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and only when all possible sets of variations de, dv, ds of the energy, volume, and entropy of the body satisfy the inequality

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For various purposes it is desirable to employ v, 0 or p, s or p, 0 as independent variables, the remaining three of the variables e, p, v, 0, s then being functions of those chosen. The determinants of these transformations are

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and by the conditions of stability none of these determinants vanish. When functions of the new variables are defined by the equations

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differentiation and comparison with (1) yields the criterion of equilibrium in the forms

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The conditions of integrability of the second members of (1) and (3) are Maxwell's "thermodynamic relations,"

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It may be asked whether p, v and 0, s can serve as sets of independent variables. We have that

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and, as discussed below, it may be taken to be a fact of observation that these determinants vanish, if at all, only at points on a line a2e/ov2 = 0. The circumstance that any one of the variables p, v, 0, s is a function of any two of the others gives rise to the consideration of six pairs of variables that can be taken successively independent, of twelve functions of pairs of variables, and of a set of twentyfour first derivatives of these functions. The derivatives of the set may conveniently be assigned to two classes, Class I containing the derivatives of the 'work variables" p, v, and of the "heat variables" 0, s, with regard to each other, and Class II containing the derivatives of a work variable with regard to a heat variable, and the reverse. Making use of the thermodynamic relations (4)

connecting the cross derivatives, the proposed classification is exhibited in the following table:

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C22

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Using the notation en d2e/dv2, e12 = d2e/dvds, e22 = d2e/ds2, ▲ = €11€22 — the tabulated values of the derivatives in terms of second derivatives of e(v, s) are found from the equations

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obtained by differentiating the conditions of equilibrium (2). The derivatives of Class II have the values e12, f12, 912, -h12; i.e., they form the terms of the conditions of integrability of the second members of the equations (1) and (3). The tabulated derivatives and their reciprocals constitute the set of twenty-four derivatives. By the "reciprocal" of (ap/av). = A/e22, for example, is meant the derivative (av/ap). - €22/A.

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The signs of the twenty-four rates of change of the quantities p, v, 0, s with regard to one another afford extensive information concerning the thermodynamic properties of fluids. By the conditions of stability we have that, for all realizable states of equilibrium, e11, e22, ▲ are positive. It appears then that, for realizable states of equilibrium, the signs of the derivatives of Class I are determined by the conditions of stability, while the signs of the derivatives of Class II are determined by the sign of e12, which must be found by observation. From (Əv/d0)p = €12/A we observe that e12 is negative when the specific volume rises on heating at constant pressure, and that e2 changes sign if the representative point crosses a line of maximum density at constant pressure. It is clear that the fluid states ordinarily observed lie in a field A (in which e12<0), but that a field B (in which e2>0) appears if, as in the case with liquid water, a locus of maximum density occurs. The derivatives whose values (having regard to their signs) are e12, f12,912, h2 are negative in the field A, positive in the field B, and zero on the "zero line" e12 0. A moment's reflection serves to refer a given derivative, with the proper sign, to e12 or f12, etc. When "heating" is understood to mean increase of entropy or rise of temperature, and "expansion means increase of volume or fall of pressure, the circumstance that e12 is positive

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at all points in the field B requires that, in this field, heating at constant volume decreases the pressure, and heating at constant pressure decreases the volume, while isentropic expansion increases the temperature, and isothermal expansion develops heat. In the field A the contrary occurs.

Many interesting conclusions may be drawn. For example, by the definitions of the specific heats at constant volume and at constant pressure, and by reference to the table,

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It follows that, cp>c, everywhere except at points on the zero line, where cp = c. Again, choosing v, as independent variables, let us consider the specific heat c of a fluid with regard to an arbitrary assigned path (v, 0) = 0. The "heat-differential" Ods is

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wherefore, on dividing by de and using the second thermodynamic relation and the definition of c.,

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where dv do is the slope of the path. The quantity op/00, which is equal to -f12, is positive in the field A and negative in the field B. Hence at any point in A the specific heat c is an increasing linear function of dv/de, while at any point in B it is a decreasing linear function of this slope. Its value at a given point depends only on the slope of the path. At any point on the zero line ap/00 0, wherefore c is constant for all finite values of de/de. In particular cpc, as found above. When de/de is infinite the value of c must be found by evaluating an indeterminate form. Two paths for which dv/de is infinite are the isotherm, for which c∞, and the isentropic, for which c = 0.

A graphic representation of the values of the general specific heat at points on the zero line is had when paths through the point are drawn on the surface representing the region of fluid states in the v, s, space. In this surface the zero line is the locus of the minimum points of isentropic section of a trough. For at points on the zero line (00/ov), = e12 = 0.

