40 MAJOE P. A. MACMAHON: MEMOIE ON THE 



defined by the eight Diophantine equatious 



= a !0. 



= a io- 



We require all values of the quantities a which satisfy these equations. 

 To form the sum 



for all solutions, introduce the auxiliary quantities 



ft, b, c, d, e, f, f/, h 



in association with the successive Diophantine equations. The sum in question may 



be written 



1 



_ 

 f (1 -adgX,) (I -fU'X 3 ) (1 -rt/'AXs) (1 -MX,) (1-beghXJ 



where after expansion we retain that portion only which is free from the auxiliaries. 

 Remarking that 



1 1 



we eliminate the auxiliar ft and obtain 



1 



l_p222)/l_^*L)/i- 



Put now bd = A, be = B, If- C, cd = D, and we obtain 



1 





f/ AX.X.pW X^W. X 3 X 



"BOD&A era/ 1 



(1 -Br^X,) (1 -CX,) (1 -DAX 7 ) (l -^2) f 1 - CD fM 



\ A / \ A / 



an artifice which reduces the number of auxiliars to be eliminated by unity. 



