1903-4.] 
Dr Muir on General Determinants. 
69 
Jacobi (1833). 
[De binis quibuslibet functionibus homogeneis secundi ordinis 
per substitutiones lineares in alias binas transformandis, 
quae solis quadratis variabilium constant : una cum 
Crelle’s Journ., xii. pp. 1-69.] 
Jacobi’s mode of proving his theorem regarding a minor of the 
adjugate occupies § 6 (pp. 9-11). Temporarily denoting by X TO the 
left-hand member of the m th given equation 
a 1 lm) x 1 + a 2 {m) x 2 + • • • • + ai m) x n = ym i 
and by Y m the left-hand member of the derived equation 
f^mV 1 "h fimV 2 "1* * J U n A-X m . 
and explaining that by 
M ■ 
he means the coefficient of x r ~ l xf l • • • x n ^ in a certain specified 
expansion of U, he recalls his paper of the year 1829 on the 
“ discerptio singularis,” and affirms that he had there proved 
fore 
[xiX«...X»] 
ry* ry* , . . <yt 
sive etiam, quod idem est, 
[ Yl Y,..Y„] 
yilh '"Vn 
ac generalius 
r aj/iflj/z- - • x n sn i 
Lx^x/ 2 * 1 - • • • X/” +1 J 
^_r!+r 2 + • • +r n + 1 
X x X % • • • X n 
- Yj 1 Y 2 r ’ 2 • - • Y/" 
2 /i Sl+ 1 y 2 S2+1 - • • y n Sn+l - 
1 
y&f-yn 
designantibus r x , r 2 , . . . , r n ac % , s n numeros 
quoslibet integros sive positivos sive negativos.” 
