of the hi) per elliptic functions 



229 



from the first two of these 



= x — + /j?y, 



dfM r , zx + y 2 + j\ 3 y 



^-y^p 2x+±\ 



and hence 



zx + y 2 + j\ 3 y 



substituting from the differential equations the expressions for 

 ^ 2221 and g?22n we fiud that there is a common factor 

 x (xz - y 2 ) + i^xy - \\y 2 



on both sides, this being that occurring on the left in the equation 

 of the quartic ; assuming this not zero we get 



jj? = 4a? + A 4 . 



Then dx = </p™du 2 + f^du^ = A 4 (%du 2 + ydi^) 



and 



give 

 and 



dy = /i 



^tSS** 



a?(iy — ydx = fidi^ 



x (zx - y 2 ) + jX 3 xy - ^X,y 2 

 2a? + A\ 4 



zx + y 2 + iX 3 y , T 7 x (zx — y 2 ) + iX 3 xy — ^A 4 ?/ 2 



sl+ii. dx ~ ydy = " du ' ^AA 



2a? + i\ 4 

 of which the left side is 



a? (#a? — y 2 ) + xX 3 a?y — -tX 4 w 2 7 w . 7 , . 



a;(2a? + ^\ 4 ) x y J J y ' 



hence from the fundamental quartic relation connecting x, y, z 



7 xdy — ydx , dx yxdy — ydx 



JQ " ^s/4^ + X 4 * ^Q 



where Q — X x i — X^y + X 2 x 2 y 2 — X 3 xy z + A, 4 y i , 



and we have put 



. — x (xz - y 2 ) + \X 3 scy - \\,y 2 



l x = sJ±x+X i , " 4*? + \ 4 -=%s!Q- 



Choosing a particular single-valued function to satisfy 



'd§ (w)y 



dw 



= \ — \ ± <fi + X.^ 2 — A, 3 (/> 3 + A, 4 c/> 4 , 



