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L(x) + Ly (1 – x) = log (v). log (1 – 8) - . This property of the transcendent L, is only true so long as x is less than unity, as when any greater value is assigned to it log (1 – w) becomes impossible.

Euler, Commen. Petrop. 1738. (8) Again in the equation

Ly (1 – 2) = * log (1 - «), we have by changing ~ into x,

L. (1 – x) = 2 log (1 – t)

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11 til

11+ at

{log (1 +2:) - log.x}.

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Analogous properties may by the same method be demonstrated of

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and so on in succession. Generally, the student will have no difficulty in demonstrating the following propositions :

L,(1 – xo) = 2*-*L, (1 + x) + 2"-"L, (1 – »),

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Spence has extended this analysis to the investigation of the properties of transcendents defined by the general law

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the final function or 0. (a) being such that it remains un

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But for this investigation and others connected with it the reader is referred to the work before quoted.

The transcendents which we have been considering are all such that they may be derived by direct integration from known functions, but there are many other transcendents which are given only by means of differential equations. As these are frequently functions of great utility in physical researches, the study of their properties without integrating the equations in which they are involved becomes of great importance. Two examples of such investigations are subjoined.

(10) Let V be a function of a and p given by the differential equation of the second order

d1dV

-) + (gr - 1) V = 0, ............(1)

d x 1 do in which g, h, and I are functions of r, and r is a variable parameter; and if V also satisfy the conditions

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+ h, V = () when x = Uyg............ (3)

dir then will

Ja mingda V.V, = 0); V., and V, being values of "corresponding to the values P, and 1, of r.

From the given equation (1) we easily obtain

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But from the condition (2) we find on taking the limit x = x1, that

u dv. dv.

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Similarly we find from (3) that at the limit x, = X, the same relation holds; hence

(rm - rm) sdxg VV, = 0. As rm and r, are supposed not to be the same, it follows that

Sendag V.V, = 0.
Since we have

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dæg V,' =

" + h, V) when X = Xx9

dx as may be deduced by the usual method for evaluating indeterminate functions.

It is to be observed that the equation (3) involves an equation to determine r, which equation may be written as

F(r) = 0. Poisson * has shewn that this equation has an infinite number of real and unequal roots, for the demonstration of which proposition I must refer to the works cited below.

Bulletin de la Société Philomatique, 1828. Théorie de la Chaleur, p. 178.

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The function V is of great importance in the theory of beat, and the investigation of its properties has formed the subject of several elaborate memoirs by MM. Sturm and Liouville. See Journal de Mathématiques, Tome 1. pages 106, 253, 269, 373, and Tome 11. p. 16.

(11) Let Y., and 2, be integral and rational functions of Me (1 – ?), cos w and sin w determined by the equations d d Y. 1 dY. du (144) du + 1 - dw? *

" + m (m + 1) Y, = 0, d

d2 i dZ,
I (1-1) du ti-a dat

* + n (n + 1) 2, = 0,
du
then will

st' du 12* dw Y, Z, = 0,
so long as m and n are different.
Multiply both equations by (1 MTM), and assume

d d
(1 -m') h, when they become
d2Ym dY,

+ m (m + 1) (1 – 12) Y» = 0 ... (1),
dt * dw'
dz, dz

" + n(n + 1) (1 – me?) 2, = 0) ... (2). att do2 Multiply (1) by Z,,dt dw and (2) by Y,dt dw, subtract (2) from (1) and integrate with respect to t and w. Then transposing, and observing that (1 – up) dt = dy, we have {m(m + 1) – n (n + 1)} Sdu dwY,2,

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Now if we effect the integration of the first term of the right hand side with respect to t, it becomes 1. d2dY,

d2dY) dw {Y, JA ' - Z

= dw (1-1) {Y,

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2.

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