513 
1908-9.] Dr Muir on the Theory of Jacobians. 
unrestricted symbol ^ is held to be equivalent to 
A+A + .... hA 
OXi OXi 0X L 
the determinant operating on n 2 u 3 . . . u, L has the first element of each 
column equal to the sum of all the other elements of the column, and 
therefore vanishes. The identity is thus thought to be established. 
In regard to this so-called demonstration we need only remark in 
passing (1) that the subject operated on is written in too product-like a 
form ; (2) that an appropriate substitute for it would be (iq , u 2 , . . . , u n ) , 
this being explained to be such that 
and 
0 r / 
^ — (zq , u 2 , . . 
ox s 
• 5 ^ n ) 
du r 
dx s 
5 
0 / 
. , u n ) 
0zq 
du 0 
+ — 2 + 
dx s 
dx s 
(3) that the assertion (A) is unsubstantiated, the fact that 
| «i b 2 c 3 | = zqj b 2 c 3 j - a 2 \ b l c 3 \ + a 3 \ c 2 1 
being nothing more than a suggestion that 
d_ d d 
dx 1 0a? o dx 3 
b\ b 2 b 3 
C l c 2 C 3 
may be a suitable abridged notation for 
- ' I ^2 C 3 I 
bx l 
ox 
0 I 7 ,0 
b 1 c 3 \ + 
dXn 
b\ c 2 1 • 
Brioschi, F. (1854, March). 
[La Teorica dei Determinant^ ; viii + 116 pp., Pavia. French 
translation by Edouard Combescure, ix + 216 pp., Paris, 1856. 
German translation by Schellbach, vii + 102 pp., Berlin, 1856.] 
Like Spottiswoode, Brioschi devotes his tenth chapter or section (§ 10) 
to “ determinant delle funzioni ” ; but his exposition is much more extensive 
(pp. 84-106), and, although of course he follows in Jacobi’s footsteps, he 
does so less closely than Spottiswoode. 
Thus the fact that the cofactor of in R is R^ : , 
dx k dfi 
VOL. XXIX. 
a fact which we 
33 
