INDETERMINATE QUARTIC EQUATIONS 69 



i=n r=r 



(iii) We may replace x* in (3) by Sa;,, 4 2/ r 4 , whence we 

 have 



+ [26c(6 8 -c 8 )] 4 Sa; n 4 . (6) 



Equation (6) is an algebraical solution of the equation 



for all values of n and r except the case n=r=0. 



(iv) A still more general result may be obtained by replacing 

 in (4) and (6) 1x,* by Sx re x n 4 and Sy r 4 by 2fx r / r 4 . 



JV..B. There are other equations which, like (3), possess 

 the property of indefinite extension by substitution for one of 

 the variables. One other example will suffice, viz. : 

 (3z 3 +3z 2 -3z-3+z 4 ) 4 +(3z 3 -37j>-3z+3-:e 4 ) 4 +(6z 2 -6-a; 4 ) 4 

 = (3z 3 +3z 2 -3z-3-x 4 ) 4 +(3z 3 -3z 2 -3z+3+x 4 ) 4 +(6z 2 -6+a; 4 ) 4 



+J6:r(z 2 -l)j 4 . (1) 



Thus x\, z=2 gives 



8 4 + 1 7 4 + 28 4 = 10 4 + 18 4 + 19 4 + 26 4 . 



Since the quantity x only occurs in the form a; 4 in (7) it is 

 clear that we may replace it as before by Sz,, 4 Sy r 4 and obtain 

 an identity of the form. 



which holds for all values of n and r, except w=r=0. 



SECTION II On the algebraical solution of the equation 



1. As in the previous case we shall first give solutions for 

 a^few particular values of r. 



QUESTION 1. Solve in integers the equation 



xPjM-nP^xPY+txPY. (i) 



Assume as before 



X(* 1 r+o)*+ i>.(x z r+b)*=\(x l r+c)*+ v.(x z r+d)* (2) 



where Xa 4 +[A& 4 =Xc 4 +!Ad 4 (3) 



