37 
1913-14.] The Theory of Bigradients from 1859 to 1880. 
where each array on the left is got from the one that precedes it by 
deleting the first row, the first column, and the last row ; and each array 
on the right by merely deleting the last row. It is noted that the leading 
determinant of the condensed array is axisymmetric. 
Lastly, it is pointed out that cases where m>n present almost no 
additional difficulty, as they are readily brought under the foregoing. 
Thus, if the case be 
we have only to take 
I (a, b, c, d, e, f) 2 I 
I (t, u, v) 5 j 
a b c d e f 
0 0 0 t u v 
for our generating array and proceed exactly as before, the results being 
a 
b 
c 
d 
e 
f 
r 
a 
b 
c 
d 
e 
f 
at 
au 
av 
t 
u 
V 
at 
au + bt 
av + bu 
bv 
t 
u 
V 
= 
at 
au + bt 
av + bu + ct 
bv + cu 
cv 
a z 
t 
u 
V 
au 
av + bu 
bv + cu 
cv + du — et 
dv - ft 
t 
u 
V 
av 
bv 
cv 
dv - ft 
ev - fu 
u v 
a b c d e f 
t u v 
t u v . 
t u v . 
t u v 
t 
u 
V 
t 
u 
V 
= 
t 
u 
V 
au 
av + bu 
bv + cu 
cv + du — et 
dv - ft 
av 
bv 
cv 
dv -ft 
eu —fu 
- 
t 
u 
V 
• 
t 
u 
V 
t 
u 
V 
. 
au 
av + bu 
bv + cu 
Cv + du — et 
dv -ft 
The requisite division by a 3 (in general a m n ) may be performed by 
removing the a’s one at a time, or by using the divisor in the form 
a b c 
a b . 
Another theorem of a similar kind but introduced for a different 
purpose, namely, for dilatation rather than condensation (pp. 129-131), is 
(*o 
K), 
(-r 
• • • , Or 
