unsymmetrical Six-valued Function of six Letters. 375 
as is number and space of the other two. Syntax and Groups 
are each of them only special branches of Tactic. I shall on an- 
other occasion give reasons to show that the doctrine of groups 
may be treated as the arithmetic of ordinal numbers. With 
respect to the twelve varieties of the A or B aggregates, they 
may be obtained from the one given by combining the substitu- 
tions corresponding to the six permutations of the three consti- 
tuents of one nome, as 7, 8, 9, with the permutation of any two 
constituents of another, as 5,6. But I have said enough for 
my present purpose, which is to point out the boundless un- 
trodden regions of thought in the sphere of order, and especially 
in the department of syntax, which remain to be expressed,mapped 
out, and brought under cultivation. The difficulty indeed is not 
to find material, of which there is a superabundance, but to dis- 
cover the proper and principal centres of speculation that may 
serve to reduce the theory into a manageable compass. 
I put on record (as a Christmas offering on the altar of science) 
for the benefit of those studying the theory of groups, or com- 
pound permutations (to which the prize shortly to be adjudicated 
by the Institute of France for the most important addition to the 
subject may tend to give a new impulse), and with an eye to the 
geometrical and algebraical verities with which, as a constant of 
reason, we may confidently anticipate it is pregnant, an exhaust- 
ive table of the monosynthematic aggregates of the trinomial 
triads that are contained in a system of three triliteral nomes. 
Let these latter be called respectively 123; 456; 789; then 
we have the annexed :— 
Table of Synthemes of Trinomial Triads to Base 3.3. 
(1.) (2.) (3.) (4.) 
147 258 369 147 258 369 147 258 369 147 258 369 
148 259 367 148 259 367 148 259 367 148 259 367 
149 257 368 149 257 368 149 257 368 149 267 358 
157 268 349 157 268 349 157 269 348 157 268 349 
158 269 347 158 269 347 158 267 349 158 269 347 
159 267 348 159 267 348 159 268 347 159 247 368 
167 248 359 167 249 358 167 248 359 167 248 359 
168 249 357 168 247 359 168 249 357 168 249 357 
169 247 358 169 248 357 169 247 358 169 257 348 
(9.) (6.) 7.) (8.) 
147 258 369 147 258 369 147 258 369 147 258 369 
148 267 359 148 267 359 148 269 357 148 269 357 
149 268 357 149 257 368 149 257 368 149 267 358 
157 249 368 157 268 349 157 268 349 157 268 349 
158 269 347 158 269 347 158 249 367 158 249 367 
159 248 367 159 248 367 159 267 348 159 247 368 
167 259 348 167 259 348 167 248 359 167 248 359 
168 257 349 168 249 357 168 259 347 168 259 347 
169 247 358 169 247 358 169 247 358 169 257 348 
A 
