and on Algebra as the Science of Pure Time. 395 
and (a,, 0) (0, a) are singly effective step-couples, the former (a, 0) being a pure 
primary, and the latter (0, a,) being a pure secondary, if 0 denote a null step, 
and a, a ellective steps. 
On the Composition and Decomposition of Step- Couples, 
2. Having stepped from one couple of moments (A,, A») to another couple of 
moments (5, B,) by one step-couple (a,, a), we may afterwards step to a third 
couple of moments (C,, C:) by a second step-couple (b,, b,), so as to have 
(a, C.)=(b1, by.) + (Bi; B)), t (3.) 
(Bi, By) =(a1 ap) +(A1p Ag) 3 
and then we may consider ourselves as having made upon the whole a compound 
couple of steps, or a compound step-couple, from the first moment-couple (A,, A,) to 
the third moment-couple (c,, c,), and may agree to call this compound step-couple 
the swm of the two component step-couples (@,, 4), (1, b), or to say that is formed & 
by adding them, and to denote as follows, 
(ci, C2) —(A; A») =( bay by) a ( 45 a,) 3 (4.) 
as, in the language of the Preliminary Essay, the two separate compound steps, from 
A, to c, and from a, to c, are the swms of the component steps, and are denoted by 
the symbols »,+ 4, and b,+ 4, respectively. With these notations, we have evi: 
dently the equation 
(>, bo) + (41, a2)=(b, + 41, bo + a2); (5.) 
that is, the swm of two step-couples may be formed by coupling the two sum-steps. 
Hence, also, 
(>,,b.)+( a1, a)=(), ao) +(>,, b»), (6.) 
that is, the order of any two component step-couples may be changed without altering 
the result ; and 
(#1, 42)=(41, 0) + (0; 4), (7) 
that is, every doubly effective step-couple is the sum of a pure primary and a pure 
VOL. XVII, 43 
