Now the integral due-wuTMm taken from u=0 to u = ∞ has a known value, m being any positive integer. We have in The preceding equation gives then, on making q'kt = u3, This equation is the same as the preceding when we suppose a=1. We see by this that integrals which we have obtained by different processes, lead to the same convergent series, and we arrive thus at two identical results, whatever be the value of x. We might, in this problem as in the preceding, compare the quantities of heat which, in a given instant, cross different sections of the heated prism, and the general expression of these quantities contains no sign of integration; but passing by these remarks, we shall terminate this section by the comparison of the different forms which we have given to the integral of the equation which represents the diffusion of heat in an infinite line. 370. To satisfy the equation du d'u da u=e-rekt, or in general u = en enkt, whence we deduce easily (Art. 364) the integral This equation holds whatever be the value of a. We may develope the first member; and by comparison of the terms we shall obtain the already known values of the integral (dq e-* q*. This value is nothing when n is odd, and we find when ʼn is an even number 2m, 371. We have employed previously as the integral of the u = a ̧e ̃n13kt cos n ̧x + α ̧e ̄ˆ‚22 cos n ̧x+α ̧e ̄¤ ̧2kt cos n ̧¤ + &c. ; or this, u = a,e-nkt sin nữ + agent sin ng+age-nk sinng +&c. a1, a, a, &c., and n1, n,, n ̧, &c., being two series of arbitrary constants. It is easy to see that each of these expressions is equivalent to the integral fdq e ̄v sin n (x+2q √kt), or - cos n (x+2q √kt). In fact, to determine the value of the integral [** dq e- sin (x + 2q √kt) ; the integral fdq e 1 2-1 1 Jdqe-(√) − e-(q±√k)2 taken from q = ∞ to q = ∞ is √π, we have therefore for the value of the integral (dqe-" sin (æ+2q √kt), the quantity √ e- sin x, and in general √π e-n2kt sin nx=dq e- sinn (x + 2q √kt), we could determine in the same manner the integral ** dq e- cos n (x + 2q √kt), the value of which is √ e-nt cos nx. We see by this that the integral e-n2kt (a, sin n,x + b, cos n,x) + e-nk (a, sin n ̧x + b, cos n ̧æ) is equivalent to += [ dq e- (a, sin n, (x + 2q √kt) + a, sin n, (x+2q √kt) + &c. ? 2 b, cos n, (x+2q √kt) + b, cos n ̧ (x + 2q √kt) + &c. S The value of the series represents, as we have seen previously, any function whatever of x+2q √kt; hence the general integral can be expressed thus Of the free movement of heat in an infinite solid. 372. The integral of the equation dv K d'v = dt CD d (a) furnishes immediately that of the equation with four variables dv K d'v d'v d'v = + + ...... ·(t), as we have already remarked in treating the question of the propagation of heat in a solid cube. For which reason it is sufficient in general to consider the effect of the diffusion in the linear case. When the dimensions of bodies are not infinite, the distribution of heat is continually disturbed by the passage from the solid medium to the elastic medium; or, to employ the expressions proper to analysis, the function which determines the temperature must not only satisfy the partial differential equation and the initial state, but is further subjected to conditions which depend on the form of the surface. In this case the integral has a form more difficult to ascertain, and we must examine the problem with very much more care in order to pass from the case of one linear co-ordinate to that of three orthogonal co-ordinates: but when the solid mass is not interrupted, no accidental condition opposes itself to the free diffusion of heat. Its movement is the same in all directions. 1 See an article by Sir W. Thomson, "On the Linear Motion of Heat," Part I, Camb. Math. Journal, Vol. 1. pp. 170-174. [A. F.] The variable temperature v of a point of an infinite line is expressed by the equation x denotes the distance between a fixed point 0, and the point m, whose temperature is equal to v after the lapse of a time t. We suppose that the heat cannot be dissipated through the external surface of the infinite bar, and that the initial state of the bar is expressed by the equation v=f(x). The differential equation, which the value of v must satisfy, is which assumes that we employ instead of t another unknown If in f (x), a function of x and constants, we substitute x + 2n √t dn for x, and if, after having multiplied by e-n2, we integrate with satisfies, as we have proved above, the differential equation (b); that is to say the expression has the property of giving the same value for the second fluxion with respect to x, and for the first fluxion with respect to t. From this it is evident that a function of three variables ƒ (x, y, z) will enjoy a like property, if we substitute for x, y, z the quantities x+2n√t, y + 2p√t, z+2q√t, provided we integrate after having multiplied by |