700 
Proceedings of the Royal Society of Edinburgh. [Sess. 
when performed on a homogeneous linear function of a, b, c, . . . is equi- 
valent to a substitution. 
The case where the two given arrays are identical is formulated, due 
care being taken with the differential-quotients because of C becoming 
axisymmetric. 
We have only to add that the form in which this new theorem of 
Casoratis is stated obscures to some extent its significance. If we write the 
case of AB = C where m — 3, n — 4 in the form 
«i 
«2 
a 3 
a 4 
ii 
h 
h 
h 
lal 
lam 
lan 
h 
m 2 
m s 
m 4 
= 
Ibl 
Ibm 
Ibn 
c \ 
C 2 
c s 
C 4 
n i 
n 2 
n z 
n 4 
Icl 
1cm 
len 
then, freed from all reference to differentiation, the theorem for the case 
r — 2, 8 — 3 is 
1 lal 
lam 
lan 
\ Ibl 
Ibm 
Ibn i . 
lal 
lam 
Id 
1cm 
len 
Id 
1cm 
lal lam lan 
lal lam a 4 
lal lam lan 
lal lam a 2 
l 4 m 4 n Y 
Ibl Ibm b 4 
+ 
l 2 m 2 n 2 
Ibl Ibm b 2 
Icl 1cm len 
Icl 1cm c 4 
Icl 1cm len 
Icl 1cm c 2 
lal lam lan 
lal lam a 3 
lal lam lan 
lal lain a 4 
h m 3 n s 
Ibl Ibm b 3 
+ 
h m 4 n 4 
Ibl Ibm b 4 
Id 1cm len 
Icl 1cm c 3 
Icl 1cm len 
Id 1cm c 4 
Further, no change but substitution is necessary on passing to the case 
where the two arrays are identical. 
Salmon, G. (1859). 
[Lessons introductory to the Modern Higher Algebra. By the 
Rev. George Salmon, A.M. . . . xii-f 147 pp. Dublin.] 
The first three lessons (pp. 1-18) of this historically interesting text-book 
are devoted to an elementary exposition of determinants. The only fresh 
matter (§ 20) concerns the determinant formed from 
a i h 
a i Pi 
(%2 ^9 
a 2 P« 
a 3 b 3 
3 
CO 
<3. 
CO 
by row-by-row multiplication. This is shown to vanish, not by pointing out 
that it contains at least one zero determinant of the third order as a factor, 
