1903-4.] Dr Muir on the Theory of Continuants. 
139 
“ supplement ” that anything apparently new in substance is met 
with. There in § a (p. 497) the following lemma occurs : — 
“The roots of the cumulant <? 2 5 * * • > <Zi] in which 
each element is a linear function of x, and wherein the 
coefficient of x for each element has the like sign, are all 
real : and between every two of such roots is contained a 
root of the cumulant [<2i , % , • . • , J and ex converso a 
root of the cumulant [g 2 } % , . . • , Q_i\ ’• and (as an evident 
corollary) for all values of £ and £' intermediate between 1 
and i the greatest root of [gq , g 2 , . . . , q t ] will be greater, 
and the least root of the same will be less than the greatest 
and least roots respectively of [ q p , q p+x , . . . , q p ,_ x , q p i\.” 
Even this, however, may be placed under the well-known theorem 
regarding the roots of the equation 
a n - x 
a VL 
^13 • * 
®12 
a 22 -x 
CO 
CM 
e 
«13 
a 2S 
a 33 -x . . 
■ . 
0 
which had been enunciated by Cauchy in 1829.* 
The next noteworthy result occupies § i. (p. 502), As a 
preparation for it the theorem 
[®1 j ’ • ’ * J > ^1 J ^2 > * * * > = [®1 > a 2 } ’ * • J ®m] [^1 » ^2 » * * * J ^«] 
- K, ‘ & 3> * * • J 
may be recalled, the group of elements on the left being now 
viewed as consisting of two sub-groups. This theorem Sylvester 
writes in the form 
[0,0,] = [Oj[Oj - [O'JOJ 
and he succeeds in including in it a general theorem, not explicitly 
formulated, in which the number of groups is i, the next two 
cases being 
[OjOgQj = [^i][d 2 ][o 3 ] 
- [0'i]['0 2 ][0 3 ] - [Oi][0' 2 ]['0 3 ] + [O'Jf'O'Jf'Og], 
* v. The theory of orthogonants ... in Proc. Roy. Soc. Edinburgh , 
xxiv. p. 261. 
