275, 276] Spherical Harmonics 237 



a reduction formula from which we readily obtain 



These results enable us to find any integral of the type 1 1 S n S' n da>. 



Biaxal Harmonics. 



276. It is often convenient to be able to express zonal harmonics 

 referred to one axis in terms of harmonics referred to other axes i.e. to be 

 able to change the axes of reference of zonal harmonics. 



Let P n be a harmonic having OP as axis. At Q the value of this 

 is P n (cos 7), where 7 is the angle PQ, and our problem is to express 

 this harmonic of order n as a sum of zonal and tesseral harmonics referred to 

 other axes. With reference to these axes, let the coordinates of Q be 6, <, let 

 those of P be 0, 3>, and let us assume a series of the type 



P n (cos 7) = V P s n (cos 0) (A s cos s(j> + B s sin 8$). 



5 = 



Let us multiply by P s n (cos 6) cos scf) and integrate over the surface of a unit 

 sphere. We obtain 



jl P n (cos 7) {P s n (cos 0) cos S(j>} da) = A s jj [P s n (cos 0)} 2 cos 2 s< da. 

 By equation (169), 



P n (cos 7) {P s n (cos 6) cos sc/>} da> = ^-^ {P s n (cos 6) cos s(} v=0 



and ft {P s n (cos 0)} 2 cos 2 s< dto = I + * }P^ (p)} 2 dp f 



Thus 



and similarly 



This analysis needs modification when 5 = 0, but it is readily found that 



A = P n (cos), = 0, 

 so that 



P n (cos 7) = P n (cos 6) P n (cos 0) + S 2 ^^ P^ (cos 0) P^ (cos 6) cos ((/, - <E 



s=l (^ T S) 



....(196). 



