318 Professor Hamitton on Conjugate Functions, 
and then the total step from a to p may be denoted either by » +4 or by *+¥, ac- 
cording as we conceive the transition performed by passing through 8 or through c ; 
we have therefore the relation 
Aye i ea (77.) 
which becomes 
a+b=b+a, (78.) 
when we establish the analogy 
D—c=B—A, that is, 2 = 5; (79.) 
we see then that if the theorem (75.) be true, we cannot have the analogy (79.) with- 
out haying also its alternate analogy, namely 
b=, or D—B=C—A: (80.) 
because the compound steps 2+ and 2+», with the common second component =, 
could not be equivalent, if the first components b’ and » were not also equivalent 
to each other. The theorem (75.) includes, therefore, the theorem of alternation. 
Reciprocally, from the theorem of alternation considered as known, we can infer 
the theorem (75.), namely, the indifference of the order of any two successive compo- 
nents, », of a compound step: for, whatever those two component steps and > 
may be, we can always apply them successively to any one moment A, so as to gene- 
rate two other moments B and c, and may again apply the step 4 to c so as to gene- 
rate a fourth moment p, the moments thus suggested having the properties 
B=« +A, C=b-+ A; D=a +¢, (81.) 
and being therefore such that 
D—A=a+b, D—C=a=B—A; (82.) 
by alternation of which last analogy, between the two pairs of moments 4 B and c D, 
we find this other analogy, 
D—B=c—A=b, D=b+B=b+2a+A, (83.) 
and finally, 
b+sa=D—A=an+d. (84.) 
