( 267 ) 
NOTE ON A DETERMINANT WITH FACTORS LIKE THOSE 
OF THE DIFFERENCE-PRODUCT. 
By Sir Thomas Muik, F.R.S. 
1. The determinant in question, when of the 3 rd and 4 th order, is 
abc 
bed 
cde 
def , 
a+b 
c+d 
ab+ac-^-bc 
bc+bd+cd 
cd-^-ce-^de 
de+df+ef 
the scheme of construction be 
b+c+d 
c-\-d-\-e 
d+e+f 
readily grasped when the scheme for 
the first row and the scheme for the second column have been observed. 
Clearly it is such that when of the n th order the number of variables 
occurring in any row is n — 1, and the total number of variables is 2(n — 1). 
2. Recalling the effect produced on a determinant by reversing the 
order of its rows, we see that the function under consideration, F say, 
is at most altered only in sign when the order of its variables is reversed : 
for example, 
F(a bed) = ~¥{d c b a), 
~F(abcd ef) = ¥(fed c b a). 
3. Performing on the 4-line determinant the operations 
rowo— row. 
we obtain 
¥{abcdef) 
—d (b-\-c)(a—d) bc(a-d) 
b—e (c J r d){b—e) cd(b—e) 
c-f (d+e)(c-f) de(c-f) 
= —(a-d)(b—e)(c—f) . F(6 c d e), 
= (d—a)(e—b)(f—c) .¥{bcd e), 
where it will be observed that the three factors on the right are got by ; 
as it were, subtracting 
from the row def 
the row abc, 
and the F on the right from that on the left by removing from the latter 
its first and last variables. 
