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2. Draw AP, the perpendicular bisector of BD, and produce it to meet BC at Q; then

AP BP tan (B+ C);

and

and

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3. Draw PR parallel to BC, bisecting CD at R; then

4. But

therefore

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EDITED BY W. A. HURWITZ, Cornell University, Ithaca, N. Y.

DISCUSSIONS.

Professor J. E. Trevor gave an example in the December number of the MONTHLY of an instance in which a theorem of analysis is of importance in thermodynamics. Another case of this sort is contributed by him as the first discussion in the present number. Corresponding to a change of independent variable, the form of a function is altered; it is readily shown that the rate of change can be represented as an ordinary derivative of another function closely related to the first. The notion is of use in connection with the heats of dilution of a solution. Instances in which mathematical conceptions arise in connection with applied science are always of value; even those mathematicians whose personal interests are principally in so-called "abstract" fields cannot afford to neglect the phases of mathematics sometimes termed "practical."

In the study of analytic geometry beyond the more elementary portions determinants play an extremely important part; a glance at any treatise on projective or metric analytic geometry, on differential geometry, or on nonEuclidean geometry analytically treated, will reveal page after page filled with resultants, discriminants, Hessians, linear dependence, and other ideas based on determinants. In the elementary parts of the subject, however, determinants

play a negligible rôle. In most texts in common use, the determinantal form for the area of a triangle is a solitary instance. Professor Foraker in the second discussion below gives an introduction to the use of determinants in the elementary field. Most of his work consists of the statement of formulae; one theorem is proved, relating to the collinearity of the circumcenter, centroid, and orthocenter of a triangle. It is scarcely to be expected that the material would be suitable for ordinary class use, but it will be of interest to a few students in any class.

The third discussion is a plea for the more general use of the history of mathematics in connection with secondary and collegiate instruction. Attracted by an announcement of the plans of the National Committee on Mathematical Requirements, Mr. Laurin Zilliacus of the Bedales School in Petersfield, England, has written a letter on the subject to Professor H. W. Tyler, a member of the committee. With the permission of the writer, the recipient, and Professor J. W. Young, chairman of the committee, we reproduce nearly the entire letter, retaining the personal form in which it is written. The fact that Mr. Zilliacus has actual experience of the growth of interest and enthusiasm in a class through the use of historical material is of more value than any amount of mere theorizing on the question.

Another article basing pedagogic recommendations on the result of personal experience is found in the last discussion, by Professor Ettlinger, on the use of graphical methods in trigonometry. It is nearly certain that all teachers make some use, in their teaching, of the ideas suggested by Professor Ettlinger; not all, however, have carried these ideas so far, or treated them with such emphasis. One incidental point in the paper moves the editor to a comment which he has long hoped to see made. It is pointed out that a correctly drawn figure is of great aid in attacking a problem. This is undeniably true in trigonometry, and especially in elementary geometry. Is it not also true, on the other hand, that sometimes a figure drawn with carefully planned inaccuracy is of extreme importance and inspiration?

I. HEATS OF DILUTION.

By J. E. TREVOR, Cornell University.

The "heats of dilution" of a solution are mathematical curiosities, in that these quantities are defined with reference to changes of the form of a function. When (x) is a given function, and dx is an arbitrary positive increment of the independent variable x, let it be supposed that a certain operation changes the value of a quantity from the value 6

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where a is a constant. The change of value 2-1 is due to the change of

form of the function (x, dx), and the rate of change of per unit increment of x in (x) is

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The heats of dilution of a solution are defined by such limits.

Consider constant masses M1, M2 of two component substances, such as salt and water or water and alcohol, capable of forming a homogeneous liquid mixture. Let the state of the body constituted of the masses M1, M2 be such that any mass m; of the jth component exists separately from the mixture of the masses M; -m; and Mk, where Mk is the mass of the other component and Mm; ≥ 0. In the case m; = 0, i.e., when the body is a homogeneous "solution" in a state of stable thermodynamic equilibrium under the pressure p at the temperature 0, let E(p, 0, M1, M2) be the energy and V(p, 0, M1, M2) be the volume of the solution, where p, 0, M1, M2 are variable parameters. When the body is in a state σ1 with separated parts, each in stable equilibrium at p, 0, its energy E1 and volume V1 are the sums of the energies and volumes of the parts. The "enthalpy" G of the solution σ and the enthalpy G1 of the body in the state σ are defined by the equations

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When the separate parts are brought together, under the constant pressure p, the state σ is transformed into the state σ and the work - p(V V1) is absorbed by the body. Hence, by the energy law, the heat absorbed by the body is the quantity G G1. The particular case of this process realized when m; = M; is the formation of the solution from its components. If g; is the enthalpy of unit mass of the jth component, the enthalpy G1 of the separate components is M191 M292, and the "heat of mixing" AG of the solution is

(1)

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where 91, 92 are functions of p, 0.

Consider now the dilution of the solution whose mass is M1+ M2, by addition of the mass 8M; of the jth component at p, 0. The value of the enthalpy of the body composed of the masses M1+ M2 and ¿M; is

(2)

1

2

G(p, 0, M1, M2) + g;(p, 0)8M;

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after the operation. Both before and after the operation, the enthalpy of the body is a function of the constant quantities p, 0, Mj, Mk, 8Mj. In the opera

tion the enthalpy of the body changes in value because the form of the function changes from the form (2) to the form (3). Suppressing the constants p, 0, Mk, this change of value is

G(M;+ ¿M;) — G(M;) — g;ồM;.

Since this expression denotes the heat absorbed by the body (M;, Mk, ¿M;) during the dilution, we have that the heat absorbed per unit mass of diluent added is the limit

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This quantity is the "heat of dilution " A; of the solution (M1, M2) for dilution by the jth component of the mixture.

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Hence A, is equal to the derivative OAG/OM; of the heat of mixing. Further, since it is known that G(p, 0, M1, M2) is homogeneous of degree one in M1, M2, and hence by (1) that AG is homogeneous of degree one in these variables, we have

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It thus appears that the heat of mixing of a solution is a linear function of the two heats of dilution.

II. DETERMINANTS IN ELEMENTARY ANALYTIC GEOMETRY.

By F. A. FORAKER, University of Pittsburgh.

The general purpose of this paper is to indicate how some results in elementary analytic geometry may be conveniently expressed in determinant form. We shall consistently use the notation

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and shall in the usual way indicate the cofactors of elements of determinants by corresponding capital letters. The points P1, P2, P3 will be understood to have coördinates (x1, Y1), (X2, Y2), (X3, yз) respectively. It is well known that the area of the triangle P1P2P3 is A, and that therefore the points P1, P2, P3 are collinear if and only if ▲ = 0.

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in particular, the equation of a line having intercepts a, b is

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The equations of lines respectively parallel and perpendicular to (2) through

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the normal form of the equation of a line may be written

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or may be readily reduced to that type. Conversely, (9) represents a straight line unless A1 = B1 = 0. From (9) any of the usual forms may be derived; it is seen at once that the intercepts and the slope are

(10)

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a =

C1/A1, b = - C1/B1; m =

The equation of a circle through the points P1, P2, P3 is

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A1/B1.

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