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 directly that the particular value question. e 4t satisfies the equation in 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 (x − a)2 Ae whatever. Consequently the sum 4t also satisfies the differential equation (b); for this sum is composed of an infinity of particular values of the same form, multiplied by arbitrary constants. Hence we can take as a value of v in the dv d'v the following, equation dt dx2 A being a constant coefficient. If in the last integral we suppose (x − x)2 4t 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 na 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-n't cos nx Jan is transformed to fdpe-r'cos 2pu, by assuming n't = p2and në = 2pu. -2pu√-1+u2 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 value e-n't cos nx. However it may be, it is certain that the 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 (a, ß, y) of the quantities a, B, 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 to be remarked since it is applicable to a great variety of cases; it is useful chiefly when the integral must satisfy conditions relative to the surface. If we examine it attentively we perceive that the transformations which it requires are all indicated by the physical nature of the problem. We can also, in equation (j), change the variables. By taking (α-x)3 (B − y)2 = p2, 4t = n3, 4t we have, on multiplying the second member by a constant coefficient 4, v = 2oA fan 1 fån fåp fåq e ̄(w2 + p32+ 4) ƒ (x + 2n√i, y+2p√i, z+2q√i). Taking the three integrals between the limits and +∞, and making t=0 in order to ascertain the initial state, we find v = 23 Aπ 2ƒ (x, y, z). Thus, if we represent the known initial temperatures by F (x, y, z), and give to the constant A the value 8 T2, we arrive at the integral V = π 2 81 "dn [* "dp[" "dqe ̄"'e-v'e- &* F(x+2n √i, y+2p√i, z+2q√i), which is the same as that of Article 372. The integral of equation (A) may be put under several other forms, from which that is to be chosen which suits best the problem which it is proposed to solve. It must be observed in general, in these researches, that two functions (x, y, z, t) are the same when they each satisfy the differential equation (A), and when they are equal for a definite value of the time. It follows from this principle that integrals, which are reduced, when in them we make t=0, to the same arbitrary function F(x, y, z), all have the same degree of generality; they are necessarily identical. The second member of the differential equation (a) was K multiplied by and in equation (b) we supposed this coefficient CD' equal to unity. To restore this quantity, it is sufficient to write Kt instead of t, in the integral () or in the integral (). We shall now indicate some of the results which follow from these equations. CD 377. The function which serves as the exponent of the number e* can only represent an absolute number, which follows from the general principles of analysis, as we have proved explicitly in Chapter II., section IX. If in this exponent we replace Kt the unknown t by we see that the dimensions of K, C, D and t, CD' with reference to unit of length, being – 1, 0, -3, and 0, the is 2 the same as that of each term of the numerator, so that the whole dimension of the exponent is 0. Let us consider the case in which the value of t increases more and more; and to simplify this examination let us employ first the equation which represents the diffusion of heat in an infinite line. Suppose the initial heat to be contained in a given portion of the line, from x=-h to x=+g, and that we assign to x a definite value X, which fixes the position of a certain point m of that line. If the a2 4t time t increase without limit, the terms and +2aX which enter into the exponent will become smaller and smaller absolute 2ax a2 numlers, so that in the product et este at we the two last factors which sensibly coincide with unity. can omit We thus This is the expression of the variable state of the line after a very long time; it applies to all parts of the line which are less. distant from the origin than the point m. The definite integral |