and on Algebra as the Science of Pure Time. 399 
other cases in which this condition is not satisfied. ‘The spirit of the present Theory 
of Couples leads us, however, to conceive that the ratio of any one effective step- 
couple to any other may perhaps be expressed in general by a number-couple, or 
couple of numbers, a primary and a secondary ; and that with reference to this more 
general view of such ratio, the relation (20.) might be more fully written thus, 
(4) aw a, 44) = (a, 0), (21.) 
(a, ay) 
and the relation (19.) as follows, 
(a), 0) x (ay ao) = (a1, 0) (a1 a,) = (a, aj, a, a»)5 (22.) 
the single number a, being changed to the couple (a, 0), which may be called a pure 
primary number-couple. The spirit of this theory of primaries and secondaries leads 
us also to conceive that the ratio of any step-couple (b,, b») to any pure primary 
step-couple (a), 0), may be expressed by coupling the two ratios ti, ?2. which the 
ae el 
two steps b!, b, bear to the effective primary step a,; so that we may write : 
(>, ba ee b, bo (a ayy Cy a;) = z 
(a. 0) ie ee re Aly 0) a (a, a,); (23.) 
and in like manner, by the weneral connexion of multiplication with ratio 
d fo} ’ 
(@, a.) x (a, 0) = (A, G2) (ay 0) = (G 24, @ a,). (24.) 
From the relations (22.) (24.), it follows by (5.) that 
. (6, +a, 0) (a,, a2)=(b,, O) (Cai, a2) + (G1, O) (ay, a2), (25.) 
and that 
(Gy, G2) (b, + a,, O) = (Gi, a2) (by, 0) + (GQ, Az) Cai, 0); (26.) 
and the spirit of the present extension of reasonings and operations on single mo- 
ments, steps, and numbers, to moment-couples, step-couples, and number-couples, 
leads us to determine (if we can) what remains yet undetermined in the conception of 
a number-couple, as a multiplier or as a ratio, so as to satisfy the two following more 
general conditions, 
(+m, b:+a:) (41, a,)=(b, b.) (a1 a) +(@, 2) (91, 82)5 (27.) 
and 
(a, a») (>, + a), bot a,)=(M, a; ) (>, bo) +(M, a2) (41, a»), (28.) 
