa.

2.
Let $$X, \ Y, \ Z, \ T, \ U$$ be quantities that play, with respect to a second surface, the same role as $$x, \ y, \ z, \ t, \ u$$ play with respect to the first.

3.
Representing the first surface onto the second means establishing a rule whereby each point on the first surface corresponds to a specific point on the second. This is achieved by stipulating that $${T}$$ and $${U}$$ are equal to certain functions of the two variables $$t$$ and $$u.$$ If one wants the representation to satisfy certain conditions, these functions can no longer be arbitrary. As $$X, \ Y, \ Z$$ then also become functions of $$t$$ and $$u,$$ these functions, in addition to the condition prescribed by the nature of the second surface, must also satisfy those required in the representation.

4.
The problem proposed by the Royal Society of Sciences requires that the representation be similar to the original in its smallest parts. The first step is to express this condition analytically.

By differentiating the functions of $$t, \ u$$ by which $$x, \ y, \ z, \ {X}, \ {Y}, \ {Z}$$ are expressed, we can derive the following equations:

The prescribed condition firstly demands that all infinitely small lines emanating from a point on the first surface and lying on that surface be proportional to the corresponding lines on the second surface, and secondly that the former lines include the same angles among themselves as the latter.

Such a linear element on the first surface has the expression

and the corresponding element on the second surface is

If these two elements are to be in a determined ratio, regardless of $$dt$$ and $$du,$$ then the three quantities

must obviously be proportional to the following three

If the ends of a second element on the first surface correspond to the values

then the cosine of the angle that this element makes with the first is

and for the cosine of the angle between the corresponding elements on the second surface, we obtain a similar expression, where $$a, \ b, \ c, \ a', \ b', \ c'$$ are replaced by $${A}, \ {B}, \ {C}, \ {A'}, \ {B'}, \ {C'}.$$ The two expressions will be equal to each other when the proportionality in question occurs, and the second condition is therefore already included in the first, which a moment of reflection makes evident.

The analytical expression of the condition of our problem is thus as follows

and this must be a finite function of $$t$$ and $$u$$ that we will set $$= m^2.$$ Thus $$m$$ expresses the ratio by which the linear quantities on the first surface are increased or decreased in their representation on the second (depending on whether $$m$$ is greater or less than $$1$$). In general, this ratio will vary with location. In the particular case where $$m$$ is constant, there will be complete similarity even in finite parts, and when moreover $$m = 1,$$ perfect equality will occur, and each of the surfaces can be developed upon the other.

5.
If, for the sake of brevity, we set

we notice that the differential equation $$\omega =0$$ will have two integrals. Indeed, if we decompose the trinomial $$\omega$$ into two factors, linear with respect to $$dt$$ and $$du,$$ one or the other must be $$= 0,$$ which will give two different integrals. One of the integrals will correspond to the equation

(where we have written $$i$$ for brevity instead of $$\sqrt{-1},$$ since it is easy to see that the irrational part of the expression must be imaginary); the other integral corresponds to a similar equation in which $$i$$ has been replaced by $$-i$$.

Therefore, if the integral of the first equation is

with $$p$$ and $$q$$ denoting real functions of $$t$$ and $$u,$$ then the other integral will be

from which it follows by the very nature of the question that

or equivalently

must be a factor of $$\omega,$$ and hence

where $$n$$ is a finite function of $$t$$ and $$u.$$

Now let $$\Omega$$ denote the trinomial obtained by replacing $$d{X}, \ d{Y}, \ d{Z}$$ with their values at $${T}, \ {U}, \ d{T}, \ d{U} $$ in

Assume as above that the two integrals of the equation $$\Omega = 0$$ are

and that

where $${P}, \ {Q}, \ {N}$$ are real functions of $${T}$$ and $${U}.$$ These integrations can obviously be carried out (apart from the general difficulties of integration) prior to the resolution of our main problem.

If we substitute functions of $$t, \ u$$ such that the condition of our main problem is fulfilled, then $$\Omega$$ transforms into $$m^2 \omega,$$ and we will have

But it is easy to see that the numerator of the first member of this equation can only be divisible by the denominator when

and

or when

and

In the first case, it follows that $$d{P}+id{Q}$$ will vanish when $$dp+idq=0,$$ or else $${P}+i{Q}$$ will be constant whenever $$p+iq,$$ is constant; that is, $${P}+i{Q}$$ will simply be a function of $$p+iq,$$ and similarly $${P}-i{Q}$$ will be a function of $$p-iq.$$ In the other case, $${P}+i{Q}$$ will be a function of $$p-iq,$$ and $${P}-i{Q}$$ a function of $$p+iq.$$ It is also easy to see that the converses of these conclusions are valid; that is, when we take for $${P}+i{Q}, \ {P}-i{Q}$$ (either respectively, or in the reverse order) functions of $$p+iq, \ p-iq,$$ the exact divisibility of $$\Omega$$ by $$\omega,$$ and thus the required proportionality will occur.

Moreover, it is easy to see that if we set, for example,

then the nature of the function $$f'$$ is determined by that of the function $$f.$$ Indeed, when among the constant quantities that the latter may contain, none of them are imaginary, the function $$f'$$ must be completely identical to $$f,$$ so that real values of $$p, \ q$$ always correspond to real values of $${P}, \ {Q};$$ otherwise $$f'$$ will only differ from $$f$$ insofar as in the imaginary elements of $$f'$$ we must replace $$i$$ everywhere by $$-i.$$

Consequently, we have

or equivalently, when the function $$f$$ is chosen completely arbitrarily [with imaginary constant elements chosen at will], $${P}$$ will be equal to the real part and $$i{Q}$$ ($$-i{Q}$$ in the second solution) will be equal to the imaginary part of $$f(p+iq),$$ and consequently we can solve for $${T}$$ and $${U}$$ as functions of $$t$$ and $$u.$$

Therefore, the proposed problem has been solved in a complete and general manner.

6.
If $$p'+iq'$$ represents any determined function of $$p+iq$$ (with $$p', \ q'$$ being real functions of $$p, \ q$$), then it is easy to see that

also represent integrals of the differential equation $$\omega = 0,$$ since they are completely equivalent to the equations

Similarly, the integrals of the differential equation $$\Omega = 0,$$

will be completely equivalent to the equations

if $${P'}-i{Q'}$$ represents any determined function of $${P}+i{Q}$$ (where $${P'}, \ {Q'}$$ are real functions of $${P}, \ {Q}$$). Hence it is clear that in the general solution given in the previous article, $$p', \ q'$$ can replace $$p, \ q,$$ and $${P'}, \ {Q'}$$ can replace $${P}, \ {Q}.$$ Although the solution of the problem does not gain any generality in this way, it is nevertheless sometimes more convenient in applications to use one of these forms instead of the other.

7.
If the functions obtained by differentiating the arbitrary functions $$f, f'$$ are denoted by $$\varphi$$ and $$\varphi'$$ respectively, so that

then as a consequence of our general solution, we will have

and consequently

The magnification ratio will therefore be determined by the formula

8.
We will now clarify our general solution with some examples, both to fully illustrate the mode of application and to further highlight the nature of some of the facts presented.

Let us first consider two plane surfaces; we can then set

The differential equation

gives here the two integrals

and similarly the two integrals of the equation

are as follows:

Therefore, the two general solutions of the problem are

This result can also be expressed as follows: Let the letter $$f$$ denote any function, then for $${X}$$ we should take the real part of $$f(x+iy),$$ and for $${Y}$$ we should take either the real or imaginary part (after removing the factor $$i$$).

If we use the letters $$\varphi, \ \varphi'$$ in the sense explained in Article 7, and if we set

where $$\xi$$ and $$\eta$$ are obviously to be real functions of $$x$$ and $$y,$$ then for the first solution we will have

and consequently

Now, if we set

so that $$ds$$ represents a linear element on the first plane, $$g$$ its inclination to the x-axis, $$d{S}$$ the corresponding linear element on the second plane, and $${G}$$ its inclination to the x-axis, the previous equations yield

and if $$\sigma$$ is considered to be positive, which is allowed, then we have

We see, therefore (in accordance with Article 7), that $$\sigma$$ represents the magnification ratio of the element $$ds$$ in the representation $$dS$$, and that it is, as it should be, independent of $$g;$$ similarly, the fact that the angle $$\gamma$$ is independent of $$g$$ shows that all linear elements originating from a point on the first plane will be represented by elements on the second plane, forming the same angles as the former, and, we may add, in the same sense.

If we choose a linear function for $$f$$, so that $$f(v)={A}+{B}v$$, where the constant coefficients are of the form

then we will have

and consequently,

The magnification ratio is thus constant at all points, and the representation is everywhere similar to the original.

For any other function $$f$$ (as is easy to demonstrate), the magnification ratio will not be constant, and consequently, similarity will only occur in the smallest parts.

If the points that must correspond to a given number of points given in the first plane are assigned on the representation, one can easily, using the ordinary method of interpolation, find the simplest algebraic function that fulfills this condition. Indeed, if we denote the values of $$x + iy$$ for the given points by $$a, b, c $$ etc., and the corresponding values of $${X}+i{Y}$$ by $${A}, {B}, {C}$$ etc., we must set

which is an algebraic function of $$v$$ whose order is one less than the number of assigned points. In the case of two points, where the function is linear, complete similarity is thus achieved.

