INDETERMINATE QUARTIC EQUATIONS 87 



Now we have identically 

 (a 4 + 6 4 + c 4 + d 4 ) 4 (a 4 + 6 4 + c 4 - d*)*= 8d*(a*+ 6 4 + c 4 ) 



[(a 4 +6 4 +c 4 ) 2 +d 8 ] 



4 -d 4 ). (1) 

 Hence if we put 



a=x i +4:xy+y 2 , b=2x 2 +2xyy*, c=x 2 2xy2y 2 , 



d=2(x 2 +xy+y 2 ) 

 equation (1) will become 

 (a 4 + & 4 + c 4 + d 4 ) 4 = (a 4 + & 4 + c 4 -<2 4 ) 4 + (2d 3 ) 4 (a 4 + 6 4 + c 4 ) 



which gives a biquadrate equal to the sum of 16 biquadrates. 

 Again, since we have identically 



and 



we have on multiplying corresponding sides of these equations 



together 



y\ (2) 



Hence if we can express x*y* as the sum of r biquadrates, 

 then equation (2) will give a biquadrate equal to the sum of 

 l+4(r+l) biquadrates. Thus for example if 



z=(a 4 +4& 4 ) 4 + (a 4 -4& 4 ) 4 , y=(a 4 +4& 4 ) 4 -(a 4 -4& 4 ) 4 , 



then by 3, Question 2, x*y* will be equal to the sum of 5 

 biquadrates, so that equation (2) will give a biquadrate equal 

 to the sum of 25 biquadrates. 



These examples of the extension of the results of 2-4 

 must suffice, for, with the increase in the magnitude of n, 

 diminishes, naturally, the difficulty of solving the equation. 



6. We come now to the case w=4, i.e. to the equation 



P 4 =P 1 4 +P 2 4 +P 3 4 +P 4 4 . (1) 



As the assumption that P and P t are respectively the 



