can therefore express the general term a, by the equation

determining suitably the quantities A, B, and u.

First we find

A and B by supposing m equal to 0 and then equal to 1, which

sin mu =

(g+ 2) sin (m-1) u - sin (m2) u,

comparing which equation with the next,

sin mu = 2 cos u sin (m − 1) u

sin (m − 2) u,

which expresses a known property of the sines of arcs increasing in arithmetic progression, we conclude that q+2 = cos u, or q=-2 versin u; it remains only to determine the value of the

we must have, in order to satisfy the condition a, a,, the equation

sin (n + 1) u — sin u = sin nu - sin (n − 1) u,

circumference and i any integer, such as 0, 1, 2, 3, 4, ... (n − 1) ;

hm

thence we deduce the n values of q or

Thus all the roots

k

of the equation in h, which give the values of h, h', h", h"", &c.

are real and negative, and are furnished by the equations

Suppose then that we have divided the semi-circumference π into n equal parts, and that in order to form u, we take i of those parts, i being less than n, we shall satisfy the differential equations by taking a, to be any quantity whatever, and making

As there are n different arcs which we may take for u, namely,

there are also n systems of particular values for a, B, y, &c., and the general values of these variables are the sums of the particular values.

254. We see first that if the arc u is nothing, the quantities which multiply a, in the values of a, B, y, &c., become all equal to unity, since

sin u

sin Ou

sin u

takes the value 1 when the arc u

vanishes; and the same is the case with the quantities which are

found in the following equations. From this we conclude that constant terms must enter into the general values of a, ß, y, w. Further, adding all the particular values corresponding to a, B, Y, &c., we have

...

an equation whose second member is reduced to 0 provided the arc u does not vanish; but in that case we should find n to be sin nu the value of We have then in general sin u

a+B+y+&c. = na1;

now the initial values of the variables being a, b, c, &c., we must necessarily have

na1 = a+b+c+ &c.;

it follows that the constant term which must enter into each of the general values of

that is to say, the mean of all the initial temperatures.

As to the general values of a, B, y,

by the following equations:

...

w, they are expressed

sin u - sin Ou

255. To determine the constants a, b, c, d... &c., we must consider the initial state of the system. In fact, when the time is nothing, the values of a, ß, y, &c. must be equal to a, b, c, &c.; we have then n similar equations to determine the n constants. The quantities

sinu-sin Ou, sin 2u-sin u, sin 3u-sin 2u, ..., sin nu-sin (n-1) u, may be indicated in this manner,

A sin Ou, A sin u, A sin 2u, A sin 3u, ... A sin (n-1) u;

the equations proper for the determination of the constants are, if the initial mean temperature be represented by C,

The quantities a1, b1, c1, d1, and C being determined by these equations, we know completely the values of the variables

a, B, 7, 8,
,... w.

We can in general effect the elimination of the unknowns in these equations, and determine the values of the quantities a, b, c, d, &c., even when the number of equations is infinite; we shall employ this process of elimination in the following articles.

256. On examining the equations which give the general values of the variables a, B, y......w, we see that as the time increases the successive terins in the value of each variable decrease very unequally: for the values of u, u', u', u'", &c. being

the exponents versin u, versin u', versin u", versin u", &c. become greater and greater. If we suppose the time t to be infinite, the first term of each value alone exists, and the temperature of each of the masses becomes equal to the mean tempera

1

ture (a+b+c+...&c). Since the time t continually increases,

n

each of the terms of the value of one of the variables diminishes proportionally to the successive powers of a fraction which, for the

2k m

versin u

versin u

for the third term e

and so on.

second term, is e The greatest of these fractions being that which corresponds to the least of the values of u, it follows that to ascertain the law which the ultimate changes of temperature follow, we need consider only the two first terms; all the others becoming incomparably smaller according as the time t increases. The ultimate variations of the temperatures a, B, y, &c. are therefore expressed by the following equations: