826 Mr Glaisher, On theorems in elementary trigonometry. [Feb. 9, 



cos ^{A + D) 



sml(a+h+c—d)sml(a+h—c+d)s'ml{a—b^c+d)iiml{h+c+d—a) 

 cos ^ a cos ^ b cos ^ c cos ^ d ' 



whence by squaring and adding we have 



n (cos ^a) =11 (cos ^ a") + U (sin | a'), 



which on changing the sign of one of the letters a, h, c, d gives the 

 equation (C)*. 



§ 10. I obtained the results (A), ... {E) by deducing them 

 from the elliptic function theorems 



^''^ + ¥k'^ sn a sn 6 sn c sn cZ = If {k"^ + ¥k"^ sn a sn h' sn c sn d'), 

 k"^ — k^ en a en b en c en d = M [k"^ — k"^ en a en b' en c en d'), 

 k"^ + dn(xdn6 dncdncZ = ii/(/i;'^+ dna'dn6'dnc'dnd'), 

 k'^k'' sn a sn 6 sn c sn cZ — F en a en 6 en c en c? 



= if (^-'^ - dn a dn 6' dn c dn c?'), 

 A;"A;'^ sn a sn 6 sn c sn cZ + dn a dn & dn c dn d 



— M [k'^ + k^ en a en b' en c' en d'), 

 k^k'^ sn a sn 6 sn c sn c? — k'^ 



= iIi'(Fcna'cn6'cnc'cnt?'- dna'dn6'dnc'dnc^'), 

 where a', b' , c', d' are as in § 1 and 



\l-k''&n'\{a'-\-b')snma'-b')]\\-kHnm^c+d')^nnAc-d')] 

 ~ [1 - ^■'sn'|(« + ^} sa'i (a - b)] [1 - A;^sn'i (c +(Z) sn' i (c - ^ )} ' 



Expanding the expressions in these formulse up to powers of k"^ 

 inclusive, we have 



1 - F + Fn (sin a) = [1 + MJ<?] [1 - F+ FH (sin a')}, 



1 - F - Fn (cos a) = {1 +i)/oF} {1 - F - F n (cos a')], 



2 - k' - 1 ^"' S sin' a =[1 + MJc'] [2 - F - P' 2 sin' a!}, 



Fn (sin a) - FH (cos a) = (1 +M^k''] {- F + ^Ft sin' a'}, 



Fn(sina)+l-iF2sinV= {1 +lfoF} (l-F+FH (cos a')}. 



Fn(sina)-1+F = {1 + MJ^] (FH (cos a')-l + JF2 sin' a'}, 



where M= 1 +MJ<f up' to powers of F inclusive. 



* By a misprint in Lexell's paper, in the expression for sin J (4 +D) the square- 

 root sign only applies to the nnmerator; and Buzengeiger, who quotes the two 

 formula from Lexell in Vol. vi. (1818) of Lindenau and Bohnenberger's Zeitschrift 

 fiir Astronomic, p. 327, accidentally omits the square root sign from both the de- 

 nominators. 



