394. Professor Hamiiton om Conjugate Functions, 
so that, comparing this with the former way of representing it, we may establish the - 
written equation, 
(B, By) — (Ay Ay) =(Bi— Aj B,— Ay). (1.) 
instead of conceiving thus a couple of ordinal relations between moments, or a 
relation between two couples of moments, discovered by the (analytic) comparison of 
one such couple of moments with another, we may conceive a couple of steps in the 
progression of time, from moment to moment respectively, or a single complex step 
which we may call a step-couple from one moment-couple to another, serving to 
generate (synthetically) one of these moment-couples from the other; and if we 
denote the two separate steps by a,, a2, (a, being the step from a, to B,, and a, being 
the step from a, to B,,) so that in the notation of the Preliminary Essay, 
Bj=a,;+A), By= ag+ Ay, 
B,=(B)—A;) +41, B= (By— Ag) + As, 
we may now establish this analogous notation for couples, 
(8, B») =(a, + Aly a + A») } 
=e ay) +(A, Ay) i (2.) 
), 
= $ (Bi, Bo) — (Ai, Ao)} +(A1, Ay 
the symbol (#,, B,) —(a,, 4.) corresponding now to the conception of the step-couple by 
which we may pass from the moment-couple (A,, A) to the moment-couple (B,, Bs), and 
the equivalent symbol (a), a) or (Bj —A,, B:—A,) corresponding now to the conception 
of the couple of steps a,, », from the two moments Aj, A,, to the two moments B,, B,, 
respectively. The step ,, or B,—A, may be called the primary step of the couple 
(a1, a), and the step a, or B,—A, may be called the secondary step. 
A step-couple may be said to be effective when it actually changes the moment- 
couple to which it is applied; that is, when one at least of its two coupled steps is 
effective : and in the contrary case, that is, when both those coupled steps are sepa- 
rately null, the step-couple itself may be said to be null also. And effective step-couples 
may be distinguished into singly effective and doubly effective step-couples, according 
as they alter one or both of the two moments of the moment-couples to which they 
are applied. Finally, a singly effective step-couple may be called a pure primary or 
pure secondary step-couple, according as only its primary or only its secondary step 
is effective, that is, according as it alters only the primary or only the secondary 
moment. Thus (0, 0) is a null step-couple, (a,, ») is a doubly effective step-couple, 
a 
2 ead 
