INDETERMINATE QUARTIC EQUATIONS 87 

 Now we have identically 



C 4 )[( 4 



6 4 +c 4 +^ 4 ) 



d 4 ). (1) 



Hence if we put 

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



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

 equation (1) will become 



4 )(a 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=( 4 +4& 4 ) 4 + (a 4 -46 4 ) 4 , /=(a 4 +46 4 ) 4 -(a 4 -46 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 n=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 1 are respectively the 



