630 J. DOUGLAS HAMILTON DICKSON ON A CLASS OF DETERMINANTS. 
as BoyeOuG, = Urei@an| os) ek es On eae ; ; : . &F 
DirgerDa) Or sOyep 
Gy, Ay Ay, MU, An+s 
Ors Bees 203s ge Ae 
&, Gn, a3, A, As ,. Ania 
by, by , bs , OS bs, Gon 

aiBiysOoeeen = — Vicide, | . . . dy, by, bs, Bare : : . @& 
Wis (On ad gids, cy 
D,, bo, bs, by, Ones 
thy Oy Os Ge Oa 
by, De, Ds, dy, 05, Ono 
OS Gye) Og Gas Gye Gs, Opes 
bi, bas bs 5 bz , bs; Bs, Dix, 
where 
@ =a,—),, B=bh—c4, MW=a—-d, &e. 
From these six equations, (1) to (6), the determinants employed may be — 
defined—as, for example, in (6)—by the symbol A, ,,., 7 being the order — 
of the determinant, and +2 the suffix of the last constituent in its first line. 
By working in like manner with the determinants A,, ,,, and As,.; ,4,, it | 
can be shown that the law holds good generally. 
(IV.) Writing 2 = 2 in the above equations, the determinants there written — 
become of the forms of those proposed to be examined, and may be represented — 
simply by A,,, where mm is the order of the determinant. % 
Substituting for a,, @,, .. . their values from equations (A), (B), . . . the ;: 
equations (1) to (6), with one more added for symmetry, become 
= Ai 
A G = As 
7) Bo ad, = the 
Oe Bo Yyat= Aa B; ' ; : (7) 
ay Bo Yo os i = AY. OG 
ap Bo Yo So M= Ac IG 
a, Bo Yo O Goda = A, 6 & aes 
&e. &e. 
where the sign depends only on the suffix of A, viz., where this is = 3 (mod. 4) 
the sign is —, but otherwise +. 

