70 REPORTS ON THE STATE OF SCIENCE. — 1919. 



Writing any function A(r) as A(rK), to distinguish the period and 

 modulus, then with 



K_oL ,_ 1-A 

 K'~ L" ~'1+A' 



these formulas may be replaced by 



B(rK)2±A(rK)2= ^/VC(2rL), B(2rL), 



C(rK)2±D(rK)2 = ^^'c(2rL), , ^^^ B(2rL), 

 1 — A. 1 — A 



E(rK)-f-F(rK)= ^^ A(2^1^ 2E(2rL) 



with A=A' = - v/2 in the Lemniscate Table I. 



A second application of this Quadric Transformation will give the 

 numbers of Table III, where 



G=4G', y'=( yi~] Y=co8 89° 34' 

 ^ VV2 + 1/ 



-V 



so that the modular angle is more than half-way through the last degree 

 of the quadrant ; and to go further does not seem of practical utility, as 

 on to K=8K'. 



The geometry of these two Quadric Transformations is shown on the 

 ellipse, of excentricity k, drawn for semi-axes 



a=50(v/2-hl) = 120-21 mm, & = 50(v' 2-1) =20-21 mm. 



The Quadric Transformations. 



To show the relation geometrically, connecting the three Tables I, 

 If, III, corresponding to 



G ^ K ^ oL 

 2G' K' L" 



2n/»c , 1—k' 



1 + K 1 + K 



the ellipse is drawn with excentricity k, taking 

 ..=(V2-1F, .= ^A^. 



Then with <^=am/K the minor excentric angle of a point P on the 

 ellipse, and a)=am (1— /)K the angle AOY of the perpendicular OY 

 on the tangent at P, 0Y= a dn(l— /)K, and the coordinates of P are 

 a, h sn/K, b cn/K. 



The longitude of P, perihelion and aphelion, is 



ASP = 2 am } (1-/)G, A'S'P = 2am J (1+/)G, 



