If in equation (B) we denote the first term of the second member by p and the second by q, we have

1

πv=p+q, tan p=e ̄(√), tan q = e ̄(*-v√−1);

This is the simplest form under which the solution of the problem can be presented.

206. This value of v or p (x, y) satisfies the conditions relative to the ends of the solid, namely, (x, π) = 0, and 4 (0, y) = 1;

† ±

tion (C) is a transformation of equation (B). Hence it represents exactly the system of permanent temperatures; and since that state is unique, it is impossible that there should be any other solution, either more general or more restricted.

The equation (C) furnishes, by means of tables, the value of one of the three unknowns v, x, y, when two of them are given; it very clearly indicates the nature of the surface whose vertical ordinate is the permanent temperature of a given point of the solid plate. Finally, we deduce from the same equation the values dv dv of the differential coefficients and which measure the velodx dy

city with which heat flows in the two orthogonal directions; and we consequently know the value of the flow in any other direction. These coefficients are expressed thus,

be remarked that in Article 194 the value of

are given by infinite series, whose sums may be easily

found, by replacing the trigonometrical quantities by imaginary

dv

dv

exponentials. We thus obtain the values of and which dx dy

we have just stated.

The problem which we have now dealt with is the first which we have solved in the theory of heat, or rather in that part of the theory which requires the employment of analysis. It furnishes very easy numerical applications, whether we make use of the trigonometrical tables or convergent series, and it represents exactly all the circumstances of the movement of heat. We pass on now to more general considerations.

SECTION VI.

Development of an arbitrary function in trigonometric series.

207. The problem of the propagation of heat in a rectd2v d2v

angular solid has led to the equation + =

dx dy3

= 0; and if it be supposed that all the points of one of the faces of the solid have a common temperature, the coefficients a, b, c, d, etc. of the series

...

a cos x + b cos 3x + c cos 5x + d cos 7x+ &c., must be determined so that the value of this function may be equal to a constant whenever the arc x is included between and +π. The value of these coefficients has just been assigned; but herein we have dealt with a single case only of a more general problem, which consists in developing any function whatever in an infinite series of sines or cosines of multiple arcs. This problem is connected with the theory of partial differential equations, and has been attacked since the origin of that analysis. It was necessary to solve it, in order to integrate suitably the equations of the propagation of heat; we proceed to explain the solution.

We shall examine, in the first place, the case in which it is required to reduce into a series of sines of multiple arcs, a function whose development contains only odd powers of the

variable. Denoting such a function by (x), we arrange the equation

(x) = a sin x + b sin 2x + c sin 3x + d sin 4x + &c.,

...

in which it is required to determine the value of the coefficients First we write the equation

a, b, c, d, &c.

If now we compare the preceding equation with the equation $(x) = = a sin x + b sin 2x + c sin 3x + d sin 4x + e sin 5x + &c., developing the second member with respect to powers of x, we have the equations

A = a + 2b+3c + 4d + 5e + &c.,

B = a+2b+3°c + 43d + 5'e + &c.,

C=a+2b+3°c + 45d+5°e + &c.,

D= a + 2b+3°c + 4'd + 5'e + &c.,

E=a+2b+3°c + 4°d +5°e + &c. ........

(a).

These equations serve to find the coefficients a, b, c, d, e, &c., whose number is infinite. To determine them, we first regard the number of unknowns as finite and equal to m; thus we suppress all the equations which follow the first m equations,

and we omit from each equation all the terms of the second member which follow the first m terms which we retain. The whole number m being given, the coefficients a, b, c, d, e, &c. have fixed values which may be found by elimination. Different values would be obtained for the same quantities, if the number of the equations and that of the unknowns were greater by one unit. Thus the value of the coefficients varies as we increase the number of the coefficients and of the equations which ought to determine them. It is required to find what the limits are towards which the values of the unknowns converge continually as the number of equations increases. These limits are the true values of the unknowns which satisfy the preceding equations when their number is infinite.

208. We consider then in succession the cases in which we should have to determine one unknown by one equation, two unknowns by two equations, three unknowns by three equations, and so on to infinity.

Suppose that we denote as follows different systems of equations analogous to those from which the values of the coefficients must be derived:

If now we eliminate the last unknown e, by means of the five equations which contain A,, В ̧, С5, D ̧, Е, &c., we find

a ̧ (52 - 1o)+2b, (5a — 2o1) + 3c, (52 — 32) + 4d, (52 — 42) = 5a A ̧—В1,

a (52 — 12) + 2% ̧ (52 — 22) + 3′c (52 — 3o) + 4′d ̧ (52 — 42) = 53 D ̧ — E ̧.

5

5

We could have deduced these four equations from the four which form the preceding system, by substituting in the latter instead of

By similar substitutions we could always pass from the case which corresponds to a number m of unknowns to that which corresponds to the number m+1. Writing in order all the relations between the quantities which correspond to one of the cases and those which correspond to the following case, we shall have

b1 = b (52 — 22), c ̧c (52-32), d1 = d ̧ (52 — 42),

4

a=a (62-1), b1 = b (62-22), c, = c, (6-3), d=d. (62 — 42),

6

e ̧ = € (62 — 52),

&c.

&c.

.....(c).