This method can be usefully applied in geodesy to improve a map made from moderately accurate measurements, good in small details but slightly distorted on a large scale, when one knows the exact position of a certain number of points. It goes without saying, however, that one can hardly venture far from the regions surrounding these points.

If we treat the second solution in the same way, we find that the only difference lies in the fact that the similarity is inverse; all elements on the representation make angles with each other equal in magnitude to those of the original, but in the opposite sense, so that what was on the right is now located on the left. This distinction is not essential, and it disappears if on one of the planes the side previously considered to be the upper one is now taken as the lower. Moreover, one can always apply this remark, when one of the two surfaces is a plane; thus, in the examples of this nature that follow, we can simply adhere to the first solution.

9.
Let us now consider (as a second example) the representation of the surface of a right cone on the plane. Let us take the equation of the first surface as

and then set

and as before,

Then the differential equation

gives the two integrals

Thus, we have the solution

or in other words, if $$f$$ denotes an arbitrary function, we take for $${X}$$ the real part of

and for $${Y}$$ the imaginary part, after removing the factor $$i$$.

For example, if we take $$f$$ to be an exponential function,

where $$h$$ is a constant, and where $$e$$ denotes the base of the hyperbolic logarithm, then we have the simplest representation

Applying the formulas of Article 7, we have here

And, since $$\varphi(v)=\varphi'(v)=he^v,$$ we have

and consequently, the magnification ratio will be

and thus constant. If we further set

then the representation becomes a complete development.

10.
Thirdly, let us represent the surface of the sphere of radius $$a$$ on a plane. Here we set

from which we obtain

The differential equation $$\omega =0$$ then gives

and integrating this gives

Therefore, if we continue to let $$f$$ denote an arbitrary function, we should take for $${X}$$ the real part and for $$i{Y}$$ the imaginary part of

We intend to treat some special cases of this general solution.

If we choose for $$f$$ a linear function, setting $$f(v) = kv,$$ we have

Applied to the Earth, where $$t$$ denotes the geographical longitude and $$90^{\circ} - u$$ the latitude, this representation is clearly nothing other than the Mercator projection. For the scale factor, the formulas from Article 7 give

If we choose for $$f$$ an imaginary exponential function, starting with the simplest one $$f(v) = ke^{iv}$$, we have

and

which is easily recognized as the stereographic projection.

If we generally set $$fv = ke^{i\lambda v},$$ we have:

For the scale factor, we obtain

hence

We see that here the representations of all points for which $$u$$ is constant occur along a circle, and the representations of all points for which $$t$$ is constant along a straight line, moreover, the circles corresponding to all different values of $$u$$ are concentric. This provides a very useful projection for maps when it is only necessary to represent a part of the surface of the sphere, and then the best thing to do is to choose $$\lambda$$ so that the scale factor is the same for the extreme values of $$u,$$ so that it takes its smallest value towards the middle. If the extreme values of $$u$$ are $$u^0$$ and $$u',$$ then we should set

The sheets of celestial maps Nos. 19 to 26 by Prof. Harding are constructed using this projection.

11.
The general solution of the example treated in the previous article can also be presented in another form, which we think should be added here due to its elegance.

According to the considerations presented in Article 6, since

is a function of

and

the general solution can also be given by

meaning that $${X}$$ should be set equal to the real part, and $$i{Y}$$ should be set equal to the imaginary part of $$f\left(\frac{x+iy}{a+z} \right),$$ where $$f$$ denotes an arbitrary function. Instead of $$f\left(\frac{x+iy}{a+z} \right)$$, it is easy to see that one can also take an arbitrary function of $$\frac{x+iz}{a+x}$$ or $$\frac{z+ix}{a+y}.$$

12.
Fourthly, let us consider the representation of the surface of an ellipsoid of revolution on the plane. Let $$a$$ and $$b$$ be the two principal semi-axes of the ellipsoid, so that we can set

Then we have

and the differential equation $$\omega = 0$$, if we set $$\sqrt{1-\tfrac{b^2}{a^2}}=\varepsilon$$ for short (with the revolution semi-axis $$b<a$$), is

If we then set

where, when applied to the terrestrial spheroid, $$90^{\circ} - w$$ represents the geographical latitude and $$t$$ the longitude, this equation transforms into

the integration of which gives

Therefore, with $$f$$ denoting an arbitrary function, we must take for $${X}$$ the real part and for $$i{Y}$$ the imaginary part of

If we choose a linear function for $$f$$, i.e., $$fv = kv,$$ we will have

which provides a projection analogous to that of Mercator.

If, on the other hand, we choose an imaginary exponential function for $$f$$, i.e., $$f(v) = ke^{i\lambda v},$$ we will have

which, for $$\lambda = 1$$, provides a projection analogous to the stereographic polar projection, which in general is very advantageous for representing a part of the earth's surface in cases where flattening needs to be taken into account.

As for what remains to be said about the other case where $$b > a,$$ it would indeed be easy to deduce it from the preceding discussion where, while retaining the same notation, epsilon is imaginary, but where $$\left( \frac{1+\varepsilon \cos w}{1-\varepsilon \cos w}\right)^{\frac{1}{2}\varepsilon }$$ will still be real. However, to be complete, let us indicate the formulas specific to this case, and to begin, let us set $$\sqrt{\tfrac{b^2}{a^2}-1}=\eta.$$ We must then determine $$w$$ using the equation

and the differential equation

yields the integral

so we must take for $${X}$$ the real part and for $$i{Y}$$ the imaginary part of

We can then immediately derive the analogues of the two particular cases considered earlier. For the first one, we must set

and for the second one

13.
As a final example, we will discuss the general representation of the surface of an ellipsoid of revolution on that of the sphere.

We will retain the notations for the ellipsoid from the previous article, and denote the semi-diameter of the sphere by $${A}$$, and we will set

Applying the general solution from Article 5, with $$f$$ denoting an arbitrary function, we find that we must set $${T}$$ equal to the real part and $$i \log \operatorname{cotang} \tfrac{1}{2} {U}$$ equal to the imaginary part of

The simplest solution is obtained by setting $$ f(v) = u, $$ hence

This provides an extremely useful transformation in higher geodesy, of which we can only indicate a few details here in passing. In particular, if we consider points on the ellipsoid's surface and on the sphere's surface as corresponding when they have the same longitude and their respective latitudes, $$90^{\circ} - {U}, \ 90^{\circ} - w,$$, are related by the equation above, then a system of relatively small triangles (which will always be the ones that can be used for actual measurements), formed on the surface of the ellipsoid by geodesic lines, will correspond to a system of triangles on the surface of the sphere, whose angles are exactly equal to the corresponding angles on the ellipsoid, and whose sides differ so little from arcs of great circles that, in most cases where the strictest accuracy is not required, they can be regarded as such; however, when the greatest accuracy is necessary, the deviation from the arc of a great circle can be easily evaluated with all the necessary precision using simple formulas. The entire system can be calculated using angles, as if it were on the sphere itself (after appropriately transferring one side of one of the triangles to the sphere, and making the necessary adjustments), then determining the values of $${T}$$ and $${U}$$ for all points in the system, and from these latter values one can go back to the corresponding values of $$w$$ (the simplest way to do this will be to use an auxiliary table, which is easy to construct).

As a network of triangles only extends over a very small part of the Earth's surface, the desired goal will be achieved even more completely if we further generalize the general solution by taking not $$f(v) = v$$ but $$f(v) = v + \text{Const.}$$ Obviously, there would be absolutely nothing gained if we attributed a real value to this constant, since then $${T}$$ and $$t$$ would simply differ by that constant, and consequently only the origins of longitudes would change. But it is quite different when the constant is attributed an imaginary value. If we set it $$= i \log k,$$ then we will have

In order to recognize the most suitable value of $$k$$ here, we must first determine the magnification ratio.

In this case, with the notation of Articles 5 and 6, we have

Moreover,

which consequently depends only on the latitude. The smallest possible deviation from complete similarity is obtained if we determine $$k$$ so that $$m$$ takes equal values for the extreme latitudes, which will clearly cause $$m$$ to take a value very close to its maximum or minimum for the average latitude. If we denote the extreme values of $$w$$ by $$w^0$$ and $$w',$$ we obtain in this way

To determine at which latitude $$m$$ reaches its maximum or minimum value, we have

from which

From this, it is clear that $$m$$ reaches its maximum or minimum value when $${U}=w ;$$ if we denote the value of $$w$$ at this point by $${W},$$ we have

from which $$W$$ can be derived, once $$k$$ has been calculated using the previous formula. However, in practice, it matters little whether there is a perfectly rigorous equality of the values of $$m$$ at the extreme latitudes, and one can be content with approximately taking the average latitude for $$90^{\circ} - {W}$$, and deriving $$k$$ from there. The general relation between $${U}$$ and $$w$$ is then given by the formula

For effective numerical computations, it is more advantageous to use series, which can take several forms, but we will not delve into their development here.

Moreover, as one can easily see, for $$w < {W},$$ we have $$\ {U} >  w,$$ so $$\cos {U} - \cos w$$ and therefore $$\frac{dm}{dw}$$ is negative; for $$w > {W},$$ we have $${U} < w,$$ and thus $$\frac{dm}{dw}$$ is positive. Thus it is clear that for $$w = {U} = {W},$$ the value of $$m$$ will always be a minimum, which moreover is

Therefore, if we take the radius of the sphere to be

then the smallest parts of the representation of the ellipsoid for latitude $$90^{\circ} - {W}$$ are not only similar but also equal to the original; but for other latitudes, it is larger.