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When the path is the curve of section of the surface by a cylinder (or plane) perpendicular to the s, 0-plane and cutting the zero line, its projection on the v, 0-plane has a horizontal tangent at the zero line. Its projection on the s, 0-plane is a curve with two coincident branches, whose slope ds/de at the point of meeting the zero line determines the value, at this point, of the specific heat c0ds/de with regard to the path. The isotherm, for which ds/de, and the isentropic, for which ds/de = 0, are such paths.

When the path is any other curve whose projection on the v, 0-plane has a horizontal tangent at the zero line, its projection on the s, 0-plane is a curve of two distinct branches that meet the zero line in a cusp. The common slope ds/de of these branches at the point of meeting determines the value, at this point, of the specific heat c with regard to the path.

When the path is any curve whose projection on the v, 0-plane has not a horizontal tangent at the point of meeting the zero line, its projection on the s, 0-plane is a curve tangent to the zero line at the point of meeting. Hence the slope ds/de of the zero line at this point determines the value there of the specific heat c with regard to the path. The isobar and the isometric (the curve of constant volume) through the point are such paths.

III. A CHECK FORMULA FOR THE AMBIGUOUS CASE IN PLANE TRIANGLES.

By W. R. RANSOM, Tufts College.

In the solution of a triangle for which sides a and b and angle A are given, two values B' and B" are first obtained for the angle opposite b; then two angles C' and C" are found, and finally two sides c' and c". The obvious relation (c' + c') = b cos A may be used to discover the presence of an error in either of the two triangles that have been computed. This formula does not appear in any text book with which I am acquainted: has it not been employed by some one?

IV. THE "KING'S CHAMBER" AND THE GEOMETRY OF THE SPHERE.
By F. J. DICK, Râja-Yoga College.

That the designers of the Great Pyramid possessed a thorough knowledge of the geometry of the sphere has been recognized by some, although the usual view1 is confined to the recognition of their knowledge of the value of T. The length of the "King's Chamber" is exactly double the breadth, while its height is exactly half the diagonal of the floor. Thus if the width be called 2, and the length 4, the "cubic diagonal" of the chamber is 5.

Attention is drawn to the significance of this in connexion with the geometry of the sphere. Let the rectangle DABC, 4 units by 2, represent the floor plan (a shape, by the way, found in many ancient temples). Let the circumscribed circle represent the diametral section of a sphere and let two other spheres touch at the center as shown forming the double vesica piscis p0nS and N0O, determining the planes JH and FG cutting the cylindrical envelope кλμv. The diagonals JG and FH coincide with the diagonals of DABC, and are each 5 in length, i.e., they are of the length of the cubic diagonal of the "King's Chamber." These are the traces of cones cutting the sphere WNES in the small circles whose diameters are AB, CD. Join AN, and draw the circle TMPQR. Then AN measures a side of the pentagon TMPQR whose diagonals only are shown. Pro1 W. M. F. Petrie, The Pyramids and Temples of Gizeh, London, 1883. 2 Op. cit., p. 195.

jecting this and its reflex on AB, CD we have of course the projection of the icosahedron on the plane containing two opposite edges AN, SC, while UVYZX is one face of the internal dodecahedron formed by joining the 12 angular points of the icosahedron.

This diagram appears to show a relation between the vesica piscis and the "length-double-the-width" features of some archaic architecture. Incidentally

it shows that the inscription of a square in a semicircle places the icosahedron in the sphere at one stroke, so to say.

Taking AN or TM, an edge of the icosahedron, as unity, the edge of the internal dodecahedron is represented by the limit of the sequence

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ing to a number of spiral circuits given by the numerator.

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It seems just possible that the geometers of ancient Egypt who, like the later Pythagorean and Platonic schools, derived their knowledge from ancient Âryâvarta, knew well what they meant when suggesting that the world-universe was built on number and the geometry of the dodecahedron. And it may be that we possess the merest fragments of what was actually taught in the temples of old. But we do have some of their mighty works in stone. Have they been read and fully understood?

RECENT PUBLICATIONS.

REVIEWS.

Euclid in Greek. Book I, with Introduction and Notes. By SIR THOMAS L. HEATH. Cambridge, 1920. Pp. x + 240. Price 10s.

Nearly ten years ago Sir George Greenhill, sitting at his baize-covered work table in Staple Inn, Holborn, in an old-world library well known to many scholars from many lands, made the remark to a visitor from over seas that he felt that the only way to teach plane geometry was by a study of Euclid in the original Greek. The remark led to an interesting discussion upon the present state of See the article by R. C. Archibald in this MONTHLY for May, 1918, vol. 25, p. 235.

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