434 
Proceedings of Royal Society of Edinburgh. [sess. 
“ II. Evanescentibus elementis omnibus. 
m (m+l) 
a k > a k ’ 
a (n) 
> a k 
in quibus respective index inferior Tc indicibus superioribus 
m, m + 1 , . . . , n minor est, fieri (vi. 7 ) 
y + an ’n ." f n) - . y + ' A™- 1 ) 
Zj±aa 1 a 2 . . . . a n - a m a m+1 . ... a n Zj±aa x . ... a m _ x . 
“ IY. Evanescentibus elementis omnibus, 
„(»») _( ot + 1 ) 
/») 
> 
in quibus indices inferiores superioribus minores sunt, si 
insuper habetur, 
a (m) = a (m+ 1) . _ . _ a W_l 
m m+l 
fit 2± Ipi 
. ... a 
As immediate deductions from tbe definition these are somewhat 
out of place, the trouble of demonstrating the first of them being 
virtually thrown away. The trouble taken by Jacobi, too, was less 
than required, the question of sign, for example, being inadequately 
discussed. 
In the course of the next section which deals with what we have 
called the recurrent law of formation, and with the vanishing 
aggregate connected with this law, Jacobi gives an expression for the 
complete differential of a determinant, the elements being viewed as 
independent variables. The passage is (p. 293) : — 
“ Determinans R est singularum quantitatum a respectu 
expressio linearis, atque ipsius a® coefficientem, qua in deter- 
minante R afficitur, vocavimus ; unde adhibita differen- 
tialium notatione ipsum A^ exhibere licet per formulam, 
3. A®- 
3B 
Hinc si quantitatibus incrementa infinite parva tribuimus, 
da<%, 
simulque B inerementum <7B capit, fit 
4. = ®<to®, 
(lvi.) 
