146 
Proceedings of tlie Royal Society of Edinburgh. [Sess. 
XIV. — Note on the Construction of an Orthogonant. 
By Sir Thomas Muir, F.R.S. 
(MS. received January 31, 1918. Read March 4, 1918.) 
(1) The essence of Cayley’s mode* of arriving at the formation of an 
orthogonal substitution lies, as is knoAvn, in the observation that if two 
sets of variables 
5 y j j ^ and ^ , Tj , £ , to 
be taken linear functions of a third set 
in such a way that the determinant of the coefficients is in the one case 
1 (the 
— a 1 li g 
— h — h 1 f 
~(J ~f 1 
and in the other the conjugate of this, then the first and second set of 
variables are orthogonally related. The indication given by Cayley at the 
outset of his paper as to how he was led to so interesting a result is far 
from satisfying, and one’s surprise is reawakened at the close when he 
gives his two examples of actual substitution, namely, 
1 + A 1 2 — /X 2 — V 2 
l+A 2 + /x 2 + v 2 
2(A/X — v) 
i + A" + /x - + v" 
2(vA + ju) 
2(A n + v) 
1 + A 2 + /X ;+ V 2 
1 + / a 2 -i/ 2 -A 2 
1 + A 2 + /n~ -f v 2 
2 (/xv — A) 
1 + A 2 + /x 2 + r 2 1 + A 2 + /x 2 + 
2(vA — /x) 
1 -T 2 + /x 2 + v 2 
2 (/xv + A) 
I +A 2 + /x 2 + v 2 
1 +v 2 -A 2 -/x 2 
1 + A 2 + /X 2 + V 2 ’ 
for the case of three variables, and something still more irregular-looking 
for the case of four. 
(2) The result naturally attracted immediate attention among mathe- 
maticians, and was soon taken over into text- books, where it has continued 
to hold its place until now, seventy years after its first appearance. Strange 
to say, however, the account of it given in such books remains practically 
unaltered and unextended; the latest advanced work on the subject, 
Kowalewski’s of 1909, devotes to it live pages that are little else than a 
paraphrase of the original. 
(3) One point connected with it that has received due attention in these 
* Journal fur Math., 32 (1846), p. 119 : Papers , vol. i, p. 332. 
