668 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
XL. — The Superadjugate Determinant and Skew Determinants 
having a Univarial Diagonal. By Thomas Muir, LL.D. 
(MS. received June 21, 1909. Read July 12, 1909.) 
1. The object of the present paper is to make some advance in the theory 
of those skew determinants that have the elements of the diagonal all equal. 
A few results on determinants of more general form and on related subjects 
are given incidentally. 
2. It is recalled as a preliminary that Cayley’s first paper on skew 
determinants ( Crelles Journ., xxxii., 1846, pp. 119-123) really concerned 
something else, namely, the construction of a general orthogonant. Taking 
a skew determinant | a n a 22 . . . a nn i , or A say, having 1 for each of its 
diagonal elements, he formed a square array with 
2ci ss K r s - n j.j ie p} ace ^ 
and ~ ( 'h'r^ rr - 1 in the (r,r) th place , 
and affirmed that these n- elements satisfied the conditions laid down for 
the coefficients of an orthogonal substitution. As a result it followed, of 
course, that 
2a 22 A 12 
2a 33 Ai 3 .... 
2cq^A 2 i 
2a 22 A 22 - A 
2<%A 23 .... 
= A w 
2a n A 3 i 
2a 22 A 32 
2a 33 A 33 — A .... 
< %rs — - Ctsr 
CLrr— 1 
Now, there is nothing in Cayley’s construction that limits its application to 
a unit-axial skew determinant, and it is therefore suggested to inquire 
what the construction leads to when the elements of the originating deter- 
minant | a n a 22 . . . a nn | have no conditions whatever attached to them. In 
other words, the determinant just written at length, and found in the 
particular case specified to be equal to A n , is like the adjugate a derivative 
of any determinant, and as such is worthy of investigation apart altogether 
from the peculiar circumstances under which it first made its appearance. 
In fact, as each element of it is a degree higher than each element of the 
