OF TWO LEGENDRE'S COEFFICIENTS. 363 



= (n + m) (n + m - I ) (n - m + 2) (n - m + ] ) I <?~ 2 Qn^dp- 



i+m l)(n + m-2)...(>i + I)n... (n m+l) I P n . 



PnPndp. 



Hence if rt and 7^ are not equal, I $!'$!!', cfyi = 0. 

 But if = n 1; then 



(14) 



19. The position of a point on. the unit sphere may be determined by 

 the coordinates 



p., ( I - /r) 4 cos (/>, (1 -p>~)^ sin<; 



277-S/A is the surface of an elementary zone and therefore S<S/* is an element 

 of the surface at the point defined by \L and (f). 



Any rational and integral function of the coordinates of the point 

 may be expressed in terms of the form 



where If is a rational and integral function of p.. 



Let <7 = cosy = ^ 1 + (l -/r) 4 (1 - prf cos<f>, 



then if V=(l -2h cosy + A 2 )~ J , 



and if Q n be the coefficient of h n in its expansion so that Q n is the same 

 function of q that P n is of /t ; then since 



we have 



and similar relations for Q n to those which have been found above for P n . 



* Note. For ^ read p.' in Articles 19 22 of this Section. 



462 



