46 REPORTS ON THE STATE OF SCIENCE. — 1919. 



5 



/ S ] '~~ S-i • So S - /Si — St) a S ~^'S'i - , / S \ tSo 



V Sj— S;, . s, — s \/ Sq — S3 . s, — s V Si — s 



s 



.K=J „ = 



sa-./^-^=cn-. /^-i^ =dn-,. /^i:^, 



V «:'— S3 V S2 — S3 \/ S,-S2 



\^ So— S \/ So — .V V^ S,-S:j.S3 — S 



V s^—s \/ Si— s V Si— s 



In the special lemniscate case, where (73=0, ^2=1) 



k2 = /c'2 = |, S =i, S2=0, .S3 = -i, n/(s,-S3) = 1, K = K' = L. 



The name arose historically in the rectification of the lemniscate 



(8) r2=2a2 cos 26, 



(9) '^=_ tan 29, ^l'=r2 sec''' 2d=2a' sec 2^, 



e 



(10) ^= f__^l-- . for the arc AP=s. 



^ ' a Jv/(icos29) 







Then putting cos 2^=cos^<^, 



'"' H:y (4 .-itw*)=l7(n|iro'.)=^<*' ^'° *^°»='^' 



and writing cl for en, . . . ., to represent the lemniscate function, 



(12) cos <&=cl eL=cl -, cos 29=cPeL, cos 6=dl eL, 



tan 0=cl(l— e)L ; 



and so the en, cl function is the first in importance compared with sn, 

 dn, or si, dl. 



