460 Colonel A. R. Clarke on the Potential of 



y^h^reel^lP-d'', e\ = c^~€?', e\^c^-]?', and P/, P/^ P/'^ 

 are what P2 becomes when a^ ^j y are respectively written 

 for yLt. 



We may write the next term thus, 



Q4 = A^* + B/ 4- C^^ + MxY + By ^2 + ^'z^x^ + W, 



where W includes all terms in which the exj)onents are odd, 

 which disappear in integration. But Q^- must satisfy the dif- 

 ferential equation 



rf^ + ^+^-" ^^^ 



Applying this to Q^, and remembering that the equation is 

 identically true, we get these equations: — 

 6A + A^ + C^ = 0, 

 6B + A^ + B' = 0, 



6C + B^+C' = 0, 

 whence 



A' + 3( A4B~C) = 0, 



B^ + 3(~A + B + C) = 0, 



C' + 3( A-B + C) = 0. 



On integrating the expression for Q4, after substituting these 

 values of A^, B^, C^, it is to be observed that 



^x^dm= ^ Ma^ ; J xydm= ^ Ma'P, 



with corresponding values for the other integrals. Also 

 A, B, C are equal respectively to P/, P/^, P/^^, where the 

 accents have the meaning already explained. Thus we get 



jQ,d« 5^^ {-P/elel + P/'.?4 + P/"^.?}. . (4) 



For the next term put Qg = Ii<7^ + Iso-y + Igo-^ + I^p^ ; and 

 then, expanding, we get 



Qg = Ax' + Bf + C^^ + A^xY + A^xY + Bjy V + Bgy^e^ 



+ Ci^V + C20V + E^y^2 



plus terms involving odd powers of x, y, z, which disappear in 

 integration. On substituting Qg in the differential equation 

 (3), which it has to satisfy, the following relations are found 

 amongst the coefficients with which we are concerned : — 



15A + A2 + Ci=0, 6Ai + 6A2 + E = 0, 



15B + Ai + B2=0, 6Bi + 6B2 + E = 0, 



15C4-C2+Bi = 0, • 6Ci + 6C2 + E = 0. 



