282 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Spottiswoode, W. (1853, August). 
[Elementary theorems relating to determinants. Second edition, re- 
written and much enlarged by the author. Crelle’s Journ ., li. 
pp. 209-271, 328-381.] 
In trying to insert in liis second edition an alternative process for 
establishing Cayley’s result of 1846, Spottiswoode is very unfortunate. 
The place selected by him is immediately after the sentence defining skew, 
and therefore immediately preceding the former process ; but in making 
the insertion (p. 260) the predicate of the important sentence in question 
has suffered excision, along with a very necessary explanation regarding 
the diagonal elements of the initial determinant. Further, at the utmost 
all that is established is the fact that the determinant of Cayley’s sub- 
stitution is equal to +1. Such neglect, however, can well be overlooked in 
view of certain deductions which he records, and which he says can be 
made from Cayley’s result. These may be enunciated in more modern 
form as follows 
If | a u a 22 . . . a nn | or A be a unit-axial skew determinant, 
I A n A 22 . . . A nn | its adjugate, and | co n oo 22 . . . oo nn \ Cayley’s orthogonant 
formed therefrom, then 
Ak + A| r + . . . + Al r = A rr . A , ) 
A^ A ls + A 2 ,A 2s + . . . + A nr A ns = i(Ars + A sr ) . A j 
and 
a lr (x) lr 4" (^ 2 r w 2 r 4" • • • 4" a nr (xi nr — d rr | (/3) 
a ir Oy ls 4" CL2 r (JL>2s + . . . + & nr (i) ns = CL rs j 
The former (a) belongs strictly to the theory of skew determinants, and 
has already been noted in its proper place. 
Cayley, A. (1853, November). 
[On the homographic transformation of a surface of the second order 
into itself. Philos. Magazine, vi. pp. 326-333 : or Collected Math. 
Papers, ii. pp. 105-112.] 
Here Cayley recalculates the general orthogonant of the 4th order, 
taking note on passing of the related identity 
(x + vy - /jlz + aw ) 2 + ( - vx + y + Az + biv ) 2 + [jxx - ky + z + cw ) 2 + ( - ax - by - cz + iv ) 2 
= x 2 + y 2 + z 2 + w 2 + (vy - yz + aw ) 2 + ( - vx + kz + bw ) 2 + (yx - \y + cw ) 2 + (ax + by + cz ) 2 