It is advantageous to expand the logarithm of $$m$$ into a series of powers of $$\cos {U} - \cos {W},$$ whose first terms, which suffice in practice, are as follows:

Therefore, if we map e.g. the Danish monarchy onto the surface of the sphere between the boundaries defined by latitudes $$53^{\circ}$$ and $$58^{\circ},$$ and if we set $${W} = 34^{\circ} 30',$$ then, with a flattening of $$\tfrac{1}{303},$$ the representation at the boundaries, linearly evaluated, will only be enlarged by $$\tfrac{1}{530000}.$$

We must be content here to have given only a brief indication of one method of representing figures in higher geodesy, reserving a properly detailed exposition for another occasion.

14.
We still need to consider here in a slightly more detailed manner a circumstance that arises in our general solution. We have shown in Article 5 that there are always two solutions; either $${P} + i{Q}$$ must be equal to a function of $$p + iq$$ and $${P} - i{Q}$$ equal to a function of $$p - iq ,$$ or $${P} + i{Q}$$ must be equal to a function of $$p - iq$$ and $${P} - i{Q}$$ to a function of $$p + iq.$$ We now want to further prove that in one of the solutions the parts of the representation have a position similar to that of the original, and that conversely in the other solution they are placed inversely. We also want to indicate a criterion by which this can be recognized a priori.

But before that, let us note that there can only be direct or inverse similarity if we distinguish two sides for each of the two surfaces, one of which will be regarded as the upper side and the other as the lower side. Since this distinction is somewhat arbitrary, the two solutions, in fact, are not essentially distinct, and inverse similarity becomes direct similarity, as soon as for one of the surfaces, the side previously considered as the upper is now regarded as the lower. Therefore, this distinction could not arise in our solution, since the surfaces are simply determined by the coordinates of their points. If one wishes to introduce this distinction, one must first determine the nature of the surface by another method that entails this distinction. To this end, we assume that the nature of the first surface is determined by an equation $$\psi=0,$$ where $$\psi$$ is a given uniform function of $$x, \ y, \ z.$$ Thus, at all points on the surface, the value of $$\psi$$ vanishes, and at all points in space that do not belong to the surface, it does not vanish. Therefore, when crossing the surface, the value of $$\psi$$ changes, in general at least, from positive to negative, and vice versa during the reverse passage; that is to say, on one side of the surface, the value of $$\psi$$ will be positive, and on the other, negative; we will consider the former side as the upper side, and the latter as the lower side. We will make the same conventions for the second surface, with its nature determined by an equation $$\Psi =0,$$ where $$\Psi$$ denotes a given uniform function of the coordinates $${X}, \ {Y}, \ {Z}.$$ Differentiation then yields

where $$e, \ g, \ h$$ are functions of $$x, \ y, \ z$$ and $${E}, \ {G}, \ {H}$$ of $${X}, \ {Y}, \ {Z}.$$

Since the considerations we must employ for our envisaged purpose, although not inherently difficult, are of a somewhat unusual nature, we will strive to provide them with the greatest clarity. Between the two representations that correspond on the surfaces $$\psi=0$$ and $$\Psi=0$$, we consider six intermediate representations on the plane, so that we have to consider eight representations, namely:

We will now compare these various representations based only on the sense of their infinitely small linear elements, leaving the magnification ratio aside entirely; thus two representations will be considered to have the same sense when, given two linear elements issuing from a point, the one on the right in one representation corresponds to the one on the right in the other: otherwise, they will be said to have the opposite sense. For planes 2 through 7, the side where the positive value of the third coordinate is found will always be regarded as the upper side; for the first and last surfaces, on the other hand, the distinction between the upper and lower sides depends on where the values of $$\psi$$ and $$\Psi$$ are positive and negative, as agreed upon earlier.

Here it is clear that at each point on the first surface, where for constant $$x$$ and $$y$$, one moves to the upper side by a positive increase in $$z,$$ the representation on 2 will have the same sense as that on 1. This will obviously happen whenever $$h$$ is positive, and for negative $$h$$, the opposite will occur, and the representations will then have the opposite sense.

Similarly, the representations on 7 and 8 will have the same or opposite sense depending on whether $${H}$$ is positive or negative.

To compare representations 2 and 3, consider on the first one an infinitely small line of length $$ds$$ drawn from the point with coordinates $$x, \ y$$ to another point with coordinates $$x + dx, \ y + dy,$$ and let $$l$$ be the angle that this line makes with the x-axis, measured in the direction from the x-axis to the y-axis, so that

In representation 3, let $$d\sigma$$ be the length of the line corresponding to $$ds,$$ and let $$\lambda$$ be the angle it makes with the x-axis, counted as before, such that $$dt = d\sigma \cos \lambda, \ du = d\sigma \sin \lambda.$$

Consequently, with the notation from Article 4, we have

and thus

If we now consider $$x$$ and $$y$$ as constants and $$l, \lambda$$ as variables, differentiation gives us

Thus we see, that depending on whether $$ab' - ba'$$ is positive or negative, $$l$$ and $$\lambda$$ always increase in the same direction or vary inversely, and thus, representations 2 and 3 have the same sense in the former case and the opposite sense in the latter.

Combining this result with the one previously found, it is recognized that representations 1 and 3 have the same or opposite sense depending on whether $$\frac{ab'-ba'}{h}$$ is positive or negative.

Since on the surface with the equation $$\psi=0$$ we have

we will also have

no matter the ratio of $$dt$$ and $$du,$$ thus it is obviously identical,

From this it follows that $$e, \ g, \ h$$ are proportional to the quantities $$bc' - cb', \ ca' - ac', \ ab' - ba'$$ respectively, and therefore

Therefore, as a criterion for whether the representations 1 and 3 have the same or opposite since, any of these three expressions, or alternatively, multiplying them by the essentially positive quantity $$e^2 + g^2 + h^2,$$ the symmetric expression thus obtained

Similarly, the representations 6 and 8 will have the same or opposite sense depending on whether

or if preferred, the symmetric expression

has a positive or negative value.

The comparison of the senses of representations 3 and 4 relies on principles entirely similar to that of 2 and 3, and depends on the positive or negative sign of the quantity

likewise, the positive or negative sign of

will determine the relationship between the senses of representations 5 and 6.

Finally, concerning the comparison of representations 4 and 5, we can refer to the analysis of Article 5, which clearly shows that they have the same or opposite sense depending on whether the first or the second solution is chosen, that is to say, depending on whether we set

or

From all this, we conclude that when the representation on the surface with equation $$\Psi=0$$ must not only be similar in its smallest parts to the original with equation $$\psi=0,$$ but must also have the same sense, one must consider the number of negative quantities among the following four:

If this number is zero or even, the first solution should be chosen; if among the above quantities one or three are negative, the second solution should be chosen. For the opposite choice, inverse similarity will always be found.

Moreover, it can be further demonstrated, denoting the four previous quantities respectively by $$r, \ s, \ S, \ R,$$ that we always have

where $$n$$ and $${N}$$ have the same meaning as in Article 5. We omit the easily found demonstration of this theorem; indeed, this theorem is not necessary for our intended purpose.


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 these old jars is very great, amounting in some instances to hundreds of dollars, and the expenses of the funeral obsequies of an opulent Dusun chief often amount to over $600, buffaloes being killed and eaten, tuak consumed in large quantities, obat (fetish ceremonies) performed, etc.

Although the Putatan cannot properly be described as a sago river, its delta would afford a large area of land suitable for planting the sago palm, the land being low-lying and swampy, and abutting on a good water-way on either side. The highlands of the interior are easily accessible up its valley, the climate is salubrious and pleasant, the population large and well-disposed, but the lands along its banks are firmly held and highly valued, and it is doubtful whether any area of such land could ever be brought into the market.

The course of the main Putatan, or Telîpuk, to the sea, from the point at which the Patâgas branch diverges from it, is somewhat tortuous but has a good depth of water. Its mouth, however, as already stated, is shoal and difficult of entry. An examination of an outcrop of the strata on its right bank, on the way down, shewed the strike to be S.E. with a dip of about 80°. A mangrove growth extends up both banks for a short distance from the kuala, and also along the coast on either side, and there is no beach available for landing on. This is not the case with the Patâgas mouth which has a sandy beach and true jungle close to the sea with however mangroves inside.

As regards the state of cultivation of the tract watered by it, the Putatan may be fairly classed as the show river of North Borneo.

S. ELPHINSTONE DALRYMPLE.


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text: Journal of the Straits Branch of the Royal Asiatic Society/Volume 13/The Tawaran and Putatan Rivers


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text: Author:Stair Elphinstone Dalrymple

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of Malay, among whom many members of this Society may be included, will learn with interest that Mr., one of the best Malay scholars in Holland, has it in contemplation to publish an English-Malay Dictionary, adapted from his new Dutch-Malay Dictionary now passing through the press.

