856 
Proceedings of Royal Society of Edinburgh. [sess. 
a k 
h l 
a k 
h l 
a h 
k l 
+ 
a l 
k h 
= o, 
or that obtained from it by replacing each minor by its com- 
plementary, 
n.lr n h n. 1 
= 0, 
a k 
a h \ 
a l . 
— 
■ + : 
\ h l j 
: k l : 
\ k h \ 
and apply to it theorem I., we get 
: a 
k : 
a h 
a l 
. h + a 
l + CL : 
— : 
+ : 
■■k + a 1 + a 
'.k + a h + a ■ 
= 0 
which when a = 1 = a becomes (5), or (A) of Muir’s paper. 
Applying both theorems I, and IV., we have 
( 7 ) 
a h + a 
k + p l + p 
a k + a 
h + p l + p 
Starting with the aggregate 
124 
356 
136 
245 
+ 
a l -H a 
k+ph+p 
= 0 
123 
123 
456 
456 
+ 
125 
346 
145 | 
236 I 
126 
345 
146 
235 
+ 
134 
256 
156 
234 
(8) 
135 
246 
and applying theorem I., we have, making a= 1, 
123 
124 
+ 
125 
126 j 
+ 
1 134 i 
567 
467 
457 
456 
367 1 
, 145 
146 
156 
+ 
347 
346 
+ 
345 
135 
357 
+ 
= 0 
136 
356 
= 0. 
( 9 ) 
Applying theorem IV. we have, making a = 1 , 
134 
456 
135 I 
356 i 
+ 
+ 
136 
137 
+ 
145 | 
346 
345 
256 
156 
157 
i + 
167 
336 
235 
1 
234 
146 
246 
+ 
147 
245 
= 0. 
( 10 ) 
Similar results are obtained from applying the other theorems. 
Starting with the aggregate 
123 
456 
123 
456 
+ 
125 
346 
126 
345 
135 
246 
+ 
136 
245 
156 
234 
= 0 
