576 Mr. stokes, on THE CRITICAL VALUES OF 



case of the parallelepiped, the shortest way would be thence to deduce the result for the case of the 

 bar infinite in one or in both directions, but if we began with considering the bar it would be best 

 to start with the integrals (50) or (65). 



50. To give one example of transformations of this kind, let us suppose b to become infinite 



in (101). Observing that v = — — , Ai/ = — , we get on passing to the limit 



-/ Zi rr^ — sin i/y sin vjrdw = - 2 (e"'' -€"•'*') £"''y sin ua^. ...(103). 



TT ^D € - 6 a 



Multiply both sides of this equation by d.vdy, and integrate from x = to x = a, and from y = 

 to y = 05 . With respect to the integration of the second side, it is only necessary to remark that 

 when y becomes greater than y, y and y' must be made to change places in the expression written 

 down in (103). As to the integration of the first side, if we first integrate from y = to y = V, 

 we get, putting f(v, -v) for the fraction involving a?. 



/ /(i/, x:) sin vy' (I - cos v Y) 



du 



Now let }' become infinite ; then the term involving cos i- Y may be omitted, not because cos v Y 

 vanishes when Y becomes infinite, which is not true, but because, as may be rigorously proved, 

 the integral in which it occurs vanishes when Y becomes infinite. If we write 1 for a, as we mav 

 without loss of generality, we get finally 



J/t rO ] g"" dv 2 1 

 risin./y'— = -2„-(i - f-"*) (104). 

 1 + e ' v' TT rf 



51. Hitherto in satisfying the general equation yw =0, together with the particular conditions 

 at the surface, the value of u has been expanded in a double series involving two of the variables, 

 and the functions of the third variable which enter as coefficients into the double series have been 

 determined by an ordinary differential equation such as (98). We might however expand u in 

 a triple series, and thus satisfy at the same time the equation yi/ = and the conditions at the 

 surface, without using an ordinary differential equation at all, but simply by means of the terms 

 introduced into the series by differentiation, which are given by the formulae at the beginning of 

 this Section ; and then by summing the triple series once, which may be done in any one of three 

 ways, we should arrive at the same results as if we had employed in succession three double 

 series, involving circular functions of x and y, y and z, z and x respectively, and the corresponding 

 ordinary differential equations. I am indebted for this method to my friend Prof. William 

 Thomson, to whom I showed the method given in Art. 4-8. 



Let us take the case of the permanent state of temperature in a rectangular parallelepiped, 

 supposing the temperature at the several points of the surface known. For more simplicity 

 suppose the temperature zero at the surface, except infinitely close to the point {x', y) in the face 

 for which x; = 0, so that all the functions/, &c. are zero, except f.^ix, y), and/3(,r, y) itself zero 

 except for values of x, y infinitely close to x, y respectively ; and let j jfzix, y) dxdy = 1, provided 

 the limits of integration include the values x = x, y = y. Let u be expanded in the triple series 



'S.'2.'S,A,„„j, sin iix sin vy sin -i^z, (105), 



where n, v, -w mean the .same as in Art. 48. Then 



(fu 2 TS 



— = 2p { - 2„2„7ir''.r<„„j, sin ,j.x sin py + - fz(a!,y)\ sin mm (106). 



4 

 Now the expansion oi f:i(x, y) in a double series is — 22 sin/nJ?' sin vy' sin fix sin vy, that is to 



ah 



