INDETERMINATE QUARTIC EQUATIONS 63 



As a particular case put x l x 2 =a=c=l, 6=2, so that 



a 3 =-5/7, r=-21/13. 

 Hence substituting in (2) the foregoing values we obtain 



8 4 +28 4 +47 4 =34 4 +34 4 +41 4 . 



In the same way we may derive other algebraical solutions 

 by putting d=b, e=c,fa, etc. 

 Second method. In the identity 



replace a by x^ and b by 



Then 



i.e. 



44 



44 



l_9-4\4 / Z ,\ 



~W> 



19.4/ 4\4 Ay._2\4 



/<y44.9*/4\4 Ay 4_0*, 4\4 / r 2\4 



/ C T^ "'/n. \ /*', *--<-t ,t \ I '*'* \ /< /I \ 



=( " Y ) ~( -% ^ -) ( -^j = 4 (!) 



Taking the first only of these equations and integrating 

 we have 



(2) 



which is an identity of the kind required. Thus we have 



of which the former is the second smallest solution which exists, 

 the smallest being 2 4 +4 4 +7 4 =3 4 +6 4 +6 4 . 



Cor. In equation (2) replace x lt y l9 x 2 , y z by their recipro- 

 cals and multiply each root by x^-y-^-x^-y^. We thereby 

 obtain 



