401 
1909-10.] Dr Muir on the Theory of Bigradients. 
With increasing tediousness a five-line determinant is reached having 
elements with s — 4 and s — 5 for second suffixes, a six-line determinant 
having elements with s — 5 and s — 6 for second suffixes, and so on. The 
form of the determinant of the (2^ + 2) th order is thus deduced, the factor 
preceding it being said to be 
( C 0 , s— 3 C 0 , s-i)( C 0 , s—5 C 0 , s- &) 2 ( c 0 , s-7 C 0 , s-s) 3 • • . • ( C 0 , s-2q+l C 0 , s-2g) ? l ( C 0 , s-Zq-lY • 
Sturm’s division-process, in which each remainder is of a lower degree 
than the remainder preceding it, and for which, therefore, the corresponding 
continued fraction has each partial denominator a linear function of x, has 
close relationship with the above division-process, because the continued 
fraction already obtained is identical with * 
Po 
X +Pi 
Vjlh 
% P2 -f- Pq 
PzPa 
v + Vi+Ps- • 
Expressions for the remainders corresponding to this continued fraction are 
thus readily obtainable, and a section (§ 3) is devoted to finding simplified 
substitutes for them. The remaining section concerns the strictly Sturmian 
case, where one of the original functions is the derivate of the other. 
Bruno, Faa di (1855, July). 
[Sulle funzioni siminetriche delle radici di un’ equazione. Annali di sci. 
mat. efis., vi. pp. 412-419.] 
As evidence of the value of a certain theorem Bruno adduces the ease 
with which the expansion of the resultant of a pair of equations may be 
calculated, and he prints at full length the resultants R 2)2 , B- 3 , 3 , R^, that 
is to say, the final expansions of 
a b c 
a b c 
. p q 
p q r 
r 
, etc., 
the arrangement of the terms being such as to make evident the fact that 
each resultant is unaltered by reversing the order of the two sets of 
coefficients of which it is a function : for example : — f 
R 2j 2 = (a 2 r 2 + c 2 p 2 ) - ( abqr + bcpq) - 2 acpr + acq 2 + b 2 pr . 
* This identity Heilerman published again separately in 1860 (see Zeitschrift f. Math, 
u. Phys v. pp. 262-263). It is included, however, in a result given by Stern in 1883 
(see Crelle's Journ., x. p. 156) ; and a still more general identity will be found in the Proc. 
Edin. Math. Soc ., xxiii. p. 37. 
t The expression for R 4i4 is full of inaccuracies. 
