and on Algebra as the Science of Pure Time. ( gir 
example, the opposite of a compound step » + must be denoted in some such 
manner as 9 (> + 2), and not merely by writing 9> +. Attending to this remark, 
we may write 
@(>+2)=O2 +Ob, (71.) 
because, in order to destroy or undo the effect of the compound step » + «, it is suf- 
ficient first to apply the step 0 which destroys the effect of the last component step», 
and afterwards to destroy the effect of the first component step a by applying its op- 
posite 92, whatever the two steps denoted by « and » may be. In like manner, 
O@(¢+b+a)=O0a +O0b+0c; (72.) 
and similarly for more steps than three. 
12. We can now express, in the language of steps, several other general theorems, 
for the most part contained under a different form in the early articles of this Essay. 
Thus, the propositions (20.) and (21.), with their reciprocals, may be expressed by 
saying that if equivalent steps be similarly combined with equivalent steps, whether in 
the way of composition or of decomposition, they generate equivalent steps ; an asser- 
tion which may be written thus : 
if a=a, then’ b = a=b fa, a ba b, 
b+Oa=b +O0a, Oa'+ b=Oa +b, (73.) 
Ob+a=Ob +a, a+ Ob=a+Ob, &. 
The proposition (25.) may be considered as expressing, that if two steps be equiva- 
lent to the same third step, they are also equivalent to anh other ; or, that 
if a'=a' and a’=a, then a’=a. ; (74.) Beg: 
The theorem of alternation of an analogy (11.) may be included in the assertion 
that in the composition of any two steps, the order of those two components may “5 
changed, without altering the compound step ; or that 
atb= ba, (75.) 
For, whatever the four moments ABcpD may be, which construct any proposed ana- 
logy or non-analogy, we may denote the step from a to B by a symbol such as 4, and 
the step from B to p by another symbol », denoting also the step from a to c by \, 
and that from c to p by #5 in such a manner that 
B—A=4, D—B=b,c—A=V, D—C=8'; (76.) 
