1904 - 5 .] Dr Muir on General Determinants. 945 
cut off from the one matrix, and a corresponding number of 
columns from the other. Each of the lines in either one of 
the matrices so reduced in width as aforesaid being then 
multiplied by each line of the other, and the results of e 
multiplication arranged as a square matrix and bordered wicn 
the two respective sets of columns cut off, arranged sym- 
metrically (the one set parallel to the new columns, the other 
set parallel to the new lines), the complete determinant 
represented by the new matrix so bordered (abstraction made 
of the algebraical sign) will be the product of the two original 
determinants. ” 
In illustration he gives three forms for the product of 
a b , 
a [ 3 
ctiid. 
c d 1 
y 
viz. — 
aa + b/3 
ay + bS 
aa 
ay b 
2 
2 a b 
Ca + d/3 
cy + dS 
> 
Ca 
cy d 
2 
2 c d 
P 
8 . 1, 
a 
/? . . 
i 
y 
8 . . 
In regard to the 2’s which occur in the last form his remark is : — 
“Any quantities might be substituted instead of 2 .... , 
as such terms do not influence the result : this figure is 
probably, however, the proper quantity arising from the 
application of the rule, because .... the value of the 
determinant represented by a matrix of no places is not zero 
but unity.” 
In the case where the two given determinants are of the third 
order, say 
a 
b 
c 
a P y 
a 
V 
c 
and 
d p' y 
n 
a 
b" 
c" 
a P" y" 
he gives only the second and third of the four (w+1) possible 
forms, viz. — 
aa + bp ad + b/3' aa" + b/3" c 
a' a + b'/3 a a + b/3' ad' + b'p" c 
a' a + b"/3 d'd + b"f3' a" a" + b" /3" c" 
y i y" 
PROC. ROY. SOC. EDIN. — YOL. XXY. 
60 
