158 ROYAL SOCIETY OF CANADA 
2. Let 
Gi TOR RE D ea, + Oye ao ee 
DE PDPERIE SL mi he FA GER on A Oop ris ie le 
D 
n, p 
Oe. ei M sXe) Ber eee ete sen Sete ye) aoe) n'es se fe, © «ec ea 'e) eae) ee mientems 
Any + LEE 3 Pm >: . Ban + Onn» + °F Pan 
let A = | ns B= We aes P = | prol, 
yn 
and let 4 a, Le cy -.+.p_ denote the determinant formed from 4, as 
follows: The first a columns are taken from À, the next # columns are 
taken from B, the next y columns are taken from C, and so on; the last 
az columns being taken from P, with the proviso that no two columns 
thus taken have the same column number, £.e., come from corresponding 
positions, and where a+ 6+ y+....2 =n; then, as is well known, 
eee et es Pe (A) 
The generalization of Muir’s theorem I is as follows : 
Dy ae REED tee s - ne De ) 
ee El SOA Was rat Maly: ARRET er NAN TT (B) 
wheren-pSasSfPsy5....5 050, 
and where, since the subscripis of a,b... . p are by definition essentially 
positive, we are to interpret 4 a, 8 +++ pas zero in case of a nega- 
n, p-2 

tive subscript which occurs when p > n. 
To prove this theorem we have but to partition all the determinants 
in (B) with polynomial elements into determinants with monomial 
elements according to (A), and then it may be seen that the complete 
coefficient of any one of these determinants with monomial elements is 
zero. 
As illustrations we have for p = 3 and n = 3 
An = Oy = Cy Aya + Oy + Cy ys + Vis + Cis | 
dy, + ba, = Ca An + Oo + Co Us + Do, + Cog 
Ay, + ds + Co Age + O52 + Co sg À Deg = Css 
a dy + bn Ay + by dis + by | 
Fa | Ay, + dy, Ax + do An, + Vs | 
Ay + by, As2 À Us A + Vs | 
| ay, Ayo a1 
a | du 922 Az 
| 43, yo gg 