Whether the English edition will, or will not, be produced, will depend upon the measure of support which may be depended on. A certain number of subscribers should be forthcoming, in the first instance, to ensure that a work involving so much labour will not result in pecuniary loss. The publication of the work, if subscribers are found in sufficient number (and among these, the Government of the Straits Settlements, the Raja of Sarawak and the Directors of the British North Borneo Company may perhaps be counted upon for substantial support ?), will be undertaken by Mr. of Leyden, whose recently issued prospectus is subjoined:—

"Mr. whose scientific and practical publications on the Malay Language and Literature have already contributed so largely to the knowledge of that language, has, as the result of several years' study, undertaken the compilation of an English-Malay Dictionary, which is to be published next year. When, during the Oriental Congress at Leyden, Mr., speaking with some competent men on this publication, was requested to make it accessible to those also, who do not understand the Dutch Language. Though a very tedious work, he would not directly decline the request, if by a sufficient subscription it should appear that the work would meet the wishes of those who are to use it.

"With regard to the manner in which he intends to compile this dictionary, he believes it to be in consonance with the


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 of the language simple and natural, and—what his long experience confirms—thoroughly practical.

"The transcription of the Malay words will be given in the dialect of the Straits-Settlements, after the manner adopted by "Messrs. and.

"The English part will be revised by a native Englishman. As soon as we have a sufficient number of subscribers-we shall begin printing. The subscription price for one copy is twenty-five shillings, bound

"To show the manner in which the author has formed his plan, and to give an idea of its completeness and extent, we give a specimen, annexed to this prospectus.

"The whole will be about one thousand—or one thousand two hundred pages.

"To lay claim to completeness the Dictionary of the English. and Malay languages now projected ought to be followed by a Dictionary of the Malay and English languages, the materials for which are in preparation. It will be published as a second volume, if the support accorded to the present volume gives sufficient encouragement."

", October, 1881.

E. J. BRILL.

A subscription list has been opened by the Honorary Secretary of the Straits Branch, Royal Asiatic Society, Singapore; and members of the Society and the public who may wish to enter their names as subscribers can do so at the Raffles. Library, Singapore, where the specimen sheet of the proposed Dictionary can be seen. As stated in the prospectus, the subscription price for one copy will be £1 5s. 0d, bound.

In the paper headed "Malacca in the Eighteenth Century" printed in Journal No. XII, p. 261, for bentang (wherever that word occurs), real benteng.


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 Gladys and Dorothea Cromwell were so essentially one that any account of either must include the other. Neither ever used the singular pronoun, and those who knew them fairly well often doubted to which sister they were speaking. Indeed when it was suggested to Gladys that "Gates of Utterance" should be dedicated to Dorothea, she answered that poets were not in the habit of dedicating their verse to themselves. So in writing even a brief sketch it is necessary to think of them as they were, an identity expressed in two terms. They were born in November, 1885, and inherited possessions, talents, and an exquisite beauty strangely poignant because in the twin sisters the charm seemed more than doubled. There are a few men and women with whom one feels a sense of spiritual mystery: one walks with them always on the road to Emmaus. It was true of these two They found their home in the unseen. In the outer, material world they existed only by an effort that cost them much, for they moved as spirits, untouched by crude desires; bending with a shy longing to meet human needs; searching for some solution that should justify their

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Rh personal immunities, their money, and the grace and luxury to which they had been born. A delicate humility made them feel debtors to life. In their eyes existence was a bond given by the soul, to be redeemed at any cost. Both had written from childhood, and in 1915 Gladys published a volume of poems that promised no uncertain music. Slight as it was, endless toil lay back of it: she had the master's sense of workmanship, and every verse and stanza was the outcome of labor that had often covered years. "Gates of Utterance" was obviously a first book: but it was the first book of a poet. Dorothea was developing more slowly, experimenting more cautiously. The short stories she left show at once more cleverness, a keener sense of epigram, of earth's hidden laughter, than any one could have guessed who saw only a graceful, fuchsia-like creature, eager to give her time and income to social experiment and investigation. But of them more was asked than selfless generosity, or will to serve. In a picture taken at the Chalons Canteen, the two girls, veiled and habited in white working uniform, stand like conventual sisters serving a group of poilus; Dorothea holds a slender pitcher from which she pours into the soldier's cup, while Gladys offers bread in a shallow basket. Clear of line like a classic bas-relief, the so fortunate and so

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 casual photograph is strangely symbolic and recalls One who said, "Take, eat; this is my body broken for you." Gladys and Dorothea Cromwell broke the bread of their bodies and poured out the wine of their spirits that others might live.

When the war drew an inerasable line across all lives, the two girls began to prepare themselves. They spent their summer months in a hospital: they learned to run a motor; they took canteen-efficiency lessons; they held themselves aloof from the over-heated speech of excitement, but their hearts burned within them. The world as they saw it demanded of them an heroic resolve.

In January, 1918, the two sisters, having rolled in the Canteen Service of the Bed Cross, sailed for France and were stationed at Chalons. For eight months they worked under fire on long day or night shifts; their free time was filled with volunteer outside service; they slept in "caves" or under trees in a field; they suffered from the exhaustion that is so acute to those who have never known physical labor; yet no one suspected until the end came that for many months they had believed their work a failure, and their efforts futile. The Chalonais called them "The Saints"; during dull evenings, the poilus, who adored the "Twin An-

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his mail one morning Tony Camonte received a unique communication, an ornate, engraved invitation to the opening of the Woodland Casino, a new roadhouse and gambling place some little distance out in the country, far beyond the jurisdiction of city authorities but not so far away as to be beyond the reach of city patrons. The invitation also conveyed the information that the opening night was to be a Bal Masque and that admittance would be by card only.

Tony didn't know what a Bal Masque was and he felt no urge to find out. But the other bit of information interested him somewhat. In common with other wealthy but socially ineligible people, he had an almost irresistible curiosity to see the inside of exclusive places. The realization that hundreds of these invitations must have been sent out did not prevent his own vanity being tickled by receiving one; the fact remained that everybody who might want to couldn't get in.

For a moment he toyed with the pleasant

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 gels," found amusement in effort, always unsuccepsful, to distinguish them apart. The workers in the Canteen loved and admired them for their courage—that finest bravery which leads fear to intrepid action; they loved them for their rare charm, but they gave them whole-souled appreciation for the tireless, efficient labor which made them invaluable as practical canteeners. In September, at their own request, they were transferred to an Evacuation Hospital, for after the rest of a "permission" they longed to work with "our own boys." Eight months overwhelming strain and fatigue had made them more weary than they realized, and the horrors of conditions near the Front broke their already overtaxed endurance. In the diaries they left, signs of mental breakdown begin to show as early as October. After the Armistice, when they returned to Chalons as guests, they showed symptoms of nervous prostration, but years of self-control and consideration for others made them conceal the black horror in which they lived—the agony through which they saw a world which they felt contained no refuge for beauty and quiet thought. In such a world they conceived they had no place, and when on their way home they jumped from the deck of the Lorraine, it was in response to a vision that promised them fulfilment and

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 ing chief makes the most important speeches. The women are completely dependent upon their husbands for their status in this village group. Once a man has been given a title, he can never go back to the Aumaga. His title may be taken away from him when he is old, or if he is inefficient, but a lower title will be given him that he may sit and drink his kava with his former associates. But the widow or divorced wife of a matai must go back into the Aualuma, sit with the young girls outside the house, serve the food and run the errands, entering the women's fono only as a servant or an entertainer.

The women's fonos are of two sorts: fonos which precede or follow communal work, sewing the thatch for a guest house, bringing the coral rubble for its floor or weaving fine mats for the dowry of the taupo; and ceremonial fonos to welcome visitors from another village. Each of these meetings was designated by its purpose, as a falelalaga a weaving bee, or an 'aiga fiafia tama'ita'i, ladies' feast. The women are only recognised socially by the women of a visiting village" but the taupo and her court are the centre of the recognition of both men and women in the malaga, the travelling party. And these wives of high chiefs have to treat their own taupo with great courtesy and respect, address her as "your highness," accompany her on journeys, use a separate set of nouns and verbs when speaking to her. Here then is a discrepancy in which the young girls who are kept in strict subjection within their households, outrank their aunts and mothers in


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 the social life between villages. This ceremonial undercutting of the older women's authority might seriously jeopardise the discipline of the household, if it were not for two considerations. The first is the tenuousness of the girls' organisation, the fact that within the village their chief raison d'être is to dance attendance upon the older women, who have definite industrial tasks to perform for the village; the second is the emphasis upon the idea of service as the chief duty of the taupo. The village princess is also the village servant. It is she who waits upon strangers, spreads their beds and makes their kava, dances when they wish it, and rises from her sleep to serve either the visitors or her own chicf. And she is compelled to serve the social needs of the women as well as the men. Do they decide to borrow thatch in another village, they dress their taupo in her best and take her along to decorate. the malaga. Her marriage is a village matter, planned and carried through by the talking chiefs and their wives who are her counsellors and chaperons. So that the rank of the taupo is really a further daily inroad upon her freedom as an individual, while the incessant chaperonage to which she is subjected and the way in which she is married without regard to her own wishes are a complete denial of her personality. And similarly, the slighter prestige of her untitled sisters, whose chief group activity is waiting upon their elders, has even less real significance in the daily life of the village.

