ON THE CALCULATION OF MATHEMATICAL TABLES. 45 



The Lemniscate Function. 



1. This is the name given to the elliptic function when the modular 

 angle 61=45°, and K=K'=L. 



It arises in the Weierstrass form when (/:!=0 in S = 4.s''-;7vS-(7:, ; 

 and then taking ^7.2=1. -^=1' W|=o)3=L, and the period parallelogram is 

 a square ; and S\=\, So=0, §3=—^. 



Some writers prefer s, = l, So=0, 83= — !, g-.^^, A = 64, but this has 

 the disadvantage of making 



o,,=o,, = --'^ =1-B1102877714605987 . . . 



which is Stirling's A, given in Halphen's Functions elliptiques, I., p. 64. 

 But with (/2=1> 5^3=0. S=4s=»—s, 



00,0 i, -i 



ds f ds 



= 1.8540740773, 



the number employed by Legendre, Jacobi, and all subsequent writers. 

 In the general case, with S resolved into real factors, 

 (2) S=4s3— 5r2S-r73=4 . s—s^ . s—s., . S-S3, s,>s.2>S3, 



,„, , so-s, ,, s,-s, f s/{s,-s,)ds r ^{s,-s,)ds 



A" I , S^ 



and the first elliptic integral will be expressible by the inverse elliptic 

 function of Jacobi, in one of the forms 



CO 



(4) oo>s>S|,cK= -^g = 



(l-e)K=J 



S—S2 

 S-S3 



V Si-s,, V S1-S2 V Si -S3 



s 



(1-/)K'=| „ 



.So 



V Si— S2.S — S3 V Si— Sj.S— S3 V S — S3 



