24 report — 1877. 



by Q and by <f>(x), a function of \/ (1—p 2 wo? x), 



[QPQ . . . QPQ] B( P , x)=F(p, .r) + sin x cos x <f>(x), 

 [PQP . . . PQP] F(p, x) = E(j), .r)+sin x cos a 00), 

 [PQP . . . QPQ] EQ>, x) =E(>, a)+sin a: cos x $0), 

 [QPQ . . . PQP] F(j>, x)=F(p, .r)+sin x cos # <£(>')• 

 Then, again, 



Va 2,1 2 \ d * i An i\ ^ ii~It?/ ^ l + (l—p 2 ) sin 2 x 



[ 4 ^ 1 -^^ +4 ^ 1 - 2 ^ f 7P)- 1 ] F ^ -) 

 and generally 



C,7« ,7»— 1 f ?n— 2 —i 



sin 2 "" 3 ^ cos .r l"" 3 ! 2 ... " . 2 , 



(l-^ 2 sin 2 z) n -* 2" 



L,7» rT" -1 rf 11- 2 — I 



• 2»-3 „„„«,,, 1»-3|2 



[(2«_3) + { (2n-3)(2n - 5) - (2»-l> s } sin 2 x 



(I-;? 2 sin 2 *)* - * 2 " ' 

 +2p 4 sin 4 a:]. 



If the modulus /> is imaginary =^ji, then 



pW(l-^sin 2 *) = V (W) { E (7^) " E (V(TW **"*) } 

 = V(l+/)E l= E 2 , 



i 



o V(l+F 



1 



^ 8 in 2 a:) -V(l+P 2 )l F W(lV)) F (v(lW ** V } 

 Fi-F. 



~V(i+p 2 ) 



E„ E 2 , F i; F 2 being notations introduced for the sake of convenience. Then 

 E 2 =1,(E 2 -F 2 ), 



d(p*)^-2p 

 cl -m ! T 1+(1+P 2 )sin 2 a; „* „ „ ~j 



giving the symbolic equations 



fl+y * >,- ^ r 1 +/ 1 1 +?. si ° 2 f P 2 sin^ coa *+E a 1. 



