678 
Proceedings of the Royal Society of Edinburgh. [Sess. 
By way of proof we have only to note that the subtraction of a 2 from the 
diagonal elements of the determinant | r 1 r 1 r 2 r 2 . . . r n r n | is equivalent to 
putting a = 0 in the said determinant, and that 
| 1 2 ? 2 . • . r n i n la=o (A ) a= o — (A<z=o) • 
20. If | a n a 22 . . . a nn | , or A say, be a skew determinant with univarial 
diagonal , then 
Aj^ A/a A ^ 2 .... A-in 
A 2 i A 22 - A /a .... A 2n 
A n i A n2 .... A nw — A ; a 
This may be proved in two ways. We may either multiply A* by the 
determinant proposed for investigation, when we shall obtain 
. -a 21 A/a - a 31 A/a .... 
- a 12 A/a . - a 32 A/a .... 
-a 13 A/a - a 23 A/a . .... 
which evidently 
A n j 0 when n is odd, 
( — a)" I F 2 when n is even, 
whence the desired result at once follows. Or, taking a hint from the 
form of the result, we may multiply the adjugate of A by A a=0 with equally 
good effect. 
21. If | a n a 22 . . . a un | , or A say , be a skew determinant with univarial 
diagonal, then 
An — A/a ^(A-i2 + A 2 i) .... J (A Xn + A„i) 
i(A 2 i + A i2 ) A 22 — A ja .... J(A 2n + A n2 ) 
i • • 
i i(A*i + Ai„) J(A w2 + A 2n ) .... A, m — A /a 
This is obtained by squaring both sides of the result of § 20 and then 
dividing by ( — A/a) n . 
22. Skew determinants of odd order have properties to which there is 
nothing analogous in the case of those of even order. To one or two of 
these attention will now be directed. 
j 0 when n is odd , 
(xxvi.) 
( A n-2 F 4 /( - a) n when n is even. 
0 when n is odd. 
(xxv.) 
A n 1 F 2 /( - a) n when n is even. 
