pressed; but it was useful to notice this conformity in a new problem, which had not yet been submitted to analysis. 332. Suppose the half side of the square which serves as the base of the prism to be very long, and that we wish to ascertain the law according to which the temperatures, at the different points of the axis decrease; we must give to y and z nul values in the general equation, and to l a very great value. Now the construction shews in this case that the first value of e is the second 2 3πT 5п the third &c. Let us make these substitutions in the general 2 2 π equation, and replace nl, nl, nl, nl, &c. by their values 5п 7 п and also substitute the fraction a for e72; we then find We see by this result that the temperature at different points of the axis decreases rapidly according as their distance from the origin increases. If then we placed on a support heated and maintained at a permanent temperature, a prism of infinite height, having as base a square whose half side l is very great; heat would be propagated through the interior of the prism, and would be dissipated at the surface into the surrounding air which is supposed to be at temperature 0. When the solid had arrived at a fixed state, the points of the axis would have very unequal temperatures, and at a height equal to half the side of the base the temperature of the hottest point would be less than one fifth part of the temperature of the base. CHAPTER VIII. OF THE MOVEMENT OF HEAT IN A SOLID CUBE. 333. IT still remains for us to make use of the equation which represents the movement of heat in a solid cube exposed to the action of the air (Chapter II., Section v.). Assuming, in the first place, for v the very simple value et cos nx cos py cos qz, if we substitute it in the proposed equation, we have the equation of condition m = k (n2 + p2 + q), the letter k denoting the K coefficient It follows from this that if we substitute for CD n, p, q any quantities whatever, and take for m the quantity k (n2+p2+q2), the preceding value of v will always satisfy the partial differential equation. We have therefore the equation v = e−k (n2 +p2 +q3) ¢ cos nx cos py cos qz. The nature of the problem requires also that if a changes sign, and if y and z remain the same, the function should not change; and that this should also hold with respect to y or z: now the value of v evidently satisfies these conditions. 334. To express the state of the surface, we must employ the following equations: These ought to be satisfied when x = ±a, or y = ± a, or z = ± a. The centre of the cube is taken to be the origin of co-ordinates: and the side is denoted by a. The first of the equations (b) gives an equation which must hold when x = ±a. It follows from this that we cannot take any value whatever for n, but that this quantity must satisfy the condition h na tan na=a. We must therefore solve the definite equation h a Now the e tan e=a, which gives the value of e, and take n = equation in e has an infinity of real roots; hence we can find for n an infinity of different values. We can ascertain in the same manner the values which may be given to p and to q; they are all represented by the construction which was employed in the preceding problem (Art. 321). Denoting these roots by n1, n, n ̧, &c.; we can then give to v the particular value expressed by the equation v = e−kt (n2+p2+q1) cos nx cos py cos qz, provided we substitute for n one of the roots n,, n,, n,, &c., and select Ρ and q in the same manner. 335. We can thus form an infinity of particular values of v, and it evident that the sum of several of these values will also satisfy the differential equation (a), and the definite equations (b). In order to give to v the general form which the problem requires, we may unite an indefinite number of terms similar to the term - kt (n2+p2+q2) cos nx cos py cos qz. ae The value of v may be expressed by the following equation: The second member is formed of the product of the three factors written in the three horizontal lines, and the quantities a1, a, a,, &c. are unknown coefficients. Now, according to the hypothesis, if t be made = 0, the temperature must be the same at all points of the cube. We must therefore determine a,, a, a,, &c., so that the value of v may be constant, whatever be the values of x, y, and z, provided that each of these values is included between a anda. Denoting by 1 the initial temperature at all points of the solid, we shall write down the equations (Art. 323) 1 = a, cos n1x+a, cos n,x+a, cos n ̧x + &c., 3 1 = c1 cos n1z + c, cos n ̧2 + c2 cos n ̧2 + &c., in which it is required to determine a,, a,, a,, &c. After multiplying each member of the first equation by cos nx, integrate from x=0 to x=a: it follows then from the analysis formerly employed (Art. 324) that we have the equation Denoting by μ, the quantity (1+i), we have με sin 2n,a This equation holds always when we give to x a value included between a and a. From it we conclude the general value of v, which is given by the following equation 336. The expression for v is therefore formed of three similar functions, one of x, the other of y, and the third of z, which is easily verified directly. we suppose v=XYZ; denoting by X a function of x and t, by Y a function of y and t, and by Z a function of z and t, we have or + + = k + 1dX 1 ď1Y 1dZ X dt Y dt Z dt X de Y dy Z dz which implies the three separate equations dX dx dy ďY dz ᏑᏃ = k = k = k dt da dt dy' dt dz We must also have as conditions relative to the surface, It follows from this, that, to solve the problem completely, it is du h equation of condition dx+ ku=0, which must hold when x=a. We must then put in the place of a, either y or z, and we shall have the three functions X, Y, Z, whose product is the general value of v. Thus the problem proposed is solved as follows: v = $ (x, t) $ (y, t) $ (z, t) ; |