28 G Proceedings of the Royal Society of Edinburgh. [Sess. 
Baltzer, R. (1857). 
[Theorie und Anwendungen der Determent anten, . . . vi + 129 pp., 
Leipzig. French translation by J. Honel ; xii + 235 pp., Paris, 
1861.] 
Baltzer devotes a whole section (§ 15) of seventeen pages (pp. 80-96) to 
the subject of “ Die lineare, insbesondere die orthogonale Substitutionen.” 
The section, like its fellows, is noteworthy, not for freshness of matter, but 
for good arrangement, clearness and compactness. 
In treating of Cayley’s orthogonant (§ 15, 6) he takes l, not 1, as the 
constant element of the basic determinant : and, when in the course of the 
proof he obtains the two values for each of Cayley’s 6’s, he does not equate 
them, but uses with each of them Hermite’s observation 
+ $i = 2 16 i , 
thus reaching the elements 
2ZL rr _ , 2lL rs 
A ’ A ’ 
of the desired substitutions without more trouble. On the other hand, he 
fails to note that Cayley’s O ’ s are so introduced as to ensure the equality of 
x W~ x i+ • • • an d ^i 2 + ^ 2 2 + • • • > an d thus he is led to prove propositions 
already established (§ 15, 5 ). 
Brioschi’s equation of August 1854 being denoted (§ 15, 9 ) by f(x) = 0, he 
multiplies f(x) by f(-x), and obtains for f(x).f(-x)/x n a skew determi- 
nant having each diagonal element equal to 1/x — x. This determinant being 
therefore expressible as a sum of squares when n is even, and as 1/x — x 
times a sum of squares when n is odd, the part of Brioschi’s proposition 
which asserts the unreality of the roots follows by a reductio ad absurdum. 
Salmon, G. (1859). 
[Lessons Introductory to the Modern Higher Algebra, . . . 
xii + 147 pp., Dublin.] 
In Salmon’s treatment of the subject (§§ 118, 139, 142, 156-7, 163-4) 
only two points call for remark. In the first place, “ orthogonal trans- 
