134 
ON A THEOREM OF ABEL’S RELATING TO A QUINTIC EQUATION. [741 
and if 0 (a, a 1} a 2 , a 3 ) contains a term a m /3 n yPy' q , then the other three functions will 
contain respectively the terms 
a m(_ /3)n y 'P(- y) q , ct m fin (— yY (_ ry')9 a m (-P) n {-yY{y) q -, 
viz. the sum of the four terms is 
= ct m /3 n [{1 + (-)p+i 1} rfy'q + {(_)«+y i + (_)»+9 1} y<iy p\ 
This obviously vanishes unless p and q are both even, or both odd; and the 
cases to be considered are 1°, n even, p and q even; 2°, n odd, p and q even; 
3°, n even, p and q odd; 4°, n odd, p and q odd. Writing, for greater distinctness, 
2n or 2n +1 for n, according as n is even or odd, and similarly for p and q, the 
term is, in the four cases respectively, 
= 2a m /3- n (y-P y 2q + y 2q y’-P), 
= 2a m /3* n+1 (y-P y- q - y- ( J y'-P), 
— 2a m /3 2n (ry 2 ’P+ 1 y' 2( i+ l — ,y29+i^'2i>+i\ 
= 2a m /3 m+1 (y 2 P+ 1 y 2 q+ 1 + y 2 Q+ 
The second, third, and fourth expressions contain the factors 
¡3 ( 7 2 _ ry' 2 ), 7 ry' (ry 2 - ry'-), f3yy, 
respectively; and the first expression as it stands, and the other three divested of 
these factors respectively are rational functions of a, /3 2 , 7 2 , y' J , that is, they are 
rational functions of m, n, e, h. But the omitted factors ¡3 (y- — y' 2 ), yy {y 2 — y' 2 ), 
¡3yy, = 2nh(l + e 2 ), 2h 2 e(l+e 2 ), nhe(l + e 2 ) are rational functions of n, h, e\ hence 
each of the original four expressions is a rational function of m, n, h, e; and the 
entire function 
0 (a, «!, a 2 , a 3 )+0(«i, a. 2 , a-., a) + (f> (a 2 , a 3 , a, a 1 ) + 0(a 3 , a, a x , a,) 
is a rational function of m, n, h, e. 
Replacing a, (3, y, y by their values, the roots of the quartic equation are 
Til + n V(1 *4* 6 2 ) + (1 + s 2 + V(1 + 6 2 ))]> 
m - n V(1 + e 2 ) + J[h (1 + e 2 - V(1 + e 2 ))], 
m + n V(1 + e 2 ) — \/[h (1 + e 2 4- \/(l + e 2 ))], 
m — n V(1 + e 2 ) — (1 + e 2 — V(1 + e 2 ))]. 
And I stop to remark that taking m, n, e, h = — \, + 2, — l respectively, the 
roots are 
-i + iV5 + V[-i(5 + V5)], 
-i-iV5 + V[-*(5-V5)], 
-i + iV6-V[-i(5 + V6)l