145 
.h 
K 
h 
m x 
12345 
21435 
34125 
43215 
23451 
14352 
41253 
32154 
34512 
43521 
12534 
21543 
45123 
35214 
25341 
15432 
51234 
52143 
53412 
54321 
J2 
K 
h 
m 2 
23145 
32415 
14235 
41325 
31452 
24153 
42351 
13254 
14523 
41532 
23514 
32541 
45231 
15324 
35142 
25413 
52314 
53241 
51423 
54132 
h 
Jc 3 
h 
m z 
31245 
13425 
24315 
42135 
12453 
34251 
43152 
21354 
24531 
42513 
31524 
13542 
45312 
25134 
15243 
35421 
53124 
51342 
52431 
54213 
The 60 negative substitutions are obtained by making the 
transposition 21 for 12 in all the preceding. 
By Xi=ji+h+h+™i 
X3=y 3 +^3+4+^3 
we mean to say that x 1 is the sum of the four subgroups written 
under j\ k x l± and m x , with a like signification for x 2 and x 3 . 
And by — x 1= — A)+(—Wi), &c., 
we mean that — x b — x 2 . — x 8 , and — -j\, &c., are what x v x 2 , x 3f 
and j lf &c., become by the transposition 21 for 12 made alike 
through their sums of substitutions. 
By = 1 J i 
Hj+H'+H* + H*+H ' 10 =& 
*4- Hi 2 -f- Hi 3 -f- Hh + H] 5 = ^ 
Hi6“b B-b -f* H^g -f- H| 9 -{- , 
H 2 i + H22+H 2 3+H^44-H 2 5 = 2 J i 
