54 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
V. — Properties of the Determinant of an Orthogonal 
Substitution. By Thomas Muir, LL.D. 
(MS. received October 21, 1914. Read November 16, 1914.) 
(1) If we seek to extend the definition of an orthogonant by making 
the square of each row equal to o- instead of 1, this fixed quantity will 
naturally be found to appear in the statement of almost all the properties 
of the determinant in question. It is therefore convenient and appropriate 
to call it the base of the orthogonant, and to speak of an orthogonant to 
the base a- or an orthogonant of base cr. 
It is a simple matter to ascertain what modifications are necessary in 
the statement of the known properties of the ordinary orthogonant to 
make them applicable to the extended definition. 
(2) The following additional theorems, which concern the effect of the 
substitution of a row for a column, need only be enunciated : they are 
deductions from more general propositions which have been established 
elsewhere ; — 
If the h^^ column of an orthogonant to the base a be deleted, and the 
p''* row be inserted in its place, the resulting determinant is equal to 
(rowp . colj,). (I) 
The determinant got by deleting the h^'* column of an orthogonant and 
inserting the p^^ row is equal to that got by deleting the p''" row and 
inserting the h^'* column. (II) 
If any column of an orthogonant to the base a- be multiplied by each 
row, the sum of the squares of the resulting products is cr^. (HI) 
The sum of the squares of the n determinants formed from any 
orthogonant of base a- by interchanging its h^^' column with all its rows in 
succession is equal to (t”. (I^) 
(3) If we proceed to the substitution of two rows for two columns, the 
following are the corresponding results : — 
If the h^^ and k^^ columns of an orthogonant to the base cr be deleted, 
and the p^^' and q^^ rows be inserted in their places, the resulting deter- 
minant is equal to 
roWp 
cob 
rowg 
cob 
(!') 
