1906-7.] Dr Muir on Axisymmetric Determinants. 
163 
Baltzer (1857). 
[Theorie und Anwendung der Determinanten, mit Beziehung au£ 
die Originalquellen, dargestellt von Dr Richard Baltzer: vi + 129 
pp., Leipzig, 1857. French translation by J. Houel, xii-j-235 pp., 
Paris, 1861.] 
In five different sections (§ 3 , 8, 9 ; § 5 , 2 ; § 6, 2, 5 ; § 7 , 5 ; § 18 , 12) 
Baltzer gives attention to determinants whose elements a ih , a u are 
identical. In § 3 he notes that conjugate minors, as they came to be 
called at a later date, are equal, and proves Jacobi’s theorem regarding 
the differential-quotient of a determinant with respect to a non-diagonal 
element by using the fact that if u be a function of x and y , and y be a 
function of x, then 
du _ du + du dy 
dx dx dy dx ’ 
the whole matter being 
a — = A« + = A a + A* = 2A a , 
0Oa oa iJc 
where it will be observed 0’s only are employed. In § 7, 5 is given the 
result which we have already seen in Lebesgue’s paper of 1837, namely, 
that for a vanishing axisymmetric determinant 
-A - *: = A ** ) 
and there is thence deduced 
Aa : Ai 2 : Ai 3 : . . . . = \/A n : \/ A 22 : s/ A 33 : . . . . 
Lastly, in § 18, 12 he applies this to Cayley’s vanishing determinants of the 
year 1841 ; for example, to the determinant 
. 1111 
1 . cZ 12 
1 • <^23 d 24 
1 d is d 2Z . d 3i 
1 d u d^ J 34 
This being equal to zero, if we write [r , s ] for the cofactor of the element 
in the place (r , s ) , we have of course 
[12] + [13] + [14] + T15] = 0, 
and consequently also 
