320 Professor Hamitton on Conjugate Functions, 
and, therefore, by alternation, 
D’—a=p—A’; (91.) 
for, whatever the three steps abe may be, we may always conceive them applied suc- 
cessively to any moment 4, so as to generate three other moments B, c, pv’, such that 
B=a+A, C=bd+B, D=c +6, (92.) 
and may also conceive two other moments a’ and p such that Bc D may be succes- 
sively generated from a’ by applying the same three steps in the order c, b, a, so that 
B=c+A, C=b+B, D=a+C3 (93.) 
and then the two first analogies of the combination (90.) will hold, and, therefore, 
also the last, together with its alternate (91.) ; that is, the step from a to D’, com- 
pounded of the three steps abc, is equivalent to the step from a’ to p, compounded of 
the same three steps in the reverse order cba, 
Since we may thus reverse the order of any three successive steps, and also the 
order of any two which immediately follow each other, it is easy to see that we may 
interchange in any manner the order of three successive steps ; thus 
pitt, et de (94.) 
=atb+crlDatet+bob+t+atea 
We might also have proved this theorem (94.), without previously establishing the © 
less general proposition (S9.), and in a manner which would extend to any number of 
component steps ; namely, by observing that when any arrangement of component 
steps is proposed, we may always reserve the first (and by still stronger reason any 
other) of those steps to be applied the last, and leave the order of the remaining steps 
unchanged, without altering the whole compound step; because the components 
which followed, in the proposed arrangement, that one which we now reserve for the 
last, may be conceived as themselves previously combined into one compound step, 
and this then interchanged in place with the reserved one, by the theorem respecting 
the arbitrary order of any two successive steps. In like manner, we might reserve 
any other step to be the last but one, and any other to be the last but two, and so on ; 
by pursuing which reasoning it becomes manifest that when any number of component 
steps are applied to any original moment, or compounded with any primary step, their 
order may be altered at pleasure, without altering the resultant moment, or the whole 
compounded step: which is, perhaps, the most important and extensive property of 
the composition of ordinal relations, or steps in the progression of time. 