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 whose title is the occasion of a great festival and enormous distribution of property by her chief to the talking chiefs who must hereafter support and confirm her rank, a Samoan girl of good family has two ways of making her début. The first, the formal entry into the Aualuma is often neglected and is more a formal fee to the community than a recognition of the girl herself. The second way is to go upon a malaga, a formal travelling party. She may go as a near relative of the taupo in which case she will be caught up in a whirl of entertainment with which the young men of the host village surround their guests; or she may travel as the only girl in a small travelling party in which case she will be treated as a taupo. (All social occasions demand the presence of a taupo, a manaia, and a talking chief; and if individuals actually holding these titles are not present, some one else has to play the rôle.) Thus it is in inter-village life, either as a member of the Aualuma who call upon and dance for the manaia of the visiting malaga, or as a visiting girl in a strange village, that the unmarried Samoan girl is honoured and recognised by her community.

But these are exceptional occasions. A malaga may come only once a year, especially in Manu'a which numbers only seven villages in the whole archipelago. And in the daily life of the village, at crises, births, deaths, marriages, the unmarried girls have no ceremonial part to play. They are simply included with the "women of the household" whose duty it is to prepare the layette for the new baby, or carry stones to


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 strew on the new grave. It is almost as if the community by its excessive recognition of the girl as a taupo or member of the Aualuma, considered itself exonerated from paying any more attention to her.

This attitude is fostered by the scarcity of taboos. In many parts of Polynesia, all women, and especially menstruating women, are considered contaminating and dangerous. A continuous rigorous social supervision is necessary, for a society can no more afford to ignore its most dangerous members than it can afford to neglect its most valuable. But in Samoa a girl's power of doing harm is very limited. She cannot make tafolo, a breadfruit pudding usually made by the young men in any case, nor make the kava while she is menstruating. But she need retire to no special house; she need not eat alone; there is no contamination in her touch or look. In common with the young men and the older women, a girl gives a wide berth to a place where chiefs are engaged in formal work, unless she has special business there. It is not the presence of a woman which is interdicted but the uncalled-for intrusion of any one of either sex. No woman can be officially present at a gathering of chiefs unless she is taupo making the kava, but any woman may bring her husband his pipe or come to deliver a message, so long as her presence need not be recognised. The only place where a woman's femininity is in itself a real source of danger is in the matter of fishing canoes and fishing tackle which she is forbidden to touch upon pain of spoiling the fishing. But the enforcement of this prohibition is in the hands of


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 individual fishermen in whose houses the fishing equipment is kept.

Within the relationship group matters are entirely different. Here women are very specifically recognised. The oldest female progenitor of the line, that is, the sister of the last holder of the title, or his predecessor's sister, has special rights over the distribution of the dowry which comes into the household. She holds the veto in the selling of land and other important family matters. Her curse is the most dreadful a man can incur for she has the power to "cut the line" and make the name extinct. If a man falls ill, it is his sister who must first take the formal oath that she has wished him no harm, as anger in her heart is most potent for evil. When a man dies, it is his paternal aunt or his sister who prepares the body for burial, anointing it with turmeric and rubbing it with oil, and it is she who sits beside the body, fanning away the flies, and keeps the fan in her possession ever after. And in the more ordinary affairs of the household, in the economic arrangements between relatives, in disputes over property or in family feuds, the women play as active a part as the men.

The girl and woman repays the general social negligence which she receives with a corresponding insouciance. She treats the lore of the village, the genealogies of the titles, the origin myths and local tales, the intricacies of the social organisation with supreme indifference. It is an exceptional girl who can give her great-grandfather's name, the exceptional boy who can-


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 not give his genealogy in traditional form for several generations. While the boy of sixteen or seventeen is eagerly trying to master the esoteric allusiveness of the talking chief whose style he most admires, the girl of the same age learns the minimum of etiquette. Yet this is in no wise due to lack of ability. The taupo must have a meticulous knowledge, not only of the social arrangements of her own village, but also of those of neighbouring villages. She must serve visitors in proper form and with no hesitation after the talking chief has chanted their titles and the names of their kava cups. Should she take the wrong post which is the prerogative of another taupo who outranks her, her hair will be soundly pulled by her rival's female attendants. She learns the intricacies of the social organisation as well as her brother does. Still more notable is the case of the wife of a talking chief. Whether she is chosen for her docility by a man who has already assumed his title, or whether, as is often the case, she marries some boy of her acquaintance who later is made a talking chief, the tausi, wife of a talking chief, is quite equal to the occasion. In the meetings of women she must be a master of etiquette and the native rules of order, she must interlard her speeches with a wealth of unintelligible traditional material and rich allusiveness, she must preserve the same even voice, the same lofty demeanour, as her husband. And ultimately, the wife of an important talking chief must qualify as a teacher as well as a performer, for it is her duty to train the taupo. But unless the community thus recognises her existence,


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 and makes formal demand upon her time and ability, a woman gives to it a bare minimum of her attention.

In like manner, women are not dealt with in the primitive penal code. A man who commits adultery with a chief's wife was beaten and banished, sometimes even drowned by the outraged community, but the woman was only cast out by her husband. The taupo who was found not to be a virgin was simply beaten by her female relatives. To-day if evil befalls the village, and it is attributed to some unconfessed sin on the part of a member of the community, the Fono and the Aumaga are convened and confession is enjoined upon any one who may have evil upon his conscience, but no such demand is made upon the Aualuma or the wives of the matais. This is in striking contrast to the family confessional where the sister is called upon first.

In matters of work the village makes a few precise demands. It is the women's work to cultivate the sugar cane and sew the thatch for the roof of the guest house, to weave the palm leaf blinds, and bring the coral rubble for the floor. When the girls have a paper mulberry plantation, the Aumaga occasionally help them in the work, the girls in turn making a feast for the boys, turning the whole affair into an industrious picnic. But between men's formal work and women's formal work there is a rigid division. Women do not enter into house-building or boat-building activities, nor go out in fishing canoes, nor may men enter the formal weaving house or the house where women are making tapa in a group. If the women's work


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 makes it necessary for them to cross the village, as is the case when rubble is brought up from the seashore to make the floor of the guest house, the men entirely disappear, either gathering in some remote house, or going away to the bush or to another village. But this avoidance is only for large formal occasions. If her husband is building the family a new cook-house, a woman may make tapa two feet away, while a chief may sit and placidly braid cinet while his wife weaves a fine mat at his elbow.

So, although unlike her husband and brothers a woman spends most of her time within the narrower circle of her household and her relationship group, when she does participate in community affairs she is treated with the punctilio which marks all phases of Samoan social life. The better part of her attention and interest is focused on a smaller group, cast in a more personal mode. For this reason, it is impossible to evaluate accurately the difference in innate social drive between men and women in Samoa. In those social spheres where women have been given an opportunity, they take their place with as much ability as the men. The wives of the talking chiefs in fact exhibit even greater adaptability than their husbands. The talking chiefs are especially chosen for their oratorical and intellectual abilities, whereas the women have a task thrust upon them at their marriage requiring great oratorical skill, a fertile imagination, tact, and a facile memory.


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 peace. To those who loved them, their death was not only heart-breaking, but brought with it a terrible sense of that most profound tragedy of war,—the bitter waste of spiritual promise. In everyday life they were of those to whom the senses carry a double message; all of us have memories of moments when a driven leaf, a slant of afternoon light, send through avenue of sight or sound an anguish no physical cause can explain—to these sisters, life was continuously bought at such a price, and the undue strain broke the too frail physiques.

It is almost a year since they died on the 19th of January, 1919. Three months later they were buried in France with military honors, and the French Government has awarded them the Croix de Guerre and the Médaille de Reconnaissance française. They gave to the world lives of shining promise and crystal purity, having followed Him who said to His other disciples: Greater love hath no man than this, that a man lay down his life for his friend.

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 you'd have made a much better lawyer than I. But you know, dear, how much I care about you; and I did hope that on that account even a prejudice, however unreasonable, might have some little weight. And I'm not alone, let me tell you, in my views. There was a fellow in court only to-day, who was saying that yours was only a succès d'estime and that woman, as a naturally talkative and hopelessly unpunctual animal, could never be more than a clever amateur in the profession you have chosen."

"That will do, Enguerrand," said Jeanne, proudly; "it seems that when argument fails, you can stoop so low as to insult me through my sex. You men are all alike—steeped in brutish masculine prejudice. Now go away, and don't mention the subject to me again till you're quite reasonable and nice."

Jeanne passed a somewhat restless night after her small scene with her cousin, waking depressed and unrefreshed. Though she had carried matters with so high a hand, and had scored so distinctly all around, she had been more agitated than she had cared to show. She liked Enguerrand; and more especially did she like his admiration for her; and that chance allusion to Clairette contained possibilities that were alarming. In embracing a professional career, she had never thought for a moment that it could militate against that due share of admiration to which, as a girl, she was justly entitled; and Enguerrand's views seemed this morning all the more narrow and inexcusable. She rose languidly, and as soon as she was dressed sent off a little note to the Mayor, saying that she had a nervous headache and felt out of sorts, and

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 begging to be excused from attendance on that day; and the missive reached the Mayor just as he was taking his usual place at the head of the Board.

"Dear, dear," said the kind-hearted old man, as soon as he had read the letter to his fellow-councilmen: "I'm very sorry. Poor girl! Here, one of you fellows, just run round and tell the gaoler there won't be any business to-day. Jeanne's seedy. It's put off till to-morrow. And now, gentlemen, the agenda"

"Really, your worship," exploded Robinet, "this is simply ridiculous!"

