574 Mr. stokes, on THE CRITICAL VALUES OF 



A and B may be easily found from these equations, and we shall have finally 



(e'' - e-"') Z =G («""-'' - e-9'' ■") + Hie"" - e"'-) + - (e''""'' - e"?''"'') f (e''' - £-'=) Tdz] 



+ - (e^-- - e-r-) f (ef'-^-'' - ^-'t^'-^') Tdz, 



T being the value of T when s = ar'. It will be observed that the letters Z, P, Q, T, A, B, G, H 

 ought properly to be affected with the double suffix mn. It would be useless to write down the 

 expression for u in terms of the known quantities /, (»/, z), &c. 



It will be observed that u might equally have been expressed by means of the double series 

 22 ^„ sin vy sm'ZB'x, or 22 F„ im /ix sin •ars?, where p is any integer. We should thus have 

 three different expressions for the same quantity u within the limits x = and a = a, y = and 

 y = 6, z = and « = c. The comparison of these three expressions when particular values are 

 assigned to the known functions /i(y, z) &c. would lead to remarkable transformations. The 

 expressions differ however in one respect which deserves notice. Their numerical values are the 

 same for values of the variables lying within the limits and a, and b, and c. The first 

 expression holds good for the extreme values of z, but fails for those of x and y : in other words, 

 in order to find from the series the value of u for the face considered, instead of first giving x or y 

 its extreme value and then summing, which would lead to a result zero, we should first have to 

 sum with respect to m or n, or conceive the summation performed, and then give x or y its extreme 

 value. The same remarks apply, mutatis mutandis, to the second and third expressions ; so that 

 the three expressions are not equivalent if we take in the extreme values of the variables. 



49- Many other remarkable transformations might be obtained from those already referred 

 to by differentiation and integration. We might for instance compare the three expressions which 



would be obtained for / / / udxdydz, and we should thus have three different expressions 



- "^0 •'o 



for the same function of the three independent variables a, b, c, which are supposed to be positive, 

 but may be of any magnitudes. Some examples of the results of transformations of this kind may 

 be seen by comparing the formulae obtained in the Supplement alluded to in Art. 44 with the 

 corresponding formula contained in the Memoir itself to which the Supplement has been added. 

 Such transformations, however, when separated from physical problems, are more curious than 

 useful. Nevertheless, it may be worth while to exhibit in its simplest shape the formula from 

 which they all flow, so long as we restrict ourselves to a function u satisfying the equation y?« = 0, 

 and expanded between the limits x = and .r = a, &c. in a double series of sines. 



The functions /, (y, z) &c., which are supposed known, are arbitrary, and enter into the 

 expression for u under the sign of double integration. Consequently we shall not lose generality, 

 so far as anything peculiar in the transformations is concerned, by considering only one element of 

 the integrals by which one of the functions is developed. Let then all the functions be zero 

 except f, ; and since in the process f^ has to be developed in the double series 



— 22 / / /a (■'-'', y ) sin fix sin vy dx dy . sin \i.x sin vy-, 

 ah Jg Jf, 



consider only the element f^ix', y) sin^''"' sin vy dx' dy of the double integral, omit the dx'dy, 

 and put/3(x', y) = 1 for the sake of simplicity. If we adopt the first expansion of u, and put 

 q- for ,u^ + /, we shall have 



4 

 Z = AU''^'-''' - e-?"^--'), uv _ e-v) J = gin fj,x' sin vy ; 



ab 



whence u = -7^2 ^^— - — ~ — sin /n.r;' sin ry' sin (u j sin vy {99). 



