950] ON THE SEXTIC RESOLVENT EQUATIONS OF JACOBI AND KRONECKER. 479 
where the terms containing 12, 13, 14, 23, 24, 34 respectively are each =0, viz. the 
coefficient of 12 is (- e + e 4 ) + (- e 4 + 1) + (- 1 + e) = 0 : that of 13 is 
1 — e 3 -f (e 2 — 1) + (— e 2 + e 3 ) = 0 : 
and so for the other coefficients. In like manner it appears that the terms multiplied 
by 1, 2, 3, 4 (= os 1 , x 2 , x 3 , # 4 ) respectively are each =0, and thus the equation in 
question is verified. And in like manner it is shown that 
fo + e [fi + /2 + e 4 / 3 + ~ 0. 
The roots / thus satisfy the relations 
fo + fi+ f 2 + fs+ / 4 =-/Vo, 
fo + e2 /l + ^ fl + e f3 + e? ’fi = 0, 
/0 + e 3 fi + efa + e 4 / 3 + e 2 / 4 = 0, 
or the equation for / 2 belongs to the class of Jacobi’s multiplier equations. Hence 
(see Brioschi’s “Appendice terza” before referred to) the form of the equation is 
(/ 2 — a) 6 - 4a (/ 2 — a) 5 + 106 (/ 2 - of — 4c (/ 2 — a) 4- ob 2 - 4>ac = 0, 
or determining the arbitrary coefficient v so that a may be =0, the form is 
/ 12 + 106/ 6 - 4c/ 2 + ob 2 = 0, 
which is Kronecker’s equation 
/ 12 - 10 <£/ 6 + 51/r 2 = yjrf\ 
As to the meaning of the coefficients a, b, c, I recall that, in virtue of the foregoing 
linear relations between the roots, these may be expressed in terms of three arbitrary 
quantities a 0 , a lt a 2 as follows: 
f =a 0 V5, 
/0 = + ®1 + 
fi = a 0 + ea x + e 4 a 2 , 
/2 = a 0 + e 2 a x + e 3 a 2 , 
/3 = a 0 + e 3 aj + e 2 a 2 , 
/4 = a 0 + e 4 a x + ea 2 , 
and a, b, c are then determinate functions of a 0 , a 1} a 2 , viz. we have 
a= a 2 + a x a 2 , 
b = 8a 0 4 aia 2 — 2 a 0 2 a 1 2 a 2 2 + a^aj — a 0 (af 3 + a 2 5 ), 
c = 80a 0 6 a 1 2 a 3 2 — 4Oa 0 4 a 1 3 a 2 3 + 5a 0 2 a 1 4 a 2 4 + a^a/' 
— a 0 (32a 0 4 — 20a 0 2 a 1 a 2 + ba 2 a 2 ) (a? + a/) 
+ l (a a 5 + a 2 5 ) 2 ; 
so that, for a = 0 and therefore a 0 = V— a^, we have 
6 = lla! 3 a 2 3 — a 0 (a 2 5 + a 2 5 ), 
c = — 44a 1 6 a 2 5 — 57a Q a^a 2 (a^ + a 2 5 ) + | (af + a 2 5 ) 2 , 
but I do not know that for Kronecker’s form the actual values of a 0 , a 1} a 2 in terms 
of the coefficients of the quintic equation have been calculated.