307 

 APPENDIX. 

 1. It is shown in Maxweij/s' Treatise that 



,^('n) ( — )" , 7,1 1 



Yc =- Lrn-^lJ), _ 



Now 



and 



,m 2n f 



n 2"+»« (n /)* " 



2n/ 



Hence 



2" {n~m) ! n ! " 



2 P„ {cos &) cos m <P ( — )" 2'» m I 

 It is also well known that 



These two equations may be combined in : 



cos m Pn {cos {}) _ {-—)» 2'« b,n ' V M 



where 



6, = 1; 6, = 6, = ... = !. 

 2. To show that: 



^m r r'P," {cos d^)cosix<P \ _ (— )"' {v + n) ! r' -« i^!i'"(co5^5') cos (m -f ,u) 

 " V ^/^ j ~ "^^^^ {v^^-n\rri)lK,^^ ^^^^ 



We consider the following cases: 



(/). ft =r m = 



We must show that 



^ (rv Pv (co« d)) = P,_„ {cos ^). 



Oz" {v — n)! 



Proof. Using Laplace's integral 



Pv{cos tV) = — I (cos >'J 4- I stn <5' . cos <pyd^ 







we have r^Pv(cos i9^) = — i {z -j- i q cos <PY <p where q = V x^ -^ y^ , 



