( 465 ) 



are in involution. This will evidently take place when ' fj and 

 cg are two roots of the covariant i A* -\- 2j a^'^, where «v* is the 

 biquadratic which has Tl^ and E^ as first polars. The quantities 

 hi and Z»2 are two roots of A'*.; if we call the other two Is and Z>4, 



^y , (x—bs){a'—bi)d.v 



=ƒ 



]/ azaj . a, uj . i- ({ ^* -\-2 j a:,'^) 



(.r— Ci) (.-r— fa) 



is a reducible integral, 



As an example we may point to the following case 



cij z= ki"- j-^' + 4 Z'l xi + 6 .r2 + 4 hx + k^^, 



i = 2 (^-1 L,-l) {ki k, - 3), i = - G (^-1 /.-, - 1)2, 



L\, = 2 (/.-i i', - 1) (2 k, r' + (ki k, + 3) x^ + 2 i-, .■) , 



o-.- a J — /l-jS ;(^^ + 3 ^-1 ^'- + 3 j'+k., at «..^ = /l-j a^+S ;p24-3 /f-, j'+V. 

 ,-^4 + 2./ «.V* = Q (3 ,r + k^) {x /■! + 3) {ki x^ f (3 - Aj k,) u: + ^-j). 



The Hessian A'* has two special roots ^3 = 0, bi = 00, the cor- 

 responding quantities c^ and c^ arc given by 3:c + /t2 = 0, j;i-i -f 3^0, 

 cj and f2 are the roots of ^1 x' + (3 — k^ kc^) ar -)- ^-^ =■ and the 

 resulting reducible integral is 



X dx 



W: 





2a;!*+3/;ia'+3.r+^-2)(^iap»+3A'^+3V+V)('fci'?H(3-i-,yfc2)^+/t2), 



where under the radical sign we may replace each of the two cubic 

 factors by any one of their linear combinations. The reduction of 

 the integral will be effected by the substitution 



ApW + B _ ^ _ *3 ^-1 + 3 :»2 

 CpW + D~^~ Zx-\-kci ' 



The case p =: 4, »• =z 3 evidently allows quite a similar treatment. 

 Seven branch-points of the integral can be assigned arbitrarily; the 

 triple condition determining the three remaining branch-points is 

 again readily obtained by the consideration of the cubic involution. 



32 



Proceedings Royal Acad. Amsterdam. Vol. I. 



