( 454 ) 



For in this sujipositiou the elliptic integral 



^1 

 Ri, 



Ri ' Ri' R^ 



is an integral of the iirst kind bolongino- to the eurvo, hecansc it 

 can assume the ibiin 



As for rerluoible non-hvperolliptic integrals the case p = 3, »• = 2 

 has been treated by Sophie Kowalevski. The curve ƒ is here the 

 general quartic upon which, »• being equal to 2, an involutory 

 correspondence one to one exists. It can be shown that in this case 

 the curve can be transformed into itself by a reciprocal projective 

 transformation of the plane. Consequently four double tangents of 

 the curve pass through the centre of the transformation , so that its 

 equation can always be thrown into the form 



ƒ = .7;V {ax + h/j) {ex + chj) — K^=0. 



Evidently the four double tangents passing through the origin 

 can be identified with the curves R. For each of these tangents 

 touches ƒ in two points, together they belong to a pencil and the 

 eight points of contact lie on the conic K. 



Accordingly we get for the elliptic integral 



pW f 1 pW f 2 pW fg 1 



y ax -j- hij ex -\- dij 



d c ex -j- d;/ 



and the integi'al itself is 



"■=/r"" 





