1910-11.] DrJMuir on Boole’s Unisignant. 453 
J ato. O 
10. Again, by|performing on the original determinant the operations 
rowj - row 4 , colj - col 4 
we obtain 
d -\ - e + f + h d + e d+f 
d + e a + c+d+e a + d a + c 
d +f a ~ d a ■+■ b + d -H f a + b 
a + c a + b a + b + c + g , 
and are thus led to elaborate the result 
(a + b + c + g ) . ( def+deh + dfh + efh) 
+ (ab + eg ) . (d + f)(e + h) 
+ ( ac + bg ) . (d + e)(f + h) 
+ (ag + be ) . (d + h)(e + /) 
+ ( abc + abg + aeg + beg ) . (d + e+f +h) 
where the factors on the left of the multiplication dot are functions of 
a ,b ,c ,g , and those on the right are functions of d ,e,f,h. As this result 
cannot be altered by the interchange 
a ,b , c , g = d , e ,f , h - a,b ,c,g 
we learn from making this interchange that 
(ab + cg ) . (d+f)(e + h) j / (de +fh ) . (a + c)(b + g) 
+ (ae + bg) . (d + e)(f+h) l = +(df+eh) . (a + b)(c + g) (XI.) 
+ (ag + be) . (d + h)(e +f) J ' + (dh + ef) . (a + g)(b + c) 
+ ±ag . ef { 
+ 4 be . dh £ 
11. By subtracting the first row of the original determinant from each 
of the other rows, thereafter increasing the first row by the sum of the 
others, and finally diminishing the first column by the sum of the others, 
we obtain the interesting alternative form 
2 (a + h) 
a- e 
a-f 
a-g 
b-h 
b+f 
b + g 
c-h 
c + e 
c + g 
d-h 
d + e 
d +f 
(XII.) 
This, by monomialising the elements of the first row and first column, 
becomes 
- ha 
. 1 1 1 
1 . b+f b+g 
1 c + e . e + g 
1 d+e d+f 
+ h 
e 
c + e 
d + e 
1 
/ g 
b+f b+g 
c + g 
d +f 
1 1 
b+f b+g 
e c+e . c+g 
d d + e d + f 
e f g 
b . b+f b + g 
c c+e . c+g 
d d+e d +f 
+ a 
