1901-2. j Dr Muir on the Theory of Jacobians. 
189 
quomodo omnibus b k evanescentibus neque ipso B evanescente, 
simul cuncta ilia Determinantia evanescant.” * 
Continuing the work of deduction, Jacobi lastly equates the 
'df 
cofactors of — in the two members of the identity 
OX 
yb + y 1 b 1 + • • • +fi m b r 
B - 2 ± a i 
+ d JL . d A 
df n 
dx m dx 
m + 1 
02L 
noting that this differential quotient does not occur at all on the 
right-hand side, nor in the /x’s on the left-hand side, but in b k occurs 
with the cofactor 
df df df 2 0/ w _ 1 
-2 
dx n+k dx x dx. 
dx, 
n - 1 
The result is the proposition! — 
“Sit jx k functionum /, f , . 
Determinante 
df df 
dx m . dx m 
, f n _ l Determinans quod in 
V ± f— 
¥n 
dx „ 
per 
¥n 
dx„ 
multiplicatur, ubi m < n , erit 
+ /h 2 j ± 
+ /V_2, ± 
The case where m — n. is specially noted 
¥ dfdf 
.. Ai 
dx n dx 1 dx 2 
df df df 2 
. . A =1 
dx n+1 dx Y dx 2 
¥ df\ df 2 
dx n+m dx 1 dx 2 
" a ~=° 
-i 
* Of course this theorem also is not limited to determinants having 
differential-quotients for their elements. The general enunciation may be 
put as follows : — If m determinants of the n th order all have the same n - 1 
columns in common , and vanish independently, then every determinant of the 
n th order whose n columns are chosen from the m + n - 1 different columns must 
vanish likewise. (Vide Proc. Roy. Soc. Edin. , xviii. pp. 73-82.) 
f This proposition, and that from which it is derived, are again propositions 
which hold regarding determinants in general, the class to which they belong 
being that which concerns aggregates of products of pairs of determinants, — a 
class, the first instances of which occur in Bezout (1779). In connection with 
Jacobi’s remark regarding the case where m — n, it is worth while to note 
Sylvester’s enunciation in Philos. Magazine (1839), xvi. p. 42. 
