41 
1913-14.] The Theory of Bigradients from 1859 to 1880. 
An important observation made in passing is that any simplified 
remainder can be condensed into a single determinant : for example, the 
three just given are equal to 
a 
b 
ex 2 + dx + e 
a b 
A 
P 
qx 2 rx + s 
or 
• V 
B 
V 
! 1 
rx 2 + sx 
p a 
Bx 
a b c 
d ex 
a b 
c d Ax 
a b 
c dx + e 
a 
be A 
p 
q rx-rS 
or 
. . p q B 
• P 2 
r sx 
• P < 
q v Bx 
p q r 
s 
p q r s B x 2 
a b c 
d e 
Ax 2 
a b 
c d 
e Ax 
a 
b c 
d A 
V d 
r B 
• . p 
q r 
s Bx 
• V 2 
r s 
. Bx 2 
p q r 
s 
. Bx 3 
, 
where, be it remarked, the ultimate forms, namely, those explicitly involv- 
ing A and B, are Cayley’s of 1848. 
Cayley’s relation between any three consecutive ‘‘simplified remainders” 
is next given (pp. 140-142), the proof arising quite naturally and being 
mainly dependent on the equality a r _ 1 a r = Dr-i- Thus, taking the equations 
that indicate the nature of the division-process, namely, 
A = QiB-R, 
B = Q 2 Bi — R 2 
b] — Q3B2 — 1*3 
b-2 = “ B 4 
and substituting p r /a r for R r , we obtain 
cq • A = cqQj-B - pj 
a l a 2 ’ B = a 2 Q2‘Pl “ a ]*P2 
a 2 a 3 -pi = a 1 a 3 Q 3 *p 2 - cqcq-pg 
a 3 a 4’P2 = a 2 a 4Q4'P3 “ a 2 a 3'P 4 
In this way there results the general equation 
Or-lOr • Pr-a. = a r-2 a rQr’Pr-l “ a r _ 2 a r _ 1 -p r , 
Hr— 1 * Pr— 2 a r— 2 a rQr’Pr— 1 — ^r—2’Pr •> 
and thence 
