426. We see clearly by this examination that the function f(x) represents, in our analysis, the system of a number n of separate quantities, corresponding to n values of x included between 0 and X, and that these n quantities have values actual, and consequently not infinite, chosen at will. All might be nothing, except one, whose value would be given.

It might happen that the series of the n values ƒ1, f2, ƒ3 ... ƒn was expressed by a function subject to a continuous law, such as x or a3, sin x, or cos x, or in general (a); the curve line OCO, whose ordinates represent the values corresponding to the abscissa x, and which is situated above the interval from x = 0 to x = X, coincides then in this interval with the curve whose ordinate is

(x), and the coefficients a,, a, a, ... a, of equation (e) determined by the preceding rule always satisfy the condition, that any value of x included between 0 and X, gives the same result when substituted in (a), and in the second member of equation (e).

F(x) represents the initial temperature of the spherical shell whose radius is x. We might suppose, for example, F(x) = bx, that is to say, that the initial heat increases proportionally to the distance, from the centre, where it is nothing, to the surface where it is bX. In this case F(x) or f(x) is equal to be; and applying to this function the rule which determines the coefficients, ba3 would be developed in a series of terms, such as

a, sin (u,x)+a, sin (μ,x) + a, sin (u,x) + ... + a, sin (μ„x).

Now each term sin (u,x), when developed according to powers of x, contains only powers of odd order, and the function ba2 is a power of even order. It is very remarkable that this function ba, denoting a series of values given for the interval from 0 to X, can be developed in a series of terms, such as a, sin (μ,x).

We have already proved the rigorous exactness of these results, which had not yet been presented in analysis, and we have shewn the true meaning of the propositions which express them. We have seen, for example, in Article 223, that the function cos is developed in a series of sines of multiple arcs, so that in the equation which gives this development, the first member contains only even powers of the variable, and the second contains only odd powers. Reciprocally, the function sin x, into

which only odd powers enter, is resolved, Art. 225, into a series "of cosines which contain only even powers.

In the actual problem relative to the sphere, the value of F(x) is developed by means of equation (e). We must then, as we see in Art. 290, write in each term the exponential factor, which contains t, and we have to express the temperature v, which is a function of x and t, the equation

х

The general solution which gives this equation (E) is wholly independent of the nature of the function F(x) since this function represents here only an infinite multitude of arbitrary constants, which correspond to as many values of x included between 0 and X.

If we supposed the primitive heat to be contained in a part only of the solid sphere, for example, from x=0 to x = {X, and that the initial temperatures of the upper layers were nothing, it would be sufficient to take the integral

fda sin (4,2) ƒ (2),

between the limits x = 0 and x = X.

In general, the solution expressed by equation (E) suits all cases, and the form of the development does not vary according to the nature of the function.

Suppose now that having written sin x instead of F(x) we have determined by integration the coefficients a,, and that we have formed the equation

x sin x = α1 sin μ1x+α2 sin Мох таз sin

It is certain that on giving to x any value whatever included between 0 and X, the second member of this equation becomes equal to a sin x; this is a necessary consequence of our process. But it nowise follows that on giving to x a value not included between 0 and X, the same equality would exist. We see the contrary very distinctly in the examples which we have cited, and,

particular cases excepted, we may say that a function subject to a continuous law, which forms the first member of equations of this kind, does not coincide with the function expressed by the second member, except for values of x included between 0 and X.

Properly speaking, equation (e) is an identity, which exists for all values which may be assigned to the variable x; each member of this equation representing a certain analytical function which coincides with a known function f(x) if we give to the variable x values included between 0 and X. With respect to the existence of functions, which coincide for all values of the variable included between certain limits and differ for other values, it is proved by all that precedes, and considerations of this kind are a necessary element of the theory of partial differential equations.

