6o ON THE ALGEBRAICAL SOLUTION OF 

 SECTION I On the algebraical solution of the equation 



1. Before attempting to solve this equation for all values 

 of n and r, it will be convenient first to give solutions for a 

 few particular values of n and r. 



QUESTION 1. Solve in integers the equation 



P 1 *+P 2 4 =PV+P' 2 4 . (1) 



Let P 1 =z 1 +2 3 , P 2 =z 2 z 4 , P'i=z 1 z 3 , P' 2 =z 2 +z 4 and the equa- 

 tion becomes 



Zjfa+XjZf^zfzi+zff (2) 



This is an indeterminate cubic in z x and z 2 and if any par- 

 ticular solution is known, another can be found, but as it 

 presents the new roots as functions of the coefficients z 3 and 

 z 4 , it will be an algebraical solution. Thus, putting z 1 =a; 1 r+?/ 1 , 

 Z 2 =x 2 r+y 2 , (2) becomes 



or, on expansion and rearrangement, 



-a; 2 z 4 3 )r+(t/ 1 3 z 3 + 2 / 1 z 3 3 )-(i/ 2 3 z 4 +7/ 2 z 4 3 )=0. (3) 



To make the term independent of r vanish, we must have 



which is equivalent to knowing a particular solution of (1) 

 since it may be written 



and to satisfy this we may evidently take 



2/i+ z 3=-2/ 2 - z 4 2/i-z 3 =:>/2- z 4> ' yi=-4 y2=-3 ( 5 )- 



In order to make the coefficient of r in (3) vanish, we must take 



i.e. ^ 2 =[(3y 1 2 z 3 +z 3 3 )/(3i/ 2 2 z 4 +z 4 3 )] a : 1 



=[z 3 (3z 4 2 +z a 2 )/z 4 (3z 3 2 +z 4 2 )] a : 1 (6) 



on substituting for y lt y 2 their values given by (5). 



