TRANSACTIONS OF SECTION A. 269 



above is different in character to an enumeration extending from zero to a given 

 high number, but Professor Tchebycheff's result gives a special interest to sepa- 

 rate enumerations of 4w + 1 and 4n + 3 primes. 



10. Formula in Elliptic Functions. By J. W. L. Glaisher, M.A., F.B.S. 



The formulae in question, which give the products of three dn's or three sn's 

 in terms of the sn, en, dn's of the four arguments %(a + b + c), %( — a+b + c), 

 $(a-b + c), %(a + b-c), are 



j j r j &' 2 + & 2 cn s cn(s - a) cn(s - b) cn(s — c) 



dn a dn b dn c = - — ^ ± ^ — ^ — ^ — -> A 



1 + & 2 sn s sn(s - a) sn(s - o) sn(s -r c) . 



Fen « en 6 en c- -^ + d" '&>(«-«) dn( g -6) dn(.-e) 

 1 + Arsn s sn(s — a) sn(s — 6) sn(s — c) 



where s = £(« + 6 + e). 



Adding the two formulae, we have 



dn a dn b dn c + ft 2 cn a en 6 en c = 

 dn s dn(s— «) dn(«-6) dn(s-e) + & 2 cn s cn(s-a) cn(s-6) cn(s — e) 

 1 + & 2 sn s sn(s — a) sn(s — b) sn(s — c) 

 As a particular case let 



a = b = c = 2x, 

 and the formulae become 



, m k' 2 + & 2 cn3.r cn 3 a- 

 1 + A-snd.i' sn 3 .r 



Fcn^=-^ + dliardn3 ' r 



dn s 2.r + k~czx 3 2x = 



1 + Wsv&x sin 3 a- ' 

 dn3.r dn 3 .r + ft 2 cn3 .rcn 3 .r 



1 + A 2 sn3.r sin s x 



and to these may be added 



, , 2 _ dn3.r dn 3 .r - & 2 cn3.r cn 3 .r 



1 + A; 2 sn3:i' sn 3 .r 



The paper in which the above formulae occur will be communicated to the Cam- 

 bridge Philosophical Society. 



11. Summation of a class of Trigonometrical series. 

 By J. W. L. Glaisher, M.A., F.B.S. 



We have 



l+.r" = l — vox . 1— vPx , . . 1 — iv- n ~ x x 



whence 



where to >= cos - + i sin -, 

 n ' n 



l + ^» = l-(f.r) 2 . l-(fV) 2 . . . l-(f 2 "-V)' 3 



where C = cos ~ +i sin _ • 

 2n 2w 



Replacing x by x I cos ~ — i sin ~ 1, this becomes 

 l_u^ = l_( p .r) 2 . l-( P 5 .r) 2 . . . l-(p 4 "- 3 .i) 2 



where p = cos ^- +i&m ^-. 

 in An 



