236 Methods for the Solution of Special Problems [OH. vm 



275. We have now found that the most general rational integral surface 

 harmonic is of the form 



n 



S n = 2 Pn (/*) (A m cos m<f> + B m sin ra</>), 

 o 



in which P(p) is interpreted to mean P n (/z), when m = 0. 

 Let us denote any tesseral harmonics of the type 



P(//,)(^cosra< + 5smm<) by S%. 



Then by 237, Jj Sf S$ da> = 



if n = n'. If n = w', then 



JJ SJ 5?' = ||PJ (/*) P?' (/*) (J TO cos ro + B m sin m</>) 



(jd TO ' cos m'<f> + 5 m ' sin m'<f)) dco, 

 and this vanishes except when m = m. 



When n n' and m = m' the value of 1 1 S% S$ dco clearly depends on 



r+i 

 that of I {P% (/x)} 2 dfji, and this we now proceed to obtain. 



We have 



-a lt . 

 - ( M ' 



ip ^ f 



-.r"^ ..-(195). 



z 

 Since ^ n = 7J is a solution of equation (191), we obtain, on taking s = m + n 



in this equation, and multiplying throughout by (1 ft 2 )' 11 " 1 , 



+ (n + m)(n 

 which, again, may be written 



In equation (195) the first term on the right-hand vanishes, so that 



(n + m) (n - m + 1) j (P- (/n)) 2 d/t, 



