60 
ON A THEOREM OF ABEL’S RELATING 
[311 
The coefficients of the quintic equation for x must of course be of the form j 
mentioned; that is, they must be functions of a, ¡3, y, 8, which remain unaltered 
the cyclic substitution a/3y8. To form the quintic equation, I write 
0 — x = a, 
fas fas 4 = 2a 2 + (as + as 4 ) 2'(lb + (as 2 + as 3 ) 2'ac, 
fas 2 fas 3 = 2a 2 + (as 2 + as 3 ) 2'ab + (as 2 4- w 3 ) 2'ac, 
where 2' is Mr Harley’s cyclical symbol, viz. 
2'ab — ab + be + cd+ de + ea ; 
and so in other cases, the order of the cycle being always abode. This gives 
fas fas 2 fas 3 fas 4 = 2a 4 + 2 arb 2 — 2a 3 b + 2 2a 2 bc — 2abcd — 52'a 2 (be + cd) ; 
and multiplying by /1, = 2a, and equating to zero, the result is found to be 
2a 5 — 5 abode — 52 'a 3 (be + cd) + 52'a (b 2 e 2 + c 2 d 2 ) = 0; 
or arranging in powers of a, this is 
+ a 3 . — 5 (be + cd) 
+ a 2 . 5(b c 2 +ce 2 +ed 2 + db 2 ) 
[+ 5 (b 2 e 2 + c 2 d 2 — becd) 
' b 5 + c 5 + e 5 + d 5 
+ ■< — 5 (b 3 de + c 3 bd + e?cb + d 3 ec) 
K + 5 (bd 2 e 2 + cb 2 d 2 + ec 2 b 2 + de 2 c 2 ) 
the several coefficients being, it will be observed, cyclical functions to the cycle 
b, c, e, d.