486 



Mr. E. H. Glaisher. 



§ 4. If A; be put equal to unity the cn and dn become identical, and 

 the formulae, as is well known, assume the simple forms 



snnu=± — ! — I ) '— (k=zl), 



(l+a>J"+(l-9)" V ; ' 



cnnu=— L -f — (& = 1). 



(1 +»)»+ (J. -a?)r v J 



Putting n = 8 in these formulae, we have 



su 8n= i — ! 1 L — , 



1 + 28^ + 70^ + 28^ + ^' 



cn 8u= ' 



1 + 28a; 2 + 70a? 4 + 28a? 6 + a? 8 



When h is put equal to 1 the formulae in § 3 should reduce to these 

 expressions : and we thus obtain an important verification of their 

 accuracy. 



Since the denominator of sn 8u, cn 8u, dn 8 26 is of the order 64 in x, 

 it is evident that a factor of the order 56 is common to the numerator 

 and denominator of these expressions when h is put equal to unity. 

 By putting h = \ in the formulae of § 3 and dividing the resulting ex- 

 pression for the numerator of sn8w by 1 + 7aP + 7x* + x 6 it is found 

 that this factor is (1 — ar) 28 , as it should be. And it was verified by 

 division that the expressions for the numerator of cn 8u and the 

 common denominator were equal to this factor multiplied by 1 — 4sa$ 

 + 6a; 4 -4a; 6 + a,' 8 and l + 28.r 2 + 70^ + 28a3 6 + ^ 8 respectively. 



§ 5. The expressions P 2 , S 2 , P 4 , S 4 are respectively the numerators 

 and denominators of sn 2 u and of sn 4 ic, and it seems worth while 

 to place on record their values, which are as follows : 



