of Edinburgh, Session 1881-82. 
415 
In this the divisor is evidently contained exactly 2^a/2y 2 times. 
Hence the complete quotient is 
^ r 3 + ^V + 2 ^7 2 - 
9. When the factors on the left in the theorems of §§ 5, 6, 7 are 
identical as to their elements, the expansions on the right become 
of course simpler in form, the simplification being more marked in 
the last of the three theorems on account of the vanishing of a de- 
terminant when two rows are identical. Thus we have 
+ + 
«1 
% 
«4 
a 1 
a 2 
<-h 
« 4 
*1 
&2 
h 
*4 
h 
C 2 
C 3 
C 4 
c i 
C 2 
C 3 
C 4 
cZj 
d'2 
(7 3 
d^ 
d x 
d 2 
d 3 
<7 4 
a\ 
a\ 
«4 
cqcj 
^2 C 2 
a sPs 
tt 4 C 4 
0 
Ct*2 
a\ 
al 
b\ 
4 
&2 
+ 2 
& 2 &2 
b 3 a s 
+ 2 
b 2 d 2 
M 3 
m 4 
<* 
4 
4 
ci 
«A 
C 2^2 
C 3^3 
C464 
«A 
C f > 2 
cf> s 
C4&4 
d\ 
d\ 
<5 
dj 
dS 
d 2 c 2 
^3 C 3 
4^4 
u 2 d 2 
ct^dg 
« 4 ^ 4 
Ojfb 2 
a s d 3 
« 4 0 4 
+ 2 
h 2 a 2 
b ? a 3 
^4 
+ 2 
m 
4 
c? 
4 
4 
C 4 
C 2 Cl 2 
C 3 a 3 
C 4 « 4 
df> 2 
df 3 
df>± 
^i c i 
d$2 
^3 C 3 
A further simplification is possible in the important case where the 
element in the r th row and s th column is ( x r - y ^ 1 . Then we have 
(*1-01 )"' (*1-02 )'* 
(* 2“01 .)'* (*2 ~ 0 s ) 1 
(* 3 -0l) _I (*3 — 02> -1 
(Xi-Vl)' 1 (*4 02> _1 
(* l - 0l )"’ (* 1 - 02) _1 
(* 2“01 )'* (* 2-02 )'* 
(*3 — 0l ) 1 (*3 - 02) 1 
(*4-0l) _1 (*4-0 2 )' 1 
(*1 — 0 3 ) — 1 (^l 0 4 ) _l 
(*2-03 )"' (*2 — 0 4 ) _ 1 
(*s -0s)' 1 (*3-04)"' 
(*4-0 3 ) _1 (*4-04)'' 
(*1 -0 3 )' 1 (*1-04 ) _1 
(*2 — 0s)* 1 (* 2 - 04 )"‘ 
(* 3“03 )'* (*3 — 04> _1 
(* 4 - 03 ) _1 (*4 " 04> -1 
