337 



quantity )., and so we have the ecjiiatioiis 



froKi which we infer 



f—Vho f-\-^bc 



or 





\ -\-tiy = 



f-V^ho 



Similarly we obtain 



1 -(- y^( = — 



(J — ^ca 



h-i^ab 

 and then by equation (10) 



f^i^bc q4-V'ca k^y^ab /^o. 



/■— l^Z>c g—y'ca h — \^ab 



Thus it appears that the reducibility of the given hyperelliptic 

 integral implies the relation (12) between the invariants common to 

 tf^i, if'g, tf^a, and conversely as soon as these invariants, with a suitable 

 determination of the surds, satisfy the relation (12) the involution J 

 can be realised, and the given integral will degenerate. 



Supposing that the condition (12) is fulfilled, we have 



abc ƒ g 



P,'a' P,'ii' P,Y P,P^(2+(ir) P,P,(24-7«) 

 h A 9x K 



(13) 



P,P,(2 + «^) P,P,(«-/?-y) P,P,(-« + /J-y) P,P,(-«_^+y) 



and 



,tPybc = iiPym — yPyab (U) 



From these equations the constants «, i^ y, P^, /\, /\, L, M,N,q^,qi,q, 

 can be successively evaluated, we can find the quantics i;,, i,, ^s,^^, 7',if\ 

 and finally the transformation that reduces the integral. 



To illustrate the method described, 1 will consider a numerical 

 example. Let the given integral be 



X dx 



Si 



\/{bw' — 12.r + 4) (5.c' — 2.V + 2.) (7a;' — 6ar + 2) 

 that is, let us assume 

 rf,, = b.v' — 12j; + 4, if?, = 5.t-» - 2;c f 2, xl\ = Ix' — &x I 2. 



Calculating the invariants, we have 



