48 REPORTS ON THE STATE OF SCIENOE.^ — 1919. 



Produce FP to meet the circle round FG, F'G' in Q. Then, since 

 FG2=2GP2=GP.GG', the circle round FPG' touches FG ; QF'G = 

 QFG=GG'F. G'GF, so that F'Q is parallel to GG', and PF'=PQ. 

 Then FP.PF=FP.FQ-GP.PG'=a2, if GP=OF=a, FG=F'G' 

 =a\/2, FF'=GG'=2a. This is the property of the lemniscate, leading 

 to the polar equation r^=2a' cos 20, with OP=r, AOP=e, FQF'=FGF' 

 = (/), FG sin ^=FF' sin 6, sm^<f>—2 sin^^, cos2</>=cos2^, 



r=0P=2PH cos ^=FG cos (^, r-=:2a2cosV=2a2cos2^. 



The rectification of the lemniscate may be considered to have 

 originated the true theory of the Elliptic Function in that it introduced 

 the First Elliptic Integral, inverse of the uniform Elliptic Function. 



The previous efforts at the rectification of the Ellipse, which gave 

 the name to the Elliptic Integral, were on the wrong track, as leading to 

 the Second Elliptic Integral, not the inverse of a uniform function. 



The lemniscate can be expressed in the vector form, in terms of a 

 parameter u, 



(15) x + iy—a seGh{u + ^iri), o-'^ = 2a^ sech 2u, for K'=^7r, k = 1, 

 degenerate case of the confocal Cassinians given by 



(16) x + iy=aGn{eK + ^K'i), or ^, dn{eK + Ui'i). 



Then 



(17) ch2tt=sec 26, sh2tt=tan 26, 

 th u =tan 6=cl(l— e)L. 



Important memoirs to consult on the Lemniscate Function are by 

 Kiepert, Crelle 75, 1873; Schwering, Crelle 107, 110; Mathews, 

 Proceedings London Mathematical Society 1896, 1915. 



Other forms of the lemniscate integral may be given, such as that 

 obtained from the Weierstrass integral with g^= —1, (73=0, and then 



with s=^z'^, 



ii_, ^=icr'l:^'=eL. 



V(l + ^^)' ' l+;j2 



(19) 



2_1 — cl2^; ^\v 



^ ~1 + q\2v' ^"^L-v)' 



dl2v=^^-^ + ^'A s\2v= ^ , tl2^;- -}- . 



