In fact, the function which we thus form, - (n2+p2+q3) π1 [ dn [dp [ dq e−(x2 + y2+q" ƒ (x + 2n√t, y + 2p√t, z+2q√t), gives three terms for the fluxion with respect to t, and these three terms are those which would be found by taking the second fluxion with respect to each of the three variables x, y, z. Hence the equation v = π • [dn [dp fdq e ̄ (w2 + p2+q? ƒ (x + 2n √t, y + 2p √t, z + 2q √ë) ..(I), gives a value of v which satisfies the partial differential equation dv d'v d'v d'v = + + (B). 373. Suppose now that a formless solid mass (that is to say one which fills infinite space) contains a quantity of heat whose actual distribution is known. Let v = F(x, y, z) be the equation which expresses this initial and arbitrary state, so that the molecule whose co-ordinates are x, y, z has an initial temperature equal to the value of the given function F(x, y, z). We can imagine that the initial heat is contained in a certain part of the mass whose first state is given by means of the equation v = F(x, y, z), and that all other points have a nul initial temperature. It is required to ascertain what the system of temperatures will be after a given time. The variable temperature v must consequently be expressed by a function (x, y, z, t) which ought to satisfy the general equation (A) and the condition (x, y, z, 0) F(x, y, z). Now the value of this function is given by the integral r+ С dn f dp [ dq e − (x2 + p2 + 4? F' (x + 2n √t, y + 2p √t, z + 2q √i). In fact, this function v satisfies the equation (4), and if in it we make t = 0, we find π * [ dn f dp ƒ dq e−(w2+r2+4) F′ (x, y, z), or, effecting the integrations, F (x, y, z). 374. Since the function v or p (x, y, z, t) represents the initial state when in it we make t=0, and since it satisfies the differential equation of the propagation of heat, it represents also that state of the solid which exists at the commencement of the second instant, and making the second state vary, we conclude that the same function represents the third state of the solid, and all the subsequent states. Thus the value of v, which we have just determined, containing an entirely arbitrary function of three variables x, y, z, gives the solution of the problem; and we cannot suppose that there is a more general expression, although otherwise the same integral may be put under very different forms. = we might give another form to the integral of the equation dv d'v and it would always be easy to deduce from it the dt da integral which belongs to the case of three dimensions. The result which we should obtain would necessarily be the same as the preceding. To give an example of this investigation we shall make use of the particular value which has aided us in forming the exponential integral. very simple value e-n't cos nx, which evidently satisfies the differential equation (b). In fact, we derive from it dv =- n2v dt belongs to the equation (b); for this value of v is formed of the sum of an infinity of particular values. Now, the integral dn e-n't cos nx 18 e 4t is known, and is known to be equivalent tot (see the following article). Hence this last function of x and t agrees also with the differential equation (b). It is besides very easy to verify e 4t satisfies the equation in directly that the particular value question. The same result will occur if we replace the variable x by x — a, a being any constant. We may then employ as a particular value the function Ae 4t the differential equation (b); for this sum is composed of an infinity of particular values of the same form, multiplied by Hence we can take as a value of v in the arbitrary constants. We see by this how the employment of the particular values 375. The relation in which these two particular values are to each other is discovered when we evaluate the integral1 To effect the integration, we might develope the factor cos nx and integrate with respect to n. We thus obtain a series which represents a known development; but the result may be derived more easily from the following analysis. The integral dn e-nt cos nx is transformed to fdpe-r'cos 2pu, by assuming n't = p2and nx = √t. [dp * cos 2pu = ↓ [dp e-p2+2pu√=1 + † fdp dp e-p2-: 2pu. ́p2 −2pu√−I+u2 = } e¬w3fdpe ̄(p¬u√=13 + ↓ e-w2 Сdp e¬6+uV=TY. Now each of the integrals which enter into these two terms is equal to √. We have in fact in general whatever be the constant b. We find then on making 1 The value is obtained by a different method in Todhunter's Integral Calculus, § 375. [A. F.] itself directly without its being necessary to deduce it from the However it may be, it is certain that the value en cos nx. 376. To pass to the case of three dimensions, it is sufficient functions, one of y and t, the other of z and t; the product will evidently satisfy the equation We shall take then for v the value thus expressed: If now we multiply the second member by da, dß, dy, and by any function whatever ƒ (α, ß, y) of the quantities a, ß, y, we find, on indicating the integration, a value of v formed of the sum of an infinity of particular values multiplied by arbitrary constants. It follows from this that the function v may be thus expressed: This equation contains the general integral of the proposed equation (A): the process which has led us to this integral ought |