1 



Va 



( 461 ) 



1 



\/ a^—cy y a^ — Cj 

 l 1 



= 



1 1 1 



l/ai — Cg l/oo— Cg l/ög — Cs 



the arguments of the nine surds being detei'mined save as to a 

 multiple of jt. The symmetry between the a's and the c's shows, 

 that, as soon as there exists the involution we started with, a 

 similar involution can be found by interchanging the c's and the 

 a's, so that reducible integrals always present themselves in pairs. 

 BüRHSIDE stated incidentally that the curve 



„ / , 2K\/ „ 2A"\ 



3-y- = 0«— 1) {.,—m~ u) I a— .v/,- (« + -^) 1 \.t—sn- (u— -)) 



admits reducible integrals. The form given here to the invariant 

 relation between the six branch-points readily provides a proof for 

 this assertion. 



Tn order to obtain this proof we introduce elliptic ai'guments in- 

 stead of the a's and the c's by putting 



«1 = «1 , t^i = pui = p {ii + y,) , 



«2 = «2 , ''2 ^^^ /^"2 ^^^ P {"■ -\- I'e) 1 



«3 = eg , Cg = /;«3 = ^ (« + yg) . 



So the invariant relation becomes 



= 



As a function of u the determinant i^s doubly periodic ; manifestly 



