THE SUMS OF PERIODIC SERIES. 573 



It is evident that u is the sum of three temperatures ?<j, u.^, tij, where m, satisfies the conditions 

 (95), and vanishes at the four remaining faces, and u^, u^ are related to the axes of y, z as Mj is 

 related to that of x, each of the quantities m,, u-,, M3 representing a possible permanent temperature. 



Now M3 may be expanded in a double series 22Z„„ sin sin — -, where Z„„ is a function 



a b 



of z which has to be determined. Let for shortness 



ottt utt p-TT 



=11, "T = "' — = ar ; 



a be 



then the substitution of the above value of u^ in the equation sju, = leads to the equation 



-d^ - 9-Z», = 0, 



where g^ = /u." + v', which gives Z,„„ = A„„e''- + B„,„e"''; and the constants A„„, B„,„ are easily 

 determined by the condition (97). We may find m, and u.^ in a similar manner, and the sum of 

 the results gives u. It is thus that such problems are usually solved. 



We may, however, expand m in a series of the form 22Z,„„ sin ixx sin vy, even though it does 

 not vanish for j; = and a? = o, and for y = and y = b; and the formulae proved in Section I. 

 enable us to make use of this expansion. 



Let then u = 22 Z sin nx sin vy, 



the suffixes of Z being omitted for the sake of simplicity. We have by the formula {B) 



^^ = 2 { - ,*^2Z sin vy + ~^ [/, - ( - I)"/-,]} sin ;u.r. 

 ax a 



Let /i(y, x) - ( - \)"'Fi{y, z) be expanded in the series 2Qsini/y by the formula (3), so that 

 Q will be a known function of z, m, and n. Then 



-r-; = 22 ) - fx'A + — Q Sin nx sin vy. 

 aar a 



The value of — may be expressed in a similar manner, and that of — — is found by direct 

 rfy^ - '^ dz' 



differentiation. We have thus, for the direct developement of yw, the double series 



[d'Z 21/ 2|a ■) 



22 {, „ - (m'' + /)Z + — - P+ — Q>sin ixx sin vy, 



*\ dz' b a ) 



where P is for x what Q is for y. The above series being the direct developement of V"! ^"^l V" 

 being equal to zero, each coefficient must be equal to zero, which gives 



d'Z 2v 2u 



-q-Z + --P+ -!^Q = (98) 



dz- I) a 



where q means the same as before. The integral of the equation (98) is 



Z = A(i' + Be-1' - - €-'' r e-'" Tdz + - e"''' [' e'- Tdx, 



2 7' denoting the sum of the last two terms of (98). It only remains to satisfy (97). If the 

 known functions /,(a', y), F,{x, y) be developed in tlie double series 22G sin ^x sin vy, 

 "^"2.11 sin iix sin vy, we shall have from (97) 



A + B = G, 



Ae''' + Be-"' - - e'" f' e-i'Tdz + -e-''' f' e^'Tdz = II. 

 Vol. VIII. Paiit V. lE 



