427 
1907-8.] Dr Muir on the Theory of Hessians. 
Determinants” Spottiswoode devotes to one or two theorems connected 
with Hessians, and mainly to Hesse’s theorem of the year 1849 (December) : 
his proof of the latter, however, is not an improvement on J acobi’s. One of 
his notations for the Hessian, U being the function and x, y, z, . . . the 
variables, is 
3 0 0 
dx 
dy dz 
0 
0 0 
dx 
'ey dz 
suggested, doubtless, by Sylvester’s general umbral notation. Further, he 
uses the word “ Hessian ” in a geometrical sense, namely, for the locus 
represented by 
H(U) = 0. 
Brioschi, Fr. (1854). 
[Solutions des questions 285, 286. Nouv. Annates de Math., xiii. 
pp. 402-409.] 
The theorems which had been set for proof were geometrical theorems 
due to Hesse, and as a foundation on which to base them and others 
Brioschi establishes a general result regarding Hessians. This with only 
slight departures from the original may be formally enunciated as follows : 
If — a be a homogeneous integral function of the m th degree in r+1 
variables x 0 , x x , x 2 , . . ., x r , the Hessian of which with respect to those 
variables is H r+1 and with respect to the variables x 15 x 2 , . . ., x r is H r , then 
the determinant which is the result of bordering H r by prefixing 
q du du du 
’ dx Y ’ dx 2 ’ ’ dx r 
as a first row and as a first column is equal to 
(_iy+i m uA r + 
to - 1 
ryt 2 
(to - 1) 
-H 
r+l • 
The bordered Hessian, B say, being equal to 
U l 
U 2 . . 
, . u r 
(m — 
i K 
u n 
u l 2 • ■ 
• • ^1 r 
(m - 
i K 
U 2l 
u 22 ‘ ' 
U 2r 
(to - 
\)u r 
u r i 
11,,, . , 
> • U rr 
1 
m— 1 
