515 
1908-9.] Dr Muir on the Theory of Jacobians. 
The next fresh paragraph (p. 91) appears, although unnecessarily, as an 
addendum to Jacobi’s solution of a set of simultaneous linear equations 
whose determinant is a functional determinant (Be determ. fund. § 8). If 
the square of R be obtained by row-by-row multiplication, and the square 
of S by column-by-column multiplication it is easily verified that 
(h th row of S 2 ) x (7t ,tb column of R) = °— , 
dh 
i.e. = (k , 7i) th element of S , 
thus incidentally giving S 2 R = S as it should do. From this it is deduced 
that 
(h th row of S' 2 ) x (& th row of R 2 ) = 1 or 0 
according as h and k are equal or unequal, * and that therefore R 2 and S 2 
as just defined are in the matter of their primary minors related as R and 
S have been shown to be. 
In the remaining fourteen pages (pp. 92-106) the only matter calling 
for attention concerns Jacobi’s theorem 
where 
d/n+m \ 
7)np / 
B=y ( 
^ \ dx dx 1 
dfn 
dx. 
n — 1 
and bW 
2 
dx dx-. 
¥n- 1 of n 
+i 
dx n - 1 dx n+T . 
From this Brioschi, by taking the /’ s to be linear functions of the x’&, 
obtains Sylvester’s theorem of March 1851 regarding a compound de- 
terminant. 
Bellavitis, G. (1857, June). 
[Sposizione elementare della teorica dei determinant!. Memorie . . . . 
Istituto Veneto .... vii. pp. 67-144.] 
To the subject of a “ Determinante formato colle derivate-prime di 
alquante funzioni di altrettanti variabili ” Bellavitis devotes nine and a 
half pages (pp. 52-61, §§65-78), that is to say, about the same as Spottis- 
woode, though his selection of theorems is not quite the same. In 
substance he gives nothing fresh. His symbolism for the determinant of 
u , v , . . . with respect to x , y , . . . resembles Cauchy’s of 1841, being 
| D x u , D y v , . . . | ; 
other changes made by him in notation are less satisfactory. 
* Viewing R and S as matrices of which the conjugates are R and S we have as an 
equivalent of this 
SS-RR-SqSR-R 
= SR=RS = 1. 
