404 
DESIDERATA AND SUGGESTIONS. 
[694 
of the order 12 upon 12 letters, where the number of independent substitutions is 
= 2. See the diagram, wherein the continuous lines represent black lines, and the 
dotted lines, red lines. 
The diagram is drawn, in the first instance, with the arrows but without the 
letters, which are then affixed at pleasure; viz. the form, of the group is quite indepen 
dent of the way in which this is done, though the group itself is of course dependent 
upon it. The diagram shows two substitutions, each of them of the third order: one is 
represented by the black triangles, and the other by the dotted triangles. It will be 
observed that there is from, each point of the diagram (that is, in the direction of 
the arrow) one and only one black line, and one and only one dotted line; hence a 
symbol B, “ move along a black line,” B 2 , “ move successively along two black lines,” 
BR (read always from right to left), “move first along a dotted line and then along a 
black line,” has in every case a perfectly definite meaning and determines the path 
when the initial point is given; any such symbol may be spoken of as a “ route.” 
a 
h 
c 
d 
e 
f 
9 
h 
i 
3 
k 
l 
1 
h 
G 
a, 
e 
f 
d 
h 
i 
9 
k 
l 
3 
abc 
. def 
ghi 
. jkl(= 
-B) 
c 
a 
h 
f 
d 
e 
i 
9 
h 
l 
3 
k 
acb 
. dfe 
gih 
. jlk 
d 
l 
h 
a 
9 
3 
e 
c 
k 
f 
i 
b 
ad . 
bl . ch 
eg . 
fj • ik 
e 
3 
i 
h 
h 
k 
f 
a 
l 
d 
9 
c 
aeh 
. bjd . 
cil 
• f k 9 
f 
k 
9 
c 
i 
l 
d 
b 
i 
e, 
h 
a 
afl 
. bkh . 
cgd 
■ ei 3 
9 
f 
k 
l 
c 
i 
3 
d 
b 
a 
e 
h 
a 93 
• hfi . 
eke 
. dlh 
h 
d 
l 
3 
a 
9 
k 
e 
c 
b 
f 
i 
ahe 
. bdj 
eli 
• fg k 
i 
e 
3 
k 
h 
h 
l 
f 
a 
c 
d 
9 
ai . 
be . cj. 
dk . 
fh . gl 
3 
i 
e 
h 
k 
b 
a 
l 
f 
9 
G 
d 
a 39 
. bif . 
cek 
ii 
^3 
R) 
k 
9 
f 
i 
l 
c 
b 
3 
d 
h 
a 
e 
ak . 
bg ■ cf 
di . 
el . lij 
l 
h 
d 
9 
3 
a 
c 
k 
e 
i 
b 
f 
alf 
. bhk . 
edg 
• eji