INDETERMINATE QUARTIC EQUATIONS 83 



+ [8% V(% 12 + 64V 2 ) 2 ( V 2 -64V 2 )] 4 [V 4 + 2 1 V 4 ] 



= [( % 12 + 64V! 12 ) 4 - (w 1 12 -64v 1 12 ) 4 ] 4 + [2(w 1 12 + 64V 2 ) 



(V 2 -64?V 2 ) 3 ] 4 



Again, since on the right-hand side of (7), and therefore 

 also on the right-hand side of (8) in its new form, the sum of 

 two of the biquadrates is w 1 8 +16v 1 8 multiplied by a certain 

 factor, it follows that the substitution u^u^, t> 1 =2v 2 3 will 

 convert u^+lGv^ into M 2 24 + 2 12 v 2 24 , which by (7) is expressible 

 as the sum of four rational biquadrates (so that 2 72 +2 12 (2v 2 3 ) 24 

 is expressible as the sum of four or of six biquadrates), and 

 since as before the sum of two of these four biquadrates will 

 be w 2 8 +16v 2 8 , multiplied by a factor, it is clear that the 

 successive substitutions u 2 =u 3 s , v z =2v 3 3 , and in general 

 u r =u r+1 3 , v r 2v r+l 3 will enable the right-hand side of (8) 

 to be expressed simultaneously as sums of biquadrates 

 successively increasing by 2, i.e. we shall have a biquadrate 

 equal to the sum of any even number of biquadrates greater 

 than four. 



There is no great difficulty in writing down, in accordance 

 with the above formulae, a biquadrate equal to the sum of 

 6+ 2n biquadrates, but such formulae will only give arithmetical 

 results of a high order of magnitude, the reason being that they 

 do not give a biquadrate merely equal to the sum of 6+ 2n 

 biquadrates, but furnish a special kind of biquadrate which 

 possesses the peculiar additional property of being expressible 

 simultaneously as the sum of every even number of biquad- 

 rates greater than four up to 6+ 2n. 