"Upon my word, Robinet," said the Mayor, "I don't know what's the matter with you. Here's a poor girl unwell—and a more hardworking girl isn't in the town—and instead of sympathising with her, and saying you're sorry, you call it ridiculous! Suppose you had a headache yourself! You wouldn't like"

"But it is ridiculous," maintained the tanner stoutly. "Who ever heard of an executioner having a nervous headache? There's no precedent for it. And 'out of sorts,' too! Suppose the criminals said they were out of sorts, and didn't feel up to being executed?"

"Well, suppose they did," replied the Mayor, "we'd try and meet them halfway, I daresay. They'd have to be executed some time or other, you know. Why on earth are you so captious about trifles? The prisoners won't mind, and I don't mind: nobody's inconvenienced, and everybody's happy!"

"You're right there, Mr. Mayor," put in another councilman. "This executing business used to give the town a lot of trouble and bother; now it's all as easy as kiss-your-hand. Instead of objecting, as they used to do, and wanting to argue the point and kick up a row, the fellows as is told off for execution come skipping along in the morning, like a lot of lambs in Maytime.

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 And then the fun there is on the scaffold! The jokes, the back-answers, the repartees! And never a word to shock a baby! Why, my little girl, as goes through the market-place every morning—on her way to school, you know—she says to me only yesterday, she says, 'Why, father,' she says, 'it's as good as the play-actors,' she says."

"There again," persisted Robinet, "I object to that too. They ought to show a properer feeling. Playing at mummers is one thing, and being executed is another, and people ought to keep 'em separate. In my father's time, that sort of thing wasn't thought good taste, and I don't hold with new-fangled notions."

"Well, really, neighbour," said the Mayor, "I think you're out of sorts yourself to-day. You must have got out of bed the wrong side this morning. As for a little joke, more or less, we all know a maiden loves a merry jest when she's certain of having the last word! But I'll tell you what I'll do, if it'll please you; I'll go round and see Jeanne myself on my way home, and tell her—quite nicely, you know—that once in a way doesn't matter, but that if she feels her health won't let her keep regular business hours, she mustn't think of going on with anything that's bad for her. Like that, don't you see? And now, gentlemen, let's read the minutes!"

Thus it came about that Jeanne took her usual walk that evening with a ruffled brow and a swelling heart; and her little hand opened and shut angrily as she paced the ramparts. She couldn't stand being found fault with. How could she help having a headache? Those clods of citizens didn't know what a highly-strung sensitive organisation was. Absorbed in her reflections, she had taken several turns up and down the grassy footway, before she became aware that she was not alone. A youth, of richer dress and more elegant bearing than the general run of

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 the Radegundians, was leaning in an embrasure, watching the graceful figure with evident interest.

"Something has vexed you, fair maiden?" he observed, coming forward deferentially as soon as he perceived he was noticed; "and care sits but awkwardly on that smooth young brow."

"Nay, it is nothing, kind sir," replied Jeanne; "we girls who work for our living must not be too sensitive. My employers have been somewhat exigent, that is all. I did wrong to take it to heart."

Tis the way of the bloated capitalist," rejoined the young man lightly, as he turned to walk by her side. "They grind us, they grind us; perhaps some day they will come under your hands in turn, and then you can pay them out. And so you toil and spin, fair lily! And yet methinks those delicate hands show little trace of labour?"

"You wrong me, indeed, sir," replied Jeanne merrily. "These hands of mine, that you are so good as to admire, do great execution!"

"I can well believe that your victims are numerous," he replied; "may I be permitted to rank myself among the latest of them?"

"I wish you a better fortune, kind sir," answered Jeanne demurely.

"I can imagine no more delightful one," he replied; "and where do you ply your daily task, fair mistress? Not entirely out of sight and access, I trust?"

"Nay, sir," laughed Jeanne, "I work in the market-place most mornings, and there is no charge for admission; and access is far from difficult. Indeed, some complain—but that is no business of mine. And now I must be wishing you a good evening. Nay"—for he would have detained her—"it is not seemly for an

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 unprotected maiden to tarry in converse with a stranger at this hour. Au revoir, sir! If you should happen to be in the marketplace any morning" And she tripped lightly away. The youth, gazing after her retreating figure, confessed himself strangely fascinated by this fair unknown, whose particular employment, by the way, he had forgotten to ask; while Jeanne, as she sped homewards, could not help reflecting that for style and distinction, this new acquaintance threw into the shade all the Enguerrands and others she had met hitherto—even in the course of business.

The next morning was bright and breezy, and Jeanne was early at her post, feeling quite a different girl. The busy little marketplace was full of colour and movement, and the gay patches of flowers and fruit, the strings of fluttering kerchiefs, and the piles of red and yellow pottery, formed an artistic setting to the quiet impressive scaffold which they framed. Jeanne was in short sleeves, according to the etiquette of her office, and her round graceful arms showed snowily against her dark blue skirt and scarlet tight-fitting bodice. Her assistant looked at her with admiration.

"Hope you're better, miss," he said respectfully. "It was just as well you didn't put yourself out to come yesterday; there was nothing particular to do. Only one fellow, and he said he didn't care; anything to oblige a lady!"

"Well, I wish he'd hurry up now, to oblige a lady," said Jeanne, swinging her axe carelessly to and fro: "ten minutes past the hour; I shall have to talk to the Mayor about this."

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 His ankles rest on the earth, and his head reaches to the very sky.

It once happened that travellerstravelers [sic] on a vessel noticed a bird. As he stood in the water, it merely covered his feet, and his head knocked against the sky. The onlookers thought the water could not have any depth at that point, and they prepared to take a bath there. A heavenly voice warned them: “Alight not here! Once a carpenter’s axe slipped from his hand at this spot, and it took it seven years to touch bottom.” The bird the travellerstravelers [sic] saw was none other than the ziz. His wings are so huge that unfurled they darken the sun. They protect the earth against the storms of the south; without their aid the earth would not be able to resist the winds blowing thence. Once an egg of the ziz fell to the ground and broke. The fluid from it flooded sixty cities, and the shock crushed three hundred cedars. Fortunately such accidents do not occur frequently. As a rule the bird lets her eggs slide gently into her nest. This one mishap was due to the fact that the egg was rotten, and the bird cast it away carelessly. The ziz has another name, Renanim, because he is the celestial singer. On account of his relation to the heavenly regions he is also called Sekwi, the seer, and, besides, he is called “son of the nest,” because his fledgling birds break away from the shell without being hatched by the mother bird; they spring directly from the nest, as it were. Like leviathan, so ziz is also a delicacy to be served to the pious at the end of time, to compensate them for the privations which abstaining from the unclean fowls imposed upon them.


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 "It's a pity there ain't a better show this morning," pursued the assistant, as he leant over the rail of the scaffold and spat meditatively into the busy throng below. "They do say as how the young Seigneur arrived at the Château yesterday—him as has been finishing his education in Paris, you know. He's as likely as not to be in the market-place to-day; and if he's disappointed, he may go off to Paris again, which would be a pity, seeing the Château's been empty so long. But he may go to Paris, or anywheres else he's a mind to, he won't see better workmanship than in this here little town!"

"Well, my good Raoul," said Jeanne, colouring slightly at the obvious compliment, "quality, not quantity, is what we aim at here, you know. If a Paris education has been properly assimilated by the Seigneur, he will not fail to make all the necessary allowances. But see, the prison-doors are opening at last!"

They both looked across the little square to the prison, which fronted the scaffold; and sure enough, a small body of men, the Sheriff at their head, was issuing from the building, conveying, or endeavouring to convey, the tardy prisoner to the scaffold. That gentleman, however, seemed to be in a different and less obliging frame of mind from that of the previous day; and at every pace one or other of the guards was shot violently into the middle of the square, propelled by a vigorous kick or blow from the struggling captive. The crowd, unaccustomed of late to such demonstrations of feeling, and resenting the prisoner's want of taste, hooted loudly; but it was not until that ingenious mediæval arrangement known as la marche aux crapauds had been brought to bear on him, that the reluctant convict could be prevailed upon to present himself before the young lady he had already so unwarrantably detained.

Jeanne's profession had both accustomed her to surprises

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 and taught her the futility of considering her clients as drawn from any one particular class: yet she could hardly help feeling some astonishment on recognising her new acquaintance of the previous evening. That, with all his evident amiability of character, he should come to this end, was not in itself a special subject for wonder; but that he should have been conversing with her on the ramparts at the hour when—after courteously excusing her attendance on the scaffold—he was cooling his heels in prison for another day, seemed hardly to be accounted for, at first sight. Jeanne, however, reflected that the reconciling of apparent contradictions was not included in her official duties.

The Sheriff, wiping his heated brow, now read the formal procѐs delivering over the prisoner to the executioner's hands; "and a nice job we've had to get him here," he added on his own account. And the young man, who had remained perfectly tractable since his arrival, stepped forward and bowed politely.

"Now that we have been properly introduced," said he courteously, "allow me to apologise for any inconvenience you have been put to by my delay. The fault was entirely mine, and these gentlemen are in no way to blame. Had I known whom I was to have the pleasure of meeting, wings could not have conveyed me swiftly enough."

