that is to say the sum of the values f f f f multiplied respectively by the factors

dæ sin (μ,dx), dx sin (μ,2dx), dx sin (μ,3dx)...... dx sin (μ,ndx), would be reduced to zero. This relation would then necessarily exist between the given quantities f1, f, f... f.; and they could not be considered entirely arbitrary, contrary to hypothesis. If these quantities f,ff...f have any values whatever, the relation in question cannot exist, and we cannot satisfy the proposed conditions by omitting one or more terms, such as a, sin (u,x) in equation (e).

Hence the function f(x) remaining undetermined, that is to say, representing the system of an infinite number of arbitrary constants which correspond to the values of x included between 0 and X, it is necessary to introduce into the second member of equation (e) all the terms such as a, sin (u), which satisfy the condition

Sede sin pe sin p=0,

the indices i and j being different; but if it happen that the function f(x) is such that the n magnitudes ƒ1ƒƒ.....ƒ are connected by a relation expressed by the equation

da sin pæf (2) = 0,

it is evident that the term a, sin u might be omitted in the equation (e).

Thus there are several classes of functions ƒ (a) whose development, represented by the second member of the equation (e), does not contain certain terms corresponding to some of the roots μ. There are for example cases in which we omit all the terms whose index is even; and we have seen different examples of this in the course of this work. But this would not hold, if the function ƒ(x) had all the generality possible. In all these cases, we ought to suppose the second member of equation (e) to be complete, and the investigation shews what terms ought to be omitted, since their coefficients become nothing.

426. We see clearly by this examination that the function ƒ (x) represents, in our analysis, the system of a number n of separate quantities, corresponding to n values of a 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 f, f) £ 3 ... În was expressed by a function subject to a continuous law, such as x or x3, sin x, or cos x, or in general p(x); 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 (x), 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 xF(x) or f(x) is equal to bx2; and applying to this function the rule which determines the coefficients, b2 would be developed in a series of terms, such as

a1 sin (u,x) + a, sin (μ,x) + a ̧ sin (μ,∞) + +a, sin (ux).

...

Now each term sin (u,x), when developed according to powers of x, contains only powers of odd order, and the function be 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 x 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 xF(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

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

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

sin x = α1 sin

sin μx+α, sin μ‚x + &c.

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 x 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 ; each Ꮖ 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 fd, sin (μμ) 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 fda sin (4,2) aF (2) 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