OF THE DIFFEEENCES OF THE BOOTS OF A GIVEN EQUATION. 
51 
13. The value of is 
— 15 (j?e 
+ 6 a-hd 
4 
~ 5 aVc 
+ 1 h* 
which is a seminvariant, and requires no addition. 
14. Hence, for the quintic equation 
(«, 5, c, iZ, e,/Xw,l)®=0, 
the equation for j3, y, is 0 = 
a® X 
aW n X 
n X 
A_ 
n 4/ n X 
' ' 
' ' 
( 
' ' 
+ 1 
0 
-125 G'V' 
-50 €?df 
— 15 de 
+ 1 
+ 300 oddef 
+ 80 aV 
+ 6 cdbd 
— 160 «V 
+ 30 c?bcf 
+ 4 (dd 
+ 50 
— 54 cdbde 
~ 5 adc 
+ 132 c^hde^ 
—36 (dde 
+ 1 d 
— 130 cdhcef 
+ 27 ded’^ 
— 120 a%d'^f 
- 8 adf 
+ 40 cdc^df 
+ 42 ab^ce 
+ 88 «+V 
— 18 abdd 
— 117 (dcd‘e 
+ 4 ac^ 
+ 27 dH^ 
- 8 b^e 
+ 28 ah^ef 
— 97 aired 
+ 4 ded 
- 1 Vd 
I 
+ 6 ab^d^e 
+ 62 abdde 
+ 66 ab~cdf 
— 18 abed^ 
— 24 abdf 
— 12 ade 
+ 4 addr 
- 16 ddf 
+ 18 
— 14 b^ede 
+ 4 b'^d^ 
+ 6 b^df 
+ 3 b'^de 
- 1 b'^deP 
15. As a verification of this result, I remark that, taking for the quintic equation 
= the roots of this equation are — 1, a, —&>, — where 
i) is an imaginary cube root of unity We ought to have /B, y, S, s) 
= — \/ □ = — 36 ; and this will be the case if, for instance, ce, /3, y, S, s are respectively 
— 1, 4/, —6/^,—tu. We have then 
^(a, /3, y, ^)= — 1 — <y . — 1 — — l-f-4;*. w — — 6, 
y, S, S )— u — aP . ci}-\-ci)^ . 24> . 24;^ . — uP-\-Ci) — — 12, 
2i^(y, ^ , £ , a)= 24/^ 4/®+<y . 4/^+1 .— 4/^+4/ . —sy^+l • “^+1 = + 6(iy— 
, £ , Cl, (3)= — aj^-\-co . — — 6) . — 4;+l . — 24^ . — 1 — 4/= — 6(4/ — 4>^), 
, a, /3, y)= — u-\-\ . — 2ot) . — a — — 1 — a. — 1 — cJ^.a — oJ^ — 6. 
