1908-9.] Dr Muir on the Theory of Jacobians. 
499 
XXXII. — The Theory of Jacobians in the Historical Order of 
Development up to 1860. By Thomas Muir, LL.I). 
(MS. received March 22, 1909. Read June 7, 1909.) 
My last communication in reference to the history of Jacobians dealt with 
the period 1815-1841 ( Proc . Roy. Soc. JEdin ., xxiv. pp. 151-195). The 
present paper continues the history up to 1860. 
Jacobi, C. G. J. (1844-1845). 
[Theoria novi multiplicatoris systemati sequationum differentialium 
vulgarium applicandi. Crelles Journ., xxvii. pp. 199-268 ; xxix. 
pp. 218-279,333-376: or Math. Werke (1846), i. pp. 47-226 : or 
Gesammelte Werke, iv. pp. 317-509.] 
The portion of this long memoir which is of interest to us in the 
present connection is the first section (pp. 201-209) of the first chapter, the 
heading being “ Lemma fundamental eiusque varii usus : de determinantibus 
functionalibus partialibus.” Passing over the treatment of the first two 
cases of the lemma we come upon the general enunciation of it, which is — 
If A , A, , A 2 , . . . , A„ be the cofactors of 
K 
dx. 
V 
dx n 
in the 
ox„ 
determinant 
■■■<"), then 
'dxdxfdxf ' dx. 
BA 9A, 3A 0 
- — + — — - + — — ^ + 
dx dx 
dXn 
+ 
0A, 
dx„ 
= 0 
'1 w ^2 
Preparatory for the proof it is pointed out that since 
f A + y Al+ . . . + Aa„ I ..M 
dx OX} 0 X n \ OX OX} OX n J 
an alternative form of the lemma is 
0(/A) 0(/A 1 ) + _ _ + 0(/A b ) = y( ± hfdj\ _ _ _ 0A\ 
dx dx-} dx n \ dx dx v dx n J 
Then calling the given determinant R, and noting that A , A x , . . . , A n are 
themselves functional determinants, A* being the determinant of f \ , f 2 , ... , 
f n with respect to x, x x , ... , x { _ x , x x+l , . . . , x n , Jacobi seeks to prove the 
lemma true in the case of R from assuming it true in the case of A< . To 
be able to formulate it in the latter case he takes each element of the first 
