310 Proceedings of the Royal Society of Edinburgh. [Sess. 
the square found for it by row-by-row multiplication is, in Cayley’s notation, 
(d,c, -b, - a ) (h,g, -/, - e) (l, k, -i) (p, o, -m, -n) 
(a, b, c, d ) 
( e,f, 9, h) 
( i » j i & ? 0 
(m,n, o,p) 
which is readily seen to be zero-axial skew. 
Another expression is, of course, got by treating the conjugate of A 4 in 
the same manner. 
Brioschi’s paper of 1855 is not referred to. 
55 55 55 55 
55 55 55 55 
55 55 55 55 
55 55 55 55 
Horner, Jos. (1865, Oct.). 
[Notes on determinants. Quart Journ. of Math., viii. pp. 157-162.] 
The second of Horner’s three notes consists of a fresh proof that a zero- 
axial skew determinant of even order, A 2m say, is the square of a rational 
function of the elements. 
A 2m multiplied by the square of the product of the non-zero elements 
of the first row is evidently equal to a zero-axial skew determinant of the 
same order, A' 2m say, having 0, 1, 1, . . . , 1 for its first row. But by per- 
forming in order the operations which we may conveniently specify by 
row 2m - row 2TO _ x , row 2m _! - row 2m _ 2 5 • • • 5 row 3 - row 2 , 
col 2m - col 2m _! , col 2m _ 1 - col 2w _, , . . . , colg - col 2 , 
it is seen that for A' 2m we may substitute a zero-axial skew determinant 
of the next lower even order, A 2m _ 2 say. The factor thus shown to connect 
A 2m and A 2m _ 2 being a square, the little that needs to be added is evident. 
LIST OF AUTHORS 
whose writings are herein dealt with. 
1855. Brioschi . 
PAGE 
. 303 
1862. Trudi 
PAGE 
. 306 
1857. Bellavitis 
. 304 
1863. Janni 
. 307 
1859. SCHEIBNER 
. 305 
1864. Cremona . 
. 307 
1860. SOUILLART 
. 306 
1865. Cayley 
. 309 
1860. Cayley 
. 306 
1865. Horner . 
. 310 
(Issued separately May 13, 1908.) 