Moreover, it is evident that equations (e) and (E) apply not only to the solid sphere whose radius is X, but represent, one the initial state, the other the variable state of an infinitely extended solid, of which the spherical body forms part; and when in these equations we give to the variable x values greater than X, they refer to the parts of the infinite solid which envelops the sphere.

This remark applies also to all dynamical problems which are solved by means of partial differential equations.

427. To apply the solution given by equation (E) to the case in which a single spherical layer has been originally heated, all the other layers having nul initial temperature, it is sufficient to

take the integral (disin (a) aF (a) between two very near limits,

a=r, and a=r+u, r being the radius of the inner surface of the heated layer, and u the thickness of this layer.

We can also consider separately the resulting effect of the initial heating of another layer included between the limits r+ u and r+ 2u; and if we add the variable temperature due to this second cause, to the temperature which we found when the first layer alone was heated, the sum of the two temperatures is that which would arise, if the two layers were heated at the same time. In order to take account of the two joint causes, it is sufficient to

F. H.

29

take the integral sdz sin (μ,2) aF (a) between the limits a=r and

a = r + 2u. More generally, equation (E) being capable of being put under the form

we see that the whole effect of the heating of different layers is the sum of the partial effects, which would be determined separately, by supposing each of the layers to have been alone heated. The same consequence extends to all other problems of the theory of heat; it is derived from the very nature of equations, and the form of the integrals makes it evident. We see that the heat contained in each element of a solid body produces its distinct effect, as if that element had alone been heated, all the others having nul initial temperature. These separate states are in a manner superposed, and unite to form the general system of temperatures.

For this reason the form of the function which represents the initial state must be regarded as entirely arbitrary. The definite integral which enters into the expression of the variable temperature, having the same limits as the heated solid, shows expressly that we unite all the partial effects due to the initial heating of each element.

428. Here we shall terminate this section, which is devoted almost entirely to analysis. The integrals which we have obtained are not only general expressions which satisfy the differential equations; they represent in the most distinct manner the natural effect which is the object of the problem. This is the chief condition which we have always had in view, and without which the results of investigation would appear to us to be only useless transformations. When this condition is fulfilled, the integral is, properly speaking, the equation of the phenomenon; it expresses clearly the character and progress of it, in the same manner as the finite equation of a line or curved surface makes known all the properties of those forms. To exhibit the solutions, we do not consider one form only of the integral; we seek to obtain directly that which is suitable to the problem. Thus it is that the integral which expresses the

movement of heat in a sphere of given radius, is very different from that which expresses the movement in a cylindrical body, or even in a sphere whose radius is supposed infinite. Now each of these integrals has a definite form which cannot be replaced by another. It is necessary to make use of it, if we wish to ascertain the distribution of heat in the body in question. In general, we could not introduce any change in the form of our solutions, without making them lose their essential character, which is the representation of the phenomena.

The different integrals might be derived from each other, since they are co-extensive. But these transformations require long calculations, and almost always suppose that the form of the result is known in advance. We may consider in the first place, bodies whose dimensions are finite, and pass from this problem to that which relates to an unbounded solid. We can then substitute a definite integral for the sum denoted by the symbol E. Thus it is that equations (a) and (B), referred to at the beginning of this section, depend upon each other. The first becomes the second, when we suppose the radius R infinite. Reciprocally we may derive from the second equation (B) the solutions relating to bodies of limited dimensions.

In general, we have sought to obtain each result by the shortest way. The chief elements of the method we have followed are these:

1st. We consider at the same time the general condition given by the partial differential equation, and all the special conditions which determine the problem completely, and we proceed to form the analytical expression which satisfies all these conditions.

2nd. We first perceive that this expression contains an infinite number of terms, into which unknown constants enter, or that it is equal to an integral which includes one or more arbitrary functions. In the first instance, that is to say, when the general term is affected by the symbol Σ, we derive from the special conditions a definite transcendental equation, whose roots give the values of an infinite number of constants.

The second instance obtains when the general term becomes an infinitely small quantity; the sum of the series is then changed into a definite integral.