90 DEVELOPMENT OF A CERTAIN INFINITE DETERMINANT ARISING IN [2 

 Thus 



and 



;-tf)| + cot 2 (g + r7)^-2cot(g-?)|cot(g + 2)|j 



77 , ., , x 77 



2 + COS6C " (g + g) 2 



Hence 



- 2 cot </TT (oot(g-g) J -cot (g + g) I) I . 

 \ * ^/ J 



44 ~ - 8) , , TT 



i< / \ ** / , i \ " 7i V - 1 *"y u/u ~r u i . / \ ii , / \ \ 



(p,q)- *(p,p- = T^^rfciy^l) l COt (g ") " C (g + 5) 2 f ' 



the terms in cosec 2 (g </),-, + cosec 2 (g + q) - disappearing, as might have been 

 expected. 



Hence the determinant, to the fourth order in a, or the eighth order 

 in m, equated to zero gives 



, 77 gl 4 (l5( ? < -35(/ 2 +8) f ,, ,77 .- ,77 



+ 128 g 3 (g 2 -l) 3 4 2 -4)i COt(g " <?) 2" COt(g + g) 2 



77 2 g/ COt (?77 



64^ 2 (2 2 I) 2 

 But 



,/ \v , t \77 2 sin 077 



cot (g - q) - cot (g + o) - = - 



* 7 2 r ^ 7 2 COS ^77 - COS g77 ' 



