331 



we get 





and hence 



dt 



dx 



V{t-k,){t-k,){t-k,){t^h) V^^.y^y]^, 



X 



iR,R,'-R,'R,) 



Now the sextic (R.R^' — R^'R^) is evidently divisible by the quintic 

 -'^^i^jSi' therefore the given integral is reducible, if we take 



^ _ {R^Ri'—Rj' Rj) 



Thus it is seen that the reducibility of the integral in the first 

 instance depends upon the existence of the involution J, and on the 

 possibility of determining the quadrics ^i,|,,^,. The investigation of 

 the involution ./ and of its characteristic properties may be conducted 

 as follows. 



With three binary quadrics 



six common invariants are associated. As such we get in the first 

 place the discriminants 



and further the harmonic invariants 



^.=(«.'«;' + «o"«;-2a/a/') , A,,={a,"a,+a,a,"-2a,'\), 



The three quadrics themselves are connected by the identical relation 



-^11 -^IS ^81 ^1 



= 0, 



which T write in the form 



K=a^,^' + bi\,,' + cxp,' + 2fi^,^xp, -1- 2gx^^jp^ + 2A U^if^, = . 

 Now this relation between binary quadrics can also be conceived 

 as the equation in trilinear coordinates xp^, if?,, \p^ of a conic K, 

 and since each of the coordinates is a quadric in x, this variable 

 procures a parametric representation of the curve. From the same 

 point of view any homogeneous polynomial F{\p^ t|'s V^i) on the 



22* 



