118 
MAJOR P. A. MACMAIIOR ON A CERTAIN CLASS OF 
This proves the proposition and also shows that the determinant 
I (^3 ^ 3 ^ 2 ) ■ ■ ■ ! 5 
of the order, vanishes identically. 
Art. 9 . Hence, of order 3 , we have the identity 
1 
(l-SlXA(1 -S„Xo)(l-S3X3) 
__ 1 _ 
I (1 1 
multiplied by 
1 1 h 1^1 ■ ^2 (Aq ^^3^2) I ' ^3 (^3 I ^ i^-A I IA3 (X3 I 
-h "*■ l-vX. 1-vAA (1 - (1 - .S3X3) 
1 (X3 0^3) (X| — I I (Ai ^9^1) (Aq ^ro^-o) | 
(1 S 3 X 3 ) (1 SJX;^) (1 %X;^) (1 S 2 X 2 ) 
and, of order n, the identity 
_ 1 _ 
(1 - s^Xp (1 - S3X2)... (1 - s«X„), 
1(1 
1 
-- —5 
^£'^2^2) * * ' I 
multiplied by 
1 + 
^1% 1 (^1 ^9^1) (^2 ^3^2) I 
1 - s,x, 
+ s 
Sj3'2 
(1 — SjX^) (1 — § 2 X 3 ) 
St I (X-j ^9*^1) (Ao ^^^2) • • • h’O) I 
(1 — s^Xj) (1 — S 3 X 2 ) ... (1 — StXt) 
3 
the last batch of fractions involving, each, n — 1 denominator factors, and the numbers 
of fractions, under the summation signs, being in order 
Moreover, it has been shown that the fraction 
_ 1 _ 
I (1 — (1 - Z) 23 yr 2 ) ... (1 — n„s„x„) | 
is the condensed form of the fraction 
_ ]. 
(1 ^1-^1) (.1 ^21^2! • • • (.1 — 9 iX„y 
or we may regard the latter as a redundant form of the former. 
