(317) 
NOTE ON A THEOREM REGARDING A SUM OF DIFFE- 
RENTIAL-COEFFICIENTS OF PRINCIPAL MINORS OF A 
JACOBIAN. 
By Thomas Mum, LL.D., F.R.S. 
(Read March 17, 1909.) 
1. The theorem in question is that which plays so important a part in 
Jacobi's method" of determining the last integrating factor for a set of 
differential equations, and which he accordingly styled his "fundamental 
lemma." It may be enunciated in simple manner as follows : If A^, A^, 
A^ be the co-factors of the elevmits of any roio of a Jacobian tuhose 
independent variables then 
DA, DA 
+ 
+ 
0. 
2. Of this theorem in 1854 Donkin gave a very peculiar demonstra- 
tion.! Separating the symbols of operation 
Dx^' Dx^ 
from the subjects operated on, which in the case of ?z. = 4 may be written 
Du 
Dx^ 
Dx^ 
Dw 
Dx, 
Du 
Dx, 
Dv 
Dx^ 
Diu 
Dx, 
he was, of course, enabled to put the left-hand member in the form 
D 
D 
D 
D 
Dxi 
Dx2 
Dx^ 
Dx^ 
Du 
Du 
Dii 
DXj: 
Dx^ 
^ 
Dx^ 
Dv 
Dv 
Dv 
Dv 
Dxj_ 
Dx^ 
Dx. 
Dx^ 
Diu 
Did 
Dio 
Diu 
Dx, 
Dx^ 
Dx^ 
* -Jacobi, C. G. J., Theoria novi multiplicatoris systemati sequationum differen- 
tialium vulgarium applicandi. Crelle^s Journal, xxvii., pp. 169-268; xxix., pp. 213- 
279, 333-375. 
t Donkin, W. F., Demonstration of a theorem of Jacobi relative to functional 
determinants. Cambridge and Dub. Math. Journ., ix., pp. 161-163. 
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