INDETERMINATE QUARTIC EQUATIONS 69 



H=W r=r 



(iii) We may replace x* in (3) by So:,, 4 Sy r 4 , whence we 

 have ' 



. (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) 2x, * by 2x, t z n 4 and % r 4 % 2|*,y r 4 . 



.Af.-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. : 



3 a -l)j* (7) 



Thus x~l, z2 gives 



8 4 + 17 4 + 28 4 = 10 4 + 18 4 + 19 4 + 26 4 . 



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

 clear that we may replace it as before by ^x^^y^ and obtain 

 an identity of the form. 



Pf+p<*+ . . . +P.+J-PV+PV+ . . . +P' r+ j, 



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



XPj 4 + {xP 2 4 =XPV+ nP' 8 . (1) 



Assume as before 



Xfor+a) 4 * [x(^ 2 r+6) 4 =X(a; 1 r-f c) 4 + v.(x#+d)* (2) 



where Xa 4 +(x6 4 =Xc 4 +firf 4 (3) 



