( 450 ) 



and ('4, associated with tlie root /'.i.=oo of the Hessian Aj, is deter- 

 mined by 3 «4 X -\- a,. = 0. 

 Ilenee the iutca-ral 



J 1/ (A,x^ + 'óA.x+A^)(i 



{A,xi + 'óA^x+ ^3)CCo .r^ + a C'l 0;=^ + 63) 



proves to be reducible under the condition that tlie coefficients 

 A and C ob(>y the relation 



AoC,= A A, C, + 12 A, C,. 



The formulae of transformation will be found to be 

 2)W—Si 'pTF— fj pW—f.^ 



(>, (.*■ — Cj) (x—bi) ^ (y, (■'•— Co) (* — ^-o) 2 (.^3 (.'■ — (■3) (-f— /^s) ^ 



1 



^„*3 + 3^3... + yl3' 

 from which we can also d(>duce 



kp]V-^^> = 



/lo.i'3 + 3/l,.r + ^l3 



Wc obtain the second reducible integral by regarding 

 f'o *''M- 3 Cj .7; -f- C3 as first polar of the biquadratic «.,*•. In this 

 supposition the Hessian A* will ailmit the root b^ — 0^ the [)ola,r 

 (.*•— C4)(,r— ?»4)2 takes the form ,r~ (/Jj.*— ^13) and the required integral 

 itself is 



ƒ. 



1/ (A^ ,ca + 3 yl 2 .-f + A,) (Go .r^^ + 3 6\ .f- + C-3) 

 The icduction of the integral is obtained by putting 



^■^(Azx—As) 



X2)W -{- jii =z 



Cox^ + 5C,.T^-^C, 



Another means of constructing reducible integrals is founded on 

 a peculiar foi-m which can ahvays be given to the invariant relation 

 between the six branch-points. 



