( 810 ) 



/% 00 /-• 00 



I c-^ Sm Sn dx z= I i^j,n Sn dx = Q , m <^ n 



and 



r* 00 



An ■= Y 1 ?< ^Sn cZ.i 

 



^y 



liere : 



/CO /^ 00 



e-^' *S„ 6',, f/.c = I tf?„ .V" di 



but : 



Xpn Sn d.V = — {\pn S„) + 2 tf>„-— d:V 



J J dx 







or, as the last integral vanishes according to the conditions : 



because, by (16), only the last term has to be taken into account. 

 The expression for An then becomes : 



Hn n ft„_i w(n— 1) fi„_2 ( — 1)" 



^"~n/w/ l/w/(n— 1)/"^ 



(n— 1)/ ' 2/ n/(n— 2)/ 



. (17) 



by which the problem is solved. 



The application to special cases will be simplified by a brief sum- 

 mary of the relations existing between the different quantities intro- 

 duced which are analogous to those holding for zonal harmonics. 



We remark that for Sn and if?„ we can also write: 



/ d \i.'^) d" 



^„ = (-1)" --1 xn , ^, = {-l)n (e-^.^n) . (18) 



\dx J dx^ 



hence : 



it' dSn ^ , -, dSn 1 



«Sn = — nSn—\ H — and o„ =: {x—n) Sn—\ — x — - — 



n dx ax 



from which the recurrent formula : 



^„4.1 + (2n + 1 - x) Sn + n' Sn-i = , . . . (19) 



can be derived, wherein for Sn as well i|'„ may be written. 

 Further the functions satisfy the diff. equ. : 

 d Sn dSn 



dx^ dx 



0,- -^ + (l+.f) -^ + (n-f-l) V'« = 0. 