"Do not mention, I pray, the word inconvenience," replied Jeanne with that timid grace which so well became her: "I only trust that any slight discomfort it may be my duty to cause you before we part, will be as easily pardoned. And now—for the morning, alas! advances—any little advice or assistance that I can offer is quite at your service; for the situation is possibly new, and you may have had but little experience."

"Faith, none worth mentioning," said the prisoner, gaily.

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 ======

As the fish were formed out of water, and the birds out of boggy earth well mixed with water, so the mammals were formed out of solid earth, and as leviathan is the most notable representative of the fish kind, and ziz of the bird kind, so behemot is the most notable representative of the mammal kind. Behemot matches leviathan in strength, and he had to be prevented, like leviathan, from multiplying and increasing, else the world could not have continued to exist; after God had created him male and female, He at once deprived him of the desire to propagate his kind. He is so monstrous that he requires the produce of a thousand mountains for his daily food. All the water that flows through the bed of the Jordan in a year suffices him exactly for one gulp. It therefore was necessary to give him one stream entirely for his own use, a stream flowing forth from Paradise, called Yubal. Behemot, too, is destined to be served to the pious as an appetizing dainty, but before they enjoy his flesh, they will be permitted to view the mortal combat between leviathan and behemot, as a reward for having denied themselves the pleasures of the circus and its gladiatorial contests.

Leviathan, ziz, and behemot are not the only monsters; there are many others, and marvellous ones, like the reëm, a giant animal, of which only one couple, male and female, is in existence. Had there been more, the world could hardly have maintained itself against them. The act of copulation occurs but once in seventy years between them, for God has so ordered it that the male and female reëm are at opposite ends of the earth, the one in the east, the other in the west.


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 as “the color of light,” “the color of a material,” “spectral colors,” etc., since such expressions may be taken to imply a psychophysical linkage between stimulus and color which is reliable under normal conditions.

The purpose of the ensuing Part of this Report is to present data on certain laws or conditions which are fundamental to visual response and, in particular, to the science of colorimetrics. These data refer mainly to psychophysical relations of the two types defined in the preceding Part, but also to certain purely psychological and purely physiological laws. A great deal remains to be discovered and made definite in this field, and the following statements merely represent the best determinations available when the best are often far from good. Reference should be had to the Report of the Committee on Visual Sensitometry for a more detailed treatment of visual laws which are of interest mainly to the photometrician or the illuminating engineer.

The three-fold attributive nature of color permits the symbolic arrangement of all possible colors in the form of a geometrical, or quasi-geometrical solid, neighboring members being separated by just noticeable differences. At present there are not sufficient data to permit of an accurate construction, but the following approximations would appear to be determined.

A. .—Since the hues are cyclical in their resemblances, they will be represented most appropriately by angular measures with reference to a fixed point and line in any plane. The saturation must then be determined—on the general principle of polar coördinates—by a distance in the plane from the fixed point, or pole. The third dimension, of brilliance, consequently becomes a distance perpendicular to the selected plane.

B. .—The most natural origin of coördinates is the point on the axis of the figure which represents the median gray (vide supra), the axis itself (standing for all of the achromatic

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 "Treat me as a raw beginner. Though our acquaintance has been but brief, I have the utmost confidence in you."

"Then, sir," said Jeanne, blushing, "suppose I were to assist you in removing this gay doublet, so as to give both of us more freedom and less responsibility?"

"A perquisite of the office?" queried the prisoner with a smile, as he slipped one arm out of the sleeve.

A flush came over Jeanne's fair brow. "That was ungenerous," she said.

"Nay, pardon me, sweet one," said he, laughing: twas but a poor jest of mine—in bad taste, I willingly admit."

"I was sure you did not mean to hurt me," she replied kindly, while her fingers were busy in turning back the collar of his shirt. It was composed, she noticed, of the finest point lace; and she could not help a feeling of regret that some slight error—as must, from what she knew, exist somewhere—should compel her to take a course so at variance with her real feelings. Her only comfort was that the youth himself seemed entirely satisfied with his situation. He hummed the last air from Paris during her ministrations, and when she had quite finished, kissed the pretty fingers with a metropolitan grace.

"And now, sir," said Jeanne, "if you will kindly come this way: and please to mind the step—so. Now, if you will have the goodness to kneel here—nay, the sawdust is perfectly clean; you are my first client this morning. On the other side of the block you will find a nick, more or less adapted to the human chin, though a perfect fit cannot of course be guaranteed in every case. So! Are you pretty comfortable?"

"A bed of roses," replied the prisoner. "And what a really admirable view one gets of the valley and the river, from just this particular point!"

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 The act of copulation results in the death of the male. He is bitten by the female and dies of the bite. The female becomes pregnant and remains in this state for no less than twelve years. At the end of this long period she gives birth to twins, a male and a female. The year preceding her delivery she is not able to move. She would die of hunger, were it not that her own spittle flowing copiously from her mouth waters and fructifies the earth near her, and causes it to bring forth enough for her maintenance. For a whole year the animal can but roll from side to side, until finally her belly bursts, and the twins issue forth. Their appearance is thus the signal for the death of the mother reëm. She makes room for the new generation, which in turn is destined to suffer the same fate as the generation that went before. Immediately after birth, the one goes eastward and the other westward, to meet only after the lapse of seventy years, propagate themselves, and perish. A travellertraveler [sic] who once saw a reëm one day old described its height to be four parasangs, and the length of its head one parasang and a half. Its horns measure one hundred ells, and their height is a great deal more.

One of the most remarkable creatures is the “man of the mountain,” Adne Sadeh, or, briefly, Adam. His form is exactly that of a human being, but he is fastened to the ground by means of a navel-string, upon which his life depends. The cord once snapped, he dies. This animal keeps himself alive with what is produced by the soil around about him as far as his tether permits him to crawl. No creature may venture to approach within the radius of his cord, for he seizes and demolishes whatever comes in his reach. To kill


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 "Charming, is it not?" replied Jeanne. "I'm so glad you do justice to it. Some of your predecessors have really quite vexed me by their inability to appreciate that view. It's worth coming here to see it. And now, to return to business for one moment,—would you prefer to give the word yourself? Some people do; it's a mere matter of taste. Or will you leave yourself entirely in my hands?"

"Oh, in your fair hands," replied her client, "which I beg you to consider respectfully kissed once more by your faithful servant to command."

Jeanne, blushing rosily, stepped back a pace, moistening her palms as she grasped her axe, when a puffing and blowing behind caused her to turn her head, and she perceived the Mayor hastily ascending the scaffold.

"Hold on a minute, Jeanne, my girl," he gasped. "Don't be in a hurry. There's been some little mistake."

Jeanne drew herself up with dignity. "I'm afraid I don't quite understand you, Mr. Mayor," she replied in freezing accents. "There's been no little mistake on my part that I'm aware of."

"No, no, no," said the Mayor, apologetically; "but on somebody else's there has. You see it happened in this way: this here young fellow was going round the town last night; and he'd been dining, I should say, and he was carrying on rather free. I will only say so much in your presence, that he was carrying on decidedly free. So the town-guard happened to come across him, and he was very high and very haughty, he was, and wouldn't give his name nor yet his address—as a gentleman should, you know, when he's been dining and carrying on free. So our fellows just ran him in—and it took the pick of them all their time to do it, too. Well, then, the other chap who was in prison

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 the gentleman who obliged you yesterday, you know—what does he do but slip out and run away in the middle of all the row and confusion; and very inconsiderate and ungentlemanly it was of him to take advantage of us in that mean way, just when we wanted a little sympathy and forbearance. Well, the Sheriff comes this morning to fetch out his man for execution, and he knows there's only one man to execute, and he sees there's only one man in prison, and it all seems as simple as A B C—he never was much of a mathematician, you know—so he fetches our friend here along, quite gaily. And—and that's how it came about, you see; hinc illæ lachrymæ, as the Roman poet has it. So now I shall just give this young fellow a good talking to, and discharge him with a caution; and we shan't require you any more to-day, Jeanne, my girl."

"Now, look here, Mr. Mayor," said Jeanne severely, "you utterly fail to grasp the situation in its true light. All these little details may be interesting in themselves, and doubtless the press will take note of them; but they are entirely beside the point. With the muddleheadedness of your officials (which I have frequently remarked upon) I have nothing whatever to do. All I know is, that this young gentleman has been formally handed over to me for execution, with all the necessary legal requirements; and executed he has got to be. When my duty has been performed, you are at liberty to re-open the case if you like; and any 'little mistake' that may have occurred through your stupidity you can then rectify at your leisure. Meantime, you've no locus standi here at all; in fact, you've no business whatever lumbering up my scaffold. So shut up and clear out."

"Now, Jeanne, do be reasonable," implored the Mayor. "You women are so precise. You never will make any allowance for the necessary margin of error in things."

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 him, one may not go near to him, the navel-string must be severed from a distance by means of a dart, and then he dies amid groans and moans.

Once upon a time a travellertraveler [sic] happened in the region where this animal is found. He overheard his host consult his wife as to what to do to honor their guest, and resolve to serve “our man,” as he said. Thinking he had fallen among cannibals, the stranger ran as fast as his feet could carry him from his entertainer, who sought vainly to restrain him. Afterward, he found out that there had been no intention of regaling him with human flesh, but only with the flesh of the strange animal called “man.”

