1907-8.] Dr Muir on the Theory of Hessians. 
431 
Baltzer, R. (1857). 
[Theorie und Anwendungen der Determinanten, mit Beziehung 
au£ die Originalquellen, dargestellt von Dr Richard Baltzer ; 
vi+129 pp. ; Leipzig, 1857. French translation by J. Honel 
xii + 235 pp. ; Paris, 1861.] 
In Hesse’s converse theorem of 1851 (March) Baltzer (§ 13, 3) wisely 
substitutes for A = 0 the condition 
+ 2 + • • • + c n u n = 0 
(which by a property of Jacobians implies A =0), his proof being that the 
substitution 
•fijc — ^>k\V l ^2^2 ^k 5 n—lVn—l ^ kVn 
will then give du/dy n = 0 , for 
du du dXi du dx 0 du dx n 
= 1-j 2 + . . . -] — - 
dy n dx 1 dij n dx 2 dy n dx n d y n 
= u Y c Y + u 2 c 2 + • • • + u n c n . 
In the second place, from the same n + 1 equations, namely, 
/ i\ n 1 
— (mi — 1 ) -+u x x x + • •• +u n x n = 0 
m - 1 
- (m-\)u 1 + u n x 1 + ••• + u nX x n = 0 
- (m — l)u n + u Xn x x + • • • +u nn x n = 0 J , 
he obtains (§ 14, 4) 
(m- 1) : Xj : x 2 : . . 
<1 
II 
V : 
: V 2 : . . 
• : V., \ 
= V x : 
:V n 
: Y 21 : . . 
■ • : Y nl , 
% v n 
: Ym 
: V 2n : . , 
• V 
where the Y’s are the cofactors of the corresponding y?s in the resultant of 
the set of equations. The first and (r + l) th lines imply respectively 
- (m - 1 ) : x r = A : Y r , 
and 
-(m- 1) : x s = Y r : Y sr , 
the former of which gives 
V,. - - * r A , 
m — i 
(a) 
and the two together 
y - m. a 
5r (m — l) 2 ‘ 
08) 
The result (a) is essentially the same as the first result reached in Jacobi’s 
proof of 1849 (December) — a proof which Baltzer restates (§ 14, 7, 8) without 
