32 
Proceedings of the Royal Society of Edinburgh. [Sess. 
V. — The Theory of Bigradients from 1859 to 1880. 
By Thomas Muir, LL.D. 
(MS. received June 9, 1913. Read November 17, 1913.) 
My last communication in reference to the history of bigradients ( Proc . 
Roy. Soc. Edin., xxx. pp. 396-406) brought the record up to the year 
1859. The present paper continues it to the year 1880. 
Bruno, F. Faa di (1859). 
[Theorie Generale de l’Elimination, .... x + 224 pp. Paris.] 
In his section on the highest-common-divisor (pp. 47-52) Bruno, denoting 
the r th Sturmian remainder of 
%x m + a^x m ~ x + • • • •, & 0 a? m_1 + b 4 x m ~ 2 + • • • ■ 
by R r , finds for it the expression 
where \ r and R r /A r are the “ allotrious factor ” and “ simplified residue ” of 
Sylvester (1853). He must, however, have overlooked the question of sign, 
for the example which he gives, namely, 
1 
% a i a 2 
X 2 + 
a 0 <h a B 
x + 
a 0 a x a 4 
b 0 \ 5 2 
K b l h 
bo \ 
( 
b 1 
. b 0 5 2 
CO 
hO 
o 
as the first Sturmian remainder of 
+ a 4 x 2 + • • . + a 4 , b 0 x s + b ^ + b 2 x + b z 
is incorrect in this particular. 
