246-249] 



Spherical Harmonics 



215 



If P is the point whose polar coordinates are a, and 

 Q is the point r, 0, then /(a)=p7\- The Cartesian co- 

 ordinates of P may be taken to be a, 0, ; let those of Q be p 

 x, y, z. Then f(a) - , so that as regards 



differentiation of /(a), 



_ 

 da 



FIG. 74. 



Thus 



_ 



~^ 



so that equation (156) becomes 



,, , i 



and on comparison with expansion (153), we see that 



giving the form for P n which we have already found to exist in 245. 



249. A more convenient form for P n can be obtained as follows. 



Let 1 hy = (1 2A/U- 4- h z )^ (15*0 



so that y = /ji + h ^ (158). 



From this relation we can expand y by Lagrange's Theorem (cf. Edwards, 

 Differential Calculus, 517) in the form 



Differentiating with respect to /z, 



From equation (157), however, we find 



Equating the coefficients of h n in the two expansions, we find 



.(159). 



