Expvesslna Solutions of Laplace's Equation. 197 



6. If we write 



j,„(p^-;[,)--..=iw,„(si„^^^)cos..^ 



=K"j,„(yr^^|^)/." . . . (13) 



if /ji = co^6 ; 

 where the meaning of the operator is that the Bessel's 

 function is expanded in powers as if — , or 



clfji' d{cosO)' 



(d \" 

 — J or 



i Yi ns I is replaced by -j-— or -j-. ^— and made to 



\i/(cos^)/ ^ - dfjb'' rt(cos^)" 



operate on the quantity to the right of the original operator. 

 Hence 



is a spherical harmonic of degree w and order ?», the S 

 stopping at r = v if n is an integer. 

 Hence 



must be a constant multiple of the Legendre^s function of 

 order m and degree n, 



7. To identify this constant multiple 



ttM J Vi^^ 7 W= 1 ^Ooo3uvi=;i^d;;i^ At" COS mudu. (14) 

 But 



n . 77 Jq 



where P^(/a) is the associated Legendre's coefficient of degree 

 n and order m. (See Whittaker, ' Modern Analysis/ p. 235, 

 or Todhunter, ' Functions of Laplace, Lame, and*^ Bessel.') 



