216 Proceedings of Royal Society of Edinburgh. [sess. 
To the proof no note is appended drawing attention to the fact 
that the very same result would have been reached by taking 
( - l) r_1 , or indeed ( - l) r ”*, instead of ( - l) r for the sign-factor 
of (2,3,..., r- 1, r+ 1, . . n ). 
The very next step taken, in accordance with the above men- 
tioned dictum, is to make the substitution in the right-hand side of 
the equation 
\/ A = ®12 \/A 22 “b ^13 \/ A 38 "b . . . + A. nn , 
the first term being used to decide whether (1,2,3, . . ., n) or 
- (1,2,3, . . ., n) has to be substituted for the left-hand side, and 
the final result being 
(1,2,3, . . ., n) = a 12 ( 3, . . ., n) + a ]3 ( 4, . . ., n, 2) + ... + a in ( 2, 1). 
Since (3,4, . . n) is the cofactor of a l2 in (1,2,3, . . ., n) and 
the differential-quotient of the latter with respect to a l2 is the 
same, it immediately follows from this that 
_ A/A 
J A - a u g ai2 
+ -u S A + 
da 13 
-j- a 
In 
djA 
da 
in 
Baltzer, however, obtains a more general result by going back to 
the corresponding more general theorem in determinants, viz., the 
theorem 
A — ■A./’j 
with which he associates 
- 1 - (ty 2 A'f 2 + . . . + 
rn j 
0 b ^ 2^-52 b • • • b a^fi A-ski , 
substituting J/\ 
In the results, 
^A for j and then dividing both sides f A. 
da rs 
^ = + 
0 = arrjf— + 
da sl 
+ a rn 
+ a r n 
dj A 
da rn 
dj A 
dao n 
it has to be noticed that there is no term in dj A !da rr . 
By comparison of the first of these with the immediately pre- 
ceding result (the recurring law of development), he deduces the 
quite general identity regarding the two forms of the cofactor of 
