1888.] Prof. Anglin on Theorems mainly Alternants. 889 
Let all tlie a’s be put equal to 1, the If a equal to a, b, c, . . . , 
respectively, the c’s equal to a^, b^, c^, . . . , respectively, &c., and 
the Vs equal to . . . , respectively. Then the 
expressions 
. ' I ' --r and — V 7 i ’ 
^2^3 1 ^2^3 I ^2% 1 ^'2^3 ! 
denoted by (23), become respectively 
bc{b - c) and (b + cy{b - c ) , 
with similar expressions involving a, c and a, b for (13) and (12) re- 
spectively ; while the expression which is equal to (23) - (13) -f (12), 
VIZ,, 
1 <^2^3 i 1 ^^1^3 I I ^1^2 I 
Hence we see that 
1 a A 
1 b B 
1 c C 
, becomes, by § 4 of former paper, \a%~^c^\. 
= 1 I or l\abc) 
(T)n 
where A denotes be, or (b c)^ ; with similar corresponding expres- 
sions for B and C. 
Again, in the case of four letters a, b, c, tZ, it will be found on 
reduction that the first, third, and second expressions denoted by 
(234) become respectively 
hcd^\bccl\ (b + c + dfCibcd), 
h h 
3 
1 Aj 
H 
1 b 
1 /7i 
lh{bcd) , 
where h refers to b, c, cZ, and H 
2^S 
P 
1 b 
1 h-^ 
stands for 
1Z> 
IZ^i 
Ic 
1/q 
IcZ 
\h. 
\ab c d P 
while the expression ^ i — becomes | a%'^chl^ | or f^{abcd). 
Hence we see that 
la a‘^ A 
lb b^ B 
1 c c2 (7 
Id df D 
= I I or ll{abcd) 
(ii), 
where A denotes bed, h-^, or 
Zq /^2 
3 
1 h. 
H 
1 b 
1 Zq 
, that is to say any one of 
