1914 - 15 .] The Determinant of an Orthogonal Substitution. 55 
The determinant got by deleting the h^^' and columns of an ortho- 
gonant and inserting the and rows is equal to that got by deleting 
the and rows and inserting the and columns. (II') 
If any pair of columns of an orthogonant to the base cr be multiplied 
by each pair of rows, the sum of the squares of the resulting products 
is (III') 
The sum of the squares of the Jn(n— 1) determinants formed from an 
orthogonant of base a by interchanging a fixed pair of columns with every 
possible pair of rows is (IT') 
In proving these it is readily seen that there exists a corresponding set 
of theorems for the case where the number of substituted rows is three or 
more, the proofs being only a little more troublesome to exhibit. 
( 4 ) Connected with any 7?.-line determinant there are n entities which 
have not received as yet sufficient attention from students, namely, the 
sums of the coaxial minors of like order ; and as in the study of ortho- 
gonants these sums become more than ordinarily prominent, it is con- 
venient to have a short notation for them. Let us therefore denote the 
sum of the r-line coaxial minors of a determinant by 
Saxm,. ; 
so that, for example, in the case of the determinant | afpgd^ | we shall have 
Saxnij = + ^3 + ^^4 > 
Saxiri2 = I \ + | ] -1- . . . . + | \ , 
Saxnig = I | + . . . . + \h^cgl^ \ , 
Saxm^ = I \ . 
( 5 ) If each element of any determinant be multiplied by its conjugate 
element, — or, what comes to the same thing, if each row of any determinant 
be multiplied by the corresponding column, — the sum of the resulting 
products is Saxm^^ — 2Saxm2 . (V) 
Taking the determinant | af^cgd^ \ , the sum of the products in question 
is evidently 
af + -1- 2agCj + 
-f- hf + “t" 
4 - ef + 2 c^c ?3 
+ df, 
and the simultaneous addition and subtraction of 
^af)^ + 2ajCg -f 2a^cC 
4 - 262^3 + ^ 
+ 2cgri4 
