42 



In [113] replace s with s±l and multiply with [114] as before. We obtain, 



-t dP. 



f]pSJ-\ J pS I jps 



(l-,2)2 in^ ^£}i±l^^(l-^2)(„,l)/p. ^_f\ 



n+1 j 



-^2(„+ l)2ps±ips ^_(„_ s ^ i::pi)(^ _^g _^ j)p5±ip5 

 Substituting in / and reducing, we find that 



/„(., . + 1) = u - s) j' [(.. - s + 1) p„%i pr' - (n + « + 1) p^^tip^ '"^ 



2(71 + s + 1)! 



^/^^ 



(n - s - 1)! 



1118] 



/ (s, s - 1) = (ti + s + 1) 

 2(71 + s + D! 



I 



(^-^)p:.iPr'-^^-^-^)p:n-\ 



yrr 



== {n- s + 1)\ 

 using Equations [108], [109], [110], and [111], 



INTEGRALS J„(s, s±l) 



These integrals appeared in the expression for M. and M.^. They were defined as 



r^r r dP^—^ dP^ 



J„(s, s±l)= J+nP„^-^! (7i + l)p;±i_^ 



J -iL L dp. d(jL 



(l-"2) 



+ [7i(s±l) + s{n+ l)]P^P^-%'y 



yrr^ 



If we substitute for 



dP' 



dP, 



■■±\ 



(l-fx2)I^ and {l-^?)Zl^ 

 da a fi 



using [D9], this is immediately integrable, using [108], [109], [HI], and [112]. We find that 



(ti + s + 1)! 

 J (s, s + I) = 2 



(--!)! [119] 



(n + sY. 

 J is, s-1) = 2- 



(n - s)\ 