As the “man of the mountain” is fixed to the ground by his navel-string, so the barnacle-goose is grown to a tree by its bill. It is hard to say whether it is an animal and must be slaughtered to be fit for food, or whether it is a plant and no ritual ceremony is necessary before eating it.

Among the birds the phnix is the most wonderful. When Eve gave all the animals some of the fruit of the tree of knowledge, the phnix was the only bird that refused to eat thereof, and he was rewarded with eternal life. When he has lived a thousand years, his body shrinks, and the feathers drop from it, until he is as small as an egg. This is the nucleus of the new bird.

The phnix is also called “the guardian of the terrestrial sphere.” He runs with the sun on his circuit, and he spreads out his wings and catches up the fiery rays of the sun. If he were not there to intercept them, neither man nor any other animate being would keep alive. On his right wing the following words are inscribed in huge letters,


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 colors) serving as the reference line for saturation, while a plane passing through the axis and the extreme spectral red forms the reference plane for hue. (See .)

C. .—It is certain that the bounding surfaces of such a color solid will have an irregular contour. The axial dimension will be about thirty times the radial one at the greatest value of the latter. Both in the directions of high and low brilliance, there will be convergence towards a vertex. If the edges of planes perpendicular to the axis are regarded as determined by the saturations obtainable by spectral stimuli, these edges will be maximally near the axis at the yellow, receding on either side to find a constant value for the intermediaries of red and blue.

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 "If I were to allow the necessary margin for all your errors, Mayor," replied Jeanne, coolly, "the edition would have to be a large-paper one, and even then the text would stand a poor chance. And now, if you don't allow me the necessary margin to swing my axe, there may be another 'little mistake'"

But at this point a hubbub arose at the foot of the scaffold, and Jeanne, leaning over, perceived sundry tall fellows, clad in the livery of the Seigneur, engaged in dispersing the municipal guard by the agency of well-directed kicks, applied with heartiness and anatomical knowledge. A moment later, there strode on to the scaffold, clad in black velvet, and adorned with his gold chain of office, the stately old seneschal of the Château, evidently in a towering passion.

"Now, mark my words, you miserable little bladder-o'-lard," he roared at the Mayor (whose bald head certainly shone provokingly in the morning sun), "see if I don't take this out of your skin presently!" And he passed on to where the youth was still kneeling, apparently quite absorbed in the view.

"My lord," he said, firmly though respectfully, "your hair-brained folly really passes all bounds. Have you entirely lost your head?"

"Faith, nearly," said the young man, rising and stretching himself. "Is that you, old Thibault? Ow, what a crick I've got in my neck! But that view of the valley was really delightful!"

"Did you come here simply to admire the view, my lord?" inquired Thibault severely.

"I came because my horse would come," replied the young Seigneur lightly: "that is, these gentlemen here were so pressing; they would not hear of any refusal; and besides, they forgot to mention what my attendance was required in such a hurry for.

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 And when I got here, Thibault, old fellow, and saw that divine creature—nay, a goddess, dea certé—so graceful, so modest, so anxious to acquit herself with credit Well, you know my weakness; I never could bear to disappoint a woman. She had evidently set her heart on taking my head; and as she had my heart already"

"I think, my lord," said Thibault with some severity, "you had better let me escort you back to the Château. This appears to be hardly a safe place for light-headed and susceptible persons!"

Jeanne, as was natural, had the last word. "Understand me, Mr. Mayor," said she, "these proceedings are entirely irregular. I decline to recognise them, and when the quarter expires I shall claim the usual bonus!"

When, an hour or two later, an invitation arrived—courteously worded, but significantly backed by an escort of half-a-dozen tall archers—for both Jeanne and the Mayor to attend at the Château without delay, Jeanne for her part received it with neither surprise nor reluctance. She had felt it especially hard that the only two interviews fate had granted her with the one man who had made some impression on her heart, should be hampered, the one by considerations of propriety, the other by the conflicting claims of her profession and its duties. On this occasion, now, she would have an excellent chaperon in the Mayor; and business being over for the day, they could meet and unbend on a common social footing. The Mayor was not at all surprised either, considering what had gone before; but he was exceedingly terrified, and sought some consolation from Jeanne as they proceeded together

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 about four thousand stadia high: “Neither the earth produces me, nor the heavens, but only the wings of fire.” His food consists of the manna of heaven and the dew of the earth. His excrement is a worm, whose excrement in turn is the cinnamon used by kings and princes. Enoch, who saw the phnix birds when he was translated, describes them as flying creatures, wonderful and strange in appearance, with the feet and tails of lions, and the heads of crocodiles; their appearance is of a purple color like the rainbow; their size nine hundred measures. Their wings are like those of angels, each having twelve, and they attend the chariot of the sun and go with him, bringing heat and dew as they are ordered by God. In the morning when the sun starts on his daily course, the phoenixes and the chalkidri sing, and every bird flaps its wings, rejoicing the Giver of light, and they sing a song at the command of the Lord

Among reptiles the salamander and the shamir are the most marvellousmarvelous [sic]. The salamander originates from a fire of myrtle wood which has been kept burning for seven years steadily by means of magic arts. Not bigger than a mouse, it yet is invested with peculiar properties. One who smears himself with its blood is invulnerable, and the web woven by it is a talisman against fire. The people who lived at the deluge boasted that, were a fire flood to come, they would protect themselves with the blood of the salamander.

King Hezekiah owes his life to the salamander. His wicked father, King Ahaz, had delivered him to the fires of Moloch, and he would have been burnt, had his mother not painted him with the blood of the salamander, so that the fire could do him no harm.


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Rh *5 U. S. C. §7204(b) (Antidiscrimination; Other Prohibitions)
 * 6 U. S. C. §488f(b) (Secure Handling of Ammonium Nitrate; Protection From Civil Liability)
 * 7 U. S. C. §2020(c)(1) (Supplemental Nutrition Assistance Program)
 * 8 U. S. C. §1152(a)(1)(A) (Immigration; Numerical Limitations on Individual Foreign States)
 * 8 U. S. C. §1187(c)(6) (Visa Waiver Program for Certain Visitors)
 * 8 U. S. C. §1522(a)(5) (Authorization for Programs for Domestic Resettlement of and Assistance to Refugees)
 * 10 U. S. C. §932(b)(4) (Uniform Code of Military Justice; Article 132 Retaliation)
 * 10 U. S. C. §1034(j)(3) (Protected Communications; Prohibition of Retaliatory Personnel Actions)
 * 12 U. S. C. §302 (Directors of Federal Reserve Banks; Number of Members; Classes)
 * 12 U. S. C. §1735f–5(a) (Prohibition Against Discrimination on Account of Sex in Extension of Mortgage Assistance)


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 The shamir was made at twilight on the sixth day of creation together with other extraordinary things. It is about as large as a barley corn, and it possesses the remarkable property of cutting the hardest of diamonds. For this reason it was used for the stones in the breastplate worn by the high priest. First the names of the twelve tribes were traced with ink on the stones to be set into the breastplate, then the shamir was passed over the lines, and thus they were graven. The wonderful circumstance was that the friction wore no particles from the stones. The shamir was also used for hewing into shape the stones from which the Temple was built, because the law prohibited iron tools to be used for the work in the Temple. The shamir may not be put in an iron vessel for safe-keeping, nor in any metal vessel, it would burst such a receptacle asunder. It is kept wrapped up in a woollen cloth, and this in turn is placed in a lead basket filled with barley bran. The shamir was guarded in Paradise until Solomon needed it. He sent the eagle thither to fetch the worm. With the destruction of the Temple the shamir vanished.

A similar fate overtook the tahash, which had been created only that its skin might be used for the Tabernacle. Once the Tabernacle was completed, the tahash disappeared. It had a horn on its forehead, was gaily colored like the turkeycock, and belonged to the class of clean animals.

Among the fishes there are also wonderful creatures, the sea-goats and the dolphins, not to mention leviathan. A seafaring man once saw a sea-goat on whose horns the words were inscribed: “I am a little sea-animal, yet I traversed three hundred parasangs to offer myself as food to the levia-


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68 *12 U. S. C. §1821(d)(13)(E)(iv) (Federal Deposit Insurance Corporation; Insurance Funds)
 * 12 U. S. C. §1823(d)(3)(D)(iv) (Federal Deposit Insurance Corporation; Corporation Moneys)
 * 12 U. S. C. §2277a–10c(b)(13)(E)(iv) (Farm Credit System Insurance Corporation; Corporation as Conservator or Receiver; Certain Other Powers)
 * 12 U. S. C. §3015(a)(4) (National Consumer Cooperative Bank; Eligibility of Cooperatives)
 * 12 U. S. C. §§3106a(1)(B) and (2)(B) (Foreign Bank Participation in Domestic Markets)
 * 12 U. S. C. §4545(1) (Fair Housing)
 * 12 U. S. C. §5390(a)(9)(E)(v) (Wall Street Reform and Consumer Protection; Powers and Duties of the Corporation)
 * 15 U. S. C. §631(h) (Aid to Small Business)
 * 15 U. S. C. §633(b)(1) (Small Business Administration)
 * 15 U. S. C. §719 (Alaska Natural Gas Transportation; Civil Rights)
 * 15 U. S. C. §775 (Federal Energy Administration; Sex Discrimination; Enforcement; Other Legal Remedies)