W

=

1

da

- (2)* [dz fdß ¥ (1, B) fdp fdq cos (px – pa) cos (qy − 98)

If we make t=0 in u and in W, the first function becomes (x, y), and the second nothing; and if we also make

equal to

d t=0 in dt

d u and in dt

W, the first function becomes nothing,

and the second becomes equal to (x, y): hence vu+W is the general integral of the proposed equation.

412. We may give to the value of u a simpler form by effecting the two integrations with respect to p and q. For this purpose we use the two equations (1) and (2) which we have proved in Art. 407, and we obtain the following integral,

Denoting by u the first part of the integral, and by W the second, which ought to contain another arbitrary function, we have

If we denote by μ and v two new unknowns, such that we

and if we substitute for a, ß, da, dẞ their values

x+2μ √t, y+2v St, 2dp √t, 2dv √t,

we have this other form of the integral,

αμ dv sin (u2 + v3) $ (x + 2μ √t, y + 2v √i) + W.

We could not multiply further these applications of our formula without diverging from our chief subject. The preceding examples relate to physical phenomena, whose laws were unknown and difficult to discover; and we have chosen them because

the integrals of these equations, which have hitherto been fruitlessly sought for, have a remarkable analogy with those which express the movement of heat.

413. We might also, in the investigation of the integrals, consider first series developed according to powers of one variable, and sum these series by means of the theorems expressed by the equations (B), (BB). The following example of this analysis, taken from the theory of heat itself, appeared to us to be worthy of notice.

We have seen, Art. 399, that the general value of u derived from the equation

developed in series, according to increasing powers of the variable t, contains one arbitrary function only of x; and that when developed in series according to increasing powers of x, it contains two completely arbitrary functions of t.

represents the sum of this series, and contains the single arbitrary function (a).

The value of v, developed according to powers of x, contains two arbitrary functions ƒ (t) and F(t), and is thus expressed :

There is therefore, independently of equation (8), another form of the integral which represents the sum of the last series, and which contains two arbitrary functions, ft) and F(t).

It is required to discover this second integral of the proposed equation, which cannot be more general than the preceding, but which contains two arbitrary functions.

We can arrive at it by summing each of the two series which enter into equation (X). Now it is evident that if we knew, in the form of a function of x and t, the sum of the first series which contains f(t), it would be necessary, after having multiplied it by dx, to take the integral with respect to x, and to change ƒ(t) into F(t). We should thus find the second series. Further, it would be enough to ascertain the sum of the odd terms which enter into the first series: for, denoting this sum by u, and the sum of all the other terms by v, we have evidently

It remains then to find the value of μ. Now the function f(t) may be thus expressed, by means of the general equation (B),

f

ƒ (1) = 1 fdx ƒ (2) √ dp cos (pt – px)

.........

It is easy to deduce from this the values of the functions

.(B).

It is evident that differentiation is equivalent to writing in

the second member of equation (B), under the sign fdp, the

respective factors —p3, +p3, —po, &c.

We have then, on writing once the common factor cos (pt-pz),

Thus the problem consists in finding the sum of the series which enters into the second member, which presents no difficulty. In fact, if y be the value of this series, we conclude

Integrating this linear equation, and determining the arbitrary constants, so that, when x is nothing, y may be 1, and

dy d'y dy

dx' dx2' di3,

may be nothing, we find, as the sum of the series,

It would be useless to refer to the details of this investigation; it is sufficient to state the result, which gives, as the integral sought,

}

qx

— sin 2q2 (t − a) (eTM − e−4o) sin qx } + W..............(BB).

The term W is the second part of the integral; it is formed by integrating the first part with respect to x, from x = 0 to x=x, and by changing f into F. Under this form the integral contains two completely arbitrary functions f(t) and F (t). If, in the value of v, we suppose a nothing, the term W becomes nothing by hypothesis, and the first part u of the integral becomes ƒ(t). If dv we make the same substitution x = 0 in the value of it is dx du dx

second,

evident that the first part
dW
dx

will become nothing, and that the

which differs only from the first by the function F being substituted for f, will be reduced to F(t). Thus the integral expressed by equation (BB) satisfies all the conditions, and represents the sum of the two series which form the second member of the equation (X).

This is the form of the integral which it is necessary to select in several problems of the theory of heat'; we see that it is very different from that which is expressed by equation (8), Art. 397.

1 See the article by Sir W. Thomson, "On the Linear Motion of Heat," Part II. Art. 1. Camb. Math. Journal, Vol. III. pp. 206—8. [A. F.]

414. We may employ very different processes of investigation to express, by definite integrals, the sums of series which represent the integrals of differential equations. The form of these expressions depends also on the limits of the definite integrals. We will cite a single example of this investigation, recalling the result of Art. 311. If in the equation which terminates that Article we write +t sin u under the sign of the function 4, we have

Denoting by the sum of the series which forms the second member, we see that, to make one of the factors 22, 42, 62, &c. disappear in each term, we must differentiate once with respect to t, multiply the result by t, and differentiate a second time with respect to t. We conclude from this that v satisfies the partial differential equation

We have therefore, to express the integral of this equation,

The second part W of the integral contains a new arbitrary function.

The form of this second part W of the integral differs very much from that of the first, and may also be expressed by definite integrals. The results, which are obtained by means of definite integrals, vary according to the processes of investigation by which they are derived, and according to the limits of the integrals.

415. It is necessary to examine carefully the nature of the general propositions which serve to transform arbitrary functions: for the use of these theorems is very extensive, and we derive from them directly the solution of several important physical problems, which could be treated by no other method. The