OF THE ROOTS OF AN EQUATION AND ITS COEFFICIENTS. 54! 
the first set of equations is, . 
2 (Caf) . | 1 , [y], [v8], [yee], Lydel] , 
the second set is, 
S{2(a8y).|1,[8] [Se], [Set], . 
abi Lig [8] , [Se] , [Sel] , 
the third set is, 
> {C(aBys). A kc Sle ere ees te 
oy te eh eC I 55. 
wid eh Baste Lies APS LG ote 
Cine i Uebel wale ist eG, 
similarly for the fourth set; and the 
of C(aBydet) :—where such a symbol 
* (par) =(p-9)(p-r)"(g-7/. 
d-, 1+, ly, [yd] . [yde] ; [dee 
or 
Leese slo aoe lsmloec| 
Or 
1,—5,10,-—10,5,-1 
0 ;=—6 b, On; Cad; g\; 
Gla, eg tae ad ,é, f 
ea 
=—6|b, 5c, 10d, 10e, 5/, 9); 
G1a,00, 10¢., 10d -5¢,/ 
t+- + —- + -— + 
P= — O10, 06 Od, 10¢, bf, 9 , . |; 
1\a,5b,10c,10d, 5e,/, . 
Gilece Ok lees Pris! 
= 5a, ov, Oe 10d be af | 
$+- + - + = + = 
Boe MO LOG. Wofogie (Gig, xia 
NG oO OC NORE HOS © fio msn Fa ine 
Gl. 5; 6, 10d, 10¢,. 5f, -9.5°. 
apis OO LOG CAL st bGi5, afer’ « 
Ole ae aoe 06> LO LOE Abe od, 
CMe nsGte able LOC AO: bets 70 
fifth set, which is the known value 
as [de] stands for d¢+6f+«¢, and 
APPLICATION TO STURM’S FUNCTIONS. 
VII. The first set of equations for a sextic is 
2i(a—B)|\1,[y],[y8],Lyse], [v6], - 
5 bs byl els [ysell sy ael] 
., write respectively 7—a,x—B,. 
For a, B,.. 
a P,,2P,,3P5,4P,,5P,,6P, | (17). 
| 6P,,5P,,4P,,3P,,2P,, P; 
.. then we have 
(a—f)’=(e@—-a—a#+ BY =(a—B)’ 
and P, becomes 
=62-P, 
6(ax+b) , 
a r) 
w-ata—B+u—y+e—6+2-€+2-C 
(18), 
* This is the &function of Professor SYLVESTER. 
