48 
Proceedings of the Royal Society of Edinburgh. [Sess. 
(»».'• 
• 5 a i)‘i 
• . w I 
I («o > • 
• • . “5)4 
l ( 6 , , • 
• • . *4)5 
s o 
*1 
s 2 
a 0 S 3 
a 0 s 4 
4 - ajSg 
s i 
’ S 2 
S 3 
% S 4 
a 0 S 5 
+ a l S 4 
S 2 
S 3 
*4 
<*0*5 
CLqSq 
+ “l S 5 
S 3 
S 4 
8^ 
a 0 s 6 
a 
0 S 7 
+ « 1«6 
S 0 
S 1 
S 2 
S 3 
S 4 
*1 
S 2 
S 3 
S 4 
% 
^2 
S 3 
S 4 
S o 
S 6 
*3 
8 4 
S 3 
**6 
S 7 
I *4 
S 5 
S 6 
S~ 
h 
5 
all in agreement with Cayley’s original result of 1846 (Hist, ii. pp. 162-164). 
Taking the second of these for proof, we multiply unity columnwise by the 
given bigradient array, obtaining 
1 
-*o 
I 
a o 
«1 
a 2 
CO 
e 
*5 • 
1 . 
-*o ~ s i 
a o 
a x 
a 2 a ?j 
a 4 a 5 
. 1 
• 
\ 
\ b 2 
CO 
1 * 
\ 
\ 
\ h 
* 4 • 
1 
K 
h 
b 2 
b 3 b 4 
a o 
a i 
a 2 
a 3 
a 4 
a 5 
. a 0 
a i 
a 2 
a 3 
a 4 
«5 
\ 
K 
\ 
^3 
% 9 i 
a 0 S 2 + <Vl 
(^0^3 4" CL]$ 2 
4- a 2 
*1 
« 0 S 4+ • • 
. a 0 s 5 + . . . 
a 0 s 2 
CLq^3 "t 
a Q S 4 "t 
+ ^2 
S 2 
% s 5+ • ‘ 
• a o s e + • • • 
% 9 0 
a 0 S l + a l S 0 
UqS 2 + tqSj + 0 2 ‘S‘o 
a 0 S 3 ’ • 
. « 0 s 4 + . . . 
a 0 S 2 + a l*l 
a 0 S 3 + Cl l S 2 + a 2 ’ S l 
% 9 4 + • • 
• a 0 s 5 + • • • 
«o « 2 
a o*z + a 4 s 2 
^o ,9 4 ”t ^ 1^3 4- ei 2 s 2 
a 0*5 4 • • 
. a 0 s 6 + . . . 
and thence the final form desired. 
The question of the existence of multiple or repeated roots in an equation 
is next taken up (pp. 178-196), the main result being: The equation A = 0 
will admit of only k distinct roots of the first determinant of each of the 
last n— k bigradient arrays arising from A, and its derivative vanishes: 
and this condition being fulfilled, the equation of the said roots will be got 
by equating to 0 the determinant formed by replacing the last column of 
D k by k zeros and the 0 th , l 6 ’*, 2 nd , . . . , k fot powers of x. For example, 
A and its derivate being 
x b + x 4 - 5a ,s - x 2 + Sx + 4 , 
5x 4 + 4a: 3 - 1 5a: 2 - 2a: + 8 , 
