450 Proceedings of the Royal Society of Edinburgh. [Sess. 
We also see from it that the proposition as above stated on the lines 
of the original is over-conditioned, it being unnecessary to stipulate that 
the multipliers in all the diagonal elements shall be positive. What is 
required is merely that this shall hold in regard to the first of these 
elements. (II.) 
Further, we learn that the number of terms in the development cannot 
exceed O n+mtn where n is the order-number of the determinant and n-\-m 
is the number of variables. (in.) 
4. Boole’s second proposition is in effect nothing more than the 
statement that a determinant of the special form exemplified in § 1, and 
which, without loss of generality, we may write 
CL-^-b-\-e^rd-\-e-\-f-\-g-{-h cl + c + d + e cl b d f cl + b + e + g 
a + c + d + e a-\- c + d-\-e a + d a + e 
cl -I - b + d + f cl + d cl + b + d -l- f cl + b 
a+b+c+g a + e a+b a+b+e+ g , 
is included as a special case of the determinant dealt with in his first 
proposition. That this statement is justified is at once seen by making it 
more definite, namely, by saying that the special case is that in which the 
values of the a’s, /3’ s, . . . are given by the matrical equation 
( 
' «i 
a 2 • • 
• • a 8 
) = ( i i i i 
1111 
ft 
. . 
00 
111. 
1 . . . 
7i 
y 2 • • 
■ ■ % 
11.1 
. 1 . 
Sx 
s 2 .. 
• • s 8 
1.11 
. . 1 . 
and in which, therefore, 
the final development is 
abed 
1 1 
1 1 
2 + 
+ efgh 
1111 
1 1 
1 . 
1 . . . 
1 1 
. 1 
. 1 . . 
1. . 
1 1 
. . 1 . 
5. The special determinant of § 4 being denoted by 
(B(a ; b,e,d ; e,f,g ; h) or B(a , bed , efg , h) 
it is readily shown by interchanging pairs of the last three rows, and 
subsequently corresponding pairs of the last three columns, that it is the 
same as any one of the five 
B(a, bde, egf, h ) , 
B(a , cbd, f eg, h ) , 
B(a , cdb, fge, h) , 
B(a , d b c, g ef , h ) , 
B(a, deb, gfe, li ) . 
