O'SuLi.iVAN — Points on the Curve of Intersection of two Quadrics. 153 



If %(,', v' be the original and transformed arguments in Jacobi's notation, 

 h, I the original and transformed moduli, 



vf = mu, Ic = —, v' = JiJ, - E3U =JEi - Ji.^u = {m + n) u 

 til/ 



, . ,s \E^ - Ei {m + n) dtc ,^ .,. sxu'cmi' 



( Pu - E3 't^uxpii "^ duu' 



which is Landen's first transformation, (v. Greenhill, p. 318.) 

 Since Fu = - Piu, PQ.z = - Pi'Qa = - Ei .: Pii^ = A',. 

 Similarly, PiQ^ = E,, FiUi = E,. 



Hence, if 20„ 2i2,, 2O3 be the periods of Pit, 



Qi = iili = ii03. Q,2 = iHi = ¥• (w3 - ot)z), Q3 = iQi = ^uo^. (6a) 



If §, i?, U, T are the coordinates of the connector passing through the 

 point on UV wliose argument is m, transformed by the substitution (3) 



Q .i + it .ZX+iYW .exp-im<j, . oiam< 

 " - " - t n7 ■ — 7 from §10 (IJ, 



1' 



9. 



-it 



R 



. r 



+ iu 



T 



1 



- it 



U 



r 



- iu 



ZX- iYW BxP + im<j>' 

 -^....-~r^.-A-.:rr^^.■ (7) 



XY-iZW _ jin Of - in-^ 

 ' ZX-iYW " V?T ■ ^ + im<f,' 



XY+ iZW . \m Ocj) + inij 



T q-it ZX-iFW \jn d4, + itn(j>' 



where 0, <p, \p are written for brevity for 0ir, <pu, ipu. 



Eegarding Q, R, U, T as the coordinates of a point on Tr whose argument 

 is U, these ratios can be written down at once from (3j by substituting v for 

 u, Id, V for miu, and Qv, $y, ^y for dii, ^u, •-//««, where 



Qv = JPv - El, ^v = JFv - E^, ^v=JPv- E,. 

 Thus, 



Q - iQv R ^v U - i'^v 



Equating these values 



6\p - im(j) 



(8) 



©<•• = - Jfiv . 



0ip + im(j) 



^>^ =Ji'(/J. + v). ^ r^, 9) 



, Q<p + intli m n + v 



'^'>' = JfJ-Ut + I') ai , ■ — 1> since — = 



