1910-11.] The Less Common Special Forms of Determinants. 329 
The final lemma used in the verification may be formulated thus : If from 
the n quantities x,, x 2> . . x„ and the Jn(n-l) others 
12, 13, .... In 
23 , . . . , 2 n 
nn 
there be formed n lineo-linear functions , namely , 
= aq.O + x 2 (\2) +# 3 (13) + . . . +2^(1^),^ 
/ 2 = -x r (12) + x 2 .0 + a? 3 (23) + . . . +x n {2 n), 
/ 3 = -05 1 (13) -x 2 (23) + x 3 .0 + . . . + a; n (3rc), y 
fn = -« 2 ( 2w ) -^ 3 ( 3w ) - • • • +X n-° 
then x 1 f l +x 2 f 2 + . . . +x n f n = 0 * It may be viewed as included in the 
identity 
X 1 
x 2 
x s . . 
x n 
12 
13 . . 
. In 
x i 
-12 
23 . . 
. . 2 n 
x 2 
-13 
-23 
3 n 
x s 
- In 
- 2 n 
-3 n . . 
X n 
or in the statement that Any quadric whose discriminant is a zero-axial 
skew determinant vanishes identically. 
Hirst, T. A. (1859, 1/9). 
[Question 489. (A determinant which vanishes for every order higher 
than the fourth.) JSfouv. Annales de Math xviii. p. 358 : (2) ix. 
pp. 561-563.] 
Hirst’s theorem is that if 
a rs = (a r + f3 r s) cos s<ji + (y r + S r s) sin s<j> 
then the determinant 
<h,s 
a l , s+1 • • 
a i , s+ra-1 
a 2,s 
®2,s+l • • 
• • ^2,s+n-l 
<*n,s 
^ n , s+1 • • 
• • ^n,s+n — 1 
* When the coefficient of x r in f r is not 0 but (rr), the result of course is 
X ifi+X 2 f 2 + • • • +x n fn = x i 2 (H) + ai 2 2 (22)+ . . . +xj(nn) ; 
and in this connection it may be well to recall a step in Hermite’s mode of effecting the 
automorphic transformation of a quadric (see under Orthogonants). 
