INDETERMINATE QUARTIC EQUATIONS 85 



simply by taking as a particular case c=a, d=b, in which case 

 it becomes 



4 + 4] 4 = [( 2 + a&+ & 2 ) 4 -4] 4 + (8ab+ 46 2 ) 4 -f (4a 2 -46 2 ) 4 



For example a =2, 6=1 gives 



2405 4 =2397 4 +784 4 +490 4 +294 4 +32 4 +20 4 +12 4 . 



Another solution, closely allied but giving smaller results, 

 may be obtained thus. We have identically 



(a 2 +46 2 ) 4 =(a 2 -46 2 ) 4 +2-2 4 a 2 6 2 (a 4 +166 4 ). (4) 



Now put b=z 2 , a=x 2 +3y 2 , and this becomes 

 [(x*+ 3t/ 2 ) 2 +42 4 ] 4 =[(z 2 + 32/ 2 ) 2 -4z 4 ] 4 + (2z) 4 [(z 2 + 3?/ 2 ) 4 + (2z 2 ) 4 ] x 



[(x+y)*+(x-y)*+(2y)*} (5) 

 For example, we have when 



x=y=z, 5 4 =4 4 +4 4 + 3 4 + 2 4 + 2 4 , 



a;=0, y= 2 , 13 4 =12 4 +8 4 +6 4 +6 4 +5 4 +4 4 +4 4 , 



x =3, y=z=l, 37 4 =35 4 +24 4 +12 4 +12 4 +4 4 +2 4 +2 4 , 

 a?=2, y=z=l, 53 4 =45 4 +42 4 +28 4 +14 4 +12 4 +8 4 +4 4 . 

 Also, since the equation 



can be solved algebraically for all values of r greater than 2 

 (see 7 infra), it follows that by putting 6=z 2 , a=<? in (4) that 

 we can get algebraical solutions of the equation 



P 4 =P 1 4 +P 2 4 + . . . 

 where s=7+2n. 



Again, write equation (5) in the form 



(a:-i/) 4 ] (6) 

 Now, an algebraical solution of the equation 



l 4 +#2 4 =^l 4 +J D 2 4 + - ' - +P, 4 (7) 



has been found l for all values of r greater than 2 by means of 

 a single formula. If then, we choose x and y so that 



we can, on replacing (x+y)*+(x y}* by P 1 4 +-P a 4 + +P r 4 , 



1 See 3 of Part II., Section I. 



