430 
Proceedings of the Royal Society of Edinburgh. [Sess. 
where the roots are a, a, a, b, b, c, and where, since 
s o = 6, Sj = 3a + 26 + c, s. 2 = Sa 2 + 2b 2 + c 2 , 
we have 
s 5 s 6 s 7 
3 2 1 
a 5 b b c b 
s 6 s 7 s 8 
= 
3 a 2 b c 
aP b 6 cP 
s 7 s 8 Sq 
3 a 2 2 b 2 c 2 
a 7 b 7 c 7 
= 6 | aPb x e 2 1 . aPb b c b 1 aPb Y cP j , 
= 6a b b b c b | aPb l c 2 | 2 . 
The other part of the proposition rests on the statement that a similar 
procedure leads in that case to factors having at least one column of zeros. 
The case where the number of rows is less than the number of different 
roots is not considered; but the first part of the proposition is used to 
obtain the modification which it is possible to make in the relation 
^n+r — 1 ^2^ w + r ’ — 2 • • • d* CL n S r 0 
when the roots cease to be all different. 
Salmon, G. (1859). 
[Lessons Introductory to the Modern Higher Algebra, .... 
xii + 147 pp., Dublin.] 
In Salmon’s treatment of the foregoing subjects (p. 14, §§ 119-126, 
162-165, p. 146) there are several points of freshness. His proof, for 
example, that if 
(a,b,c,d,e,f tyx,y ) 5 = (fx + m x yf + (l 2 x + m 2 yf + (l z x + m z y ) 5 
the persymmetric form of the canonizant is equal to 
I h W 2 I 2 I h m 3 I 2 I 2 * ( l l X + m l V)( l 2 X + m 2V)( l 3 X + m $) > 
is accomplished (§ 119) by using four differentiations to show that 
ax + by = Ifilpc + m^y) + lf(l 2 x + ui 2 y) + lf(l 3 x + m 3 y) 
= l-fu + Ifv + If io , say, 
bx + cy = Ifmpi + Ifmfl + lfm 3 w , 
substituting these trinomials for ax + by , bx + cy , . . . in the canonizant, 
and then examining * the twenty-seven determinants into which the latter 
* It is better to note at this stage that tbe determinant is the product of 
Ifu 
Ifv 
Ifw I 
7 2 
h 
7 2 
7 2 
l 3 m 3 w and 
l 1 m 1 
l 2 m 2 
l 3 m 3 
mfu 
mfv 
m 3 w 1 
mf 
mf 
ai)d therefore is equal to | If l 2 m 2 mff . uvw. 
