1888 - 89 .] Dr T. Muir on the Theory of Determinants. 
435 
siquidem sub signo summatorio utrique indici i et h valores 
0, 1, 2, . . . , n conferuntur.” 
The recurrent law of formation and its dependent neighbour 
formula he is enabled, by means of (3), to view as the partial 
differential equations which the determinant must satisfy. His 
words are (p. 295) : — 
“ Substituendo formulas (3), inventas formulas sic quoque 
exhibere licet : 
A (i) 0R A'\ 0R 
9. R - + af- m + . 
da (,) 1 daf 
0R , 3R 
= a,, + a , - — — + . 
. + d 
(*). 
da. 
k da' 
lc 
0R 
a«®’ 
n 
(n) d R 
' ’ + a t daW ’ 
k 
-I A A (V'\ 0R ft'\0R 
io. o + + ' 
. . + d 
,n 0R 
ff) 
da 
(O’ 
A 0R , 0R 
0= a - — + a — + . 
+ . 
* 
Quae sunt aequationes differentials partiales quibus deter- 
minans R satisfacit.” 
Passing over a section (7) on simultaneous linear equations, and a 
short section (8) in which Laplace’s expansion-theorem is enun- 
ciated, we come to two sections dealing with what at a later time 
would have been called the secondary minors. No name is given 
to them by Jacobi ; they only appear as co-factors of the product of 
a pair of elements, the aggregate of the terms containing ^ as 
a factor being denoted by 
<T a T- K’fi- ( xll8 ) 
From observing that the interchange of / and f or of g and g' 
alters R into - R and cannot alter P/ ( j * , it is concluded that 
A /,J { 
9, 9 
9,d’ 
and that the full co-factor of A f ’ f , is <x (/) a^P - <fP <x (/) in accord- 
s', 9 9 9 9 9 
ance with the expansion-theorem of the previous section. The 
