476 ON THE SEXTIC RESOLVENT EQUATIONS OF JACOBI AND KRONECKER. [950 
Kronecker s resolvent equation. 
Kronecker writes x m to denote x 0 , x 1} x 2 , x 3 , x 4 according as the residue of 
m (mod. 5) is = 0, 1, 2, 3 or 4: then putting 
^m+n+ VS?m'^m+n^m+zn = ® . Wl + W • m + 2/2, 
a root is 
/= (012 +123 + 234 + 340 + 412) sin — 
D 
+ (024 + 130 + 241 + 302 + 413) sin 
+ (031 +142 + 203 + 314 + 420) sin 
+ (043 +104 + 210 + 321 + 432) sin 
47r 
5 
07T 
T 
87T 
5 
and the other roots are deduced from this by changing 01234 into 03412, 14023, 
20134, 31240, 42301 respectively. 
2?r 9rjj- 
Taking e an imaginary fifth root of unity, say e = cos —- + i sin —, so that 
o o 
. 27T . 477 . 677 . 877 A ... 
sm —, sin sin -g-, sin — are as e —e 4 , e 2 — e 3 , e 3 — e 2 , e 4 — e; also writing 
01234 = 012 + 123 + 234 + 340 + 412, ..., 
so that 01234, ..., are cyclic irreversible functions of the roots x, then the expression 
for the root f is 
/= ( e _e 4 ) 01234 + (e 2 -e 3 ) 02413 
- (e - e 4 ) 04321 - (e 2 - e 3 ) 03142, 
or, as this may be written, 
/= {(e - e 4 ) (1 - £) + (e 2 - e 3 ) (1 - R) >8} 01234, 
and the expressions for the six roots are 
/ = {(e - e 4 ) (1 - R) + (e 2 - e 3 ) (1 - R) 8} 01234, 
/0 = {(e - e 4 ) (1 - R) + (e 2 - e 3 ) (1 - R) S] 03412, 
/1 = {(« “ 6 ') (1 ~ R) + (e 2 - e 3 ) (1 - R) iSf} 14023, 
/2 = {(e - e 4 ) (1 -R) + (e 2 - e 3 ) (1 - R) S j 20134, 
/3 = {(e - e 4 ) (1 - R) + (e 2 - e 3 ) (1 - R) S} 31240, 
/4 = {(e - e 4 ) (1 - R) + (e 2 - e 3 ) (1 - R) 42301. 
I write down the analogous functions 
F = {(e - e 4 ) (1 - R) + (e 2 - e 3 ) (1 - R) S} 03142, (= ib801234), 
F 0 = {(e - e 4 ) (1 - R) + (e 2 - e 3 ) (1 - R) 01324, (= ££03412), 
F 1 = {(e - e 4 ) (1 - R) + (e 2 - e 3 ) (1 - R) £} 12430, (= ££14023), 
F2 = {(e - e 4 ) (1 - £) + (e 2 - e 3 ) (1 - £) £} 23041, (= ££20134), 
£ 3 = j(e - e 4 ) (1 - £) + (e 2 - e 3 ) (1 - £) £} 34102, (= ££31240), 
Fi = {(e - e 4 ) (!-£) + (e 2 - e 3 ) (1 - £) £} 40213, (= ££42301).