316 Professor Hamiiron on Conjugate Functions, 
marks +, and to treat the enclosed set as if they formed only one single symbol ; 
thus, 
(69.) 
e+b+a+A=>e4+b+(a+a)He+(b+a)+a 
=(e+b+a)+a, &., 
the notation ¢ +(b+a)+A, for example, directing us to begin by combining (in 
thought) the two steps 4 and > into one compound step » +, and then to apply 
successively this compound step and the remaining step ¢ to the original moment A ; 
while the notation (¢ +» +a)+A suggests a previous composition (in thought) of 
all the three proposed steps 4, b, ©, into one compound step ¢ +» +a, and then the » 
application of this one step to the same original moment. It is clear that all these 
different processes must conduct to one common result ; and generally, that as, by the 
very meaning and conception of a compound step, it may be applied to any moment 
by applying in their proper order its component steps successively, so also may these 
components be compounded successively with any other step, as a mode of com- 
pounding with that other step the whole original compound. 
We may also consider decomposition as well as composition of steps, and may pro- 
pose to deduce either of two components a and > from the other component and from 
the compound » +a. For this purpose, it appears from (68.) that we have the re-. 
lations 
a=Ob+c, and /=c +Oa, if c=b +a; (70.) 
observing that a problem of decomposition is plainly a determinate problem, in the: 
sense that if any one component step, such as here the step denoted by 0» +, or 
that denoted by ¢ +9, has been found to conduct to a given compound ¢, when 
combined in a given order with a given component > or a, then no other component 
a or », essentially different from the one thus found, can conduct by the same process- 
of composition to the same given compound step. We see then that each of the two 
components a and » may be deduced from the other, and from the compound ¢, by 
compounding with that given compound the opposite of the given component, in a 
suitable order of composition, which order itself we shall shortly find to be indifferent. 
Meanwhile it is important to observe, that though we have agreed, for the sake of 
conciseness, to omit the parentheses about a complex symbol of the kind ©, when 
combined with other written signs by the interposed mark +, yet it is in general ne- 
cessary, if we would avoid confusion, to retain the parentheses, or some such con- 
necting mark or marks, for any complex symbol of a step, when we wish to form, by — 
prefixing the mark of opposition ©, a symbol for the opposite of that step: for 
