: and on Algebra as the Science of Pure Time. 411 
and, therefore, 
LPF, (4) Fx (8)— Fp (a +8)} =O, (112.) 
LL Fn (Ai @) Fn (bis 0.) — Fn ((ar, a2) + (bis b,)) } = (0, 0). (113.) 
In the foregoing investigation, a and 8 denoted positive numbers; but the theorem 
(113.) shows that the formula (112.) holds good, whatever numbers may be denoted 
by a and 23, if we still interpret the symbol me by the rule (100.). 
11. Ifa still retain the signification (94.), it results, from the foregoing rea- 
sonings, that the primary and secondary numbers of the couple 
Frntm’ (iy Qa) —Fm (diy Ae) (114.) 
are each 
+ Fram (a)—F,, (a), and ¢ F(a) —Frim (a); (115.) 
and, therefore, may each be made nearer to O (on the positive or on the contra- 
positive side) than any proposed positive number é by choosing m large enough, 
however large m’ and a may be, and however small 8 may be: because in the ex- 
pression 
a” a a a” 
Fn tm (#) — Fn (a) = 1x2x3x.. amet mE ey tt eee} (116.) 
m 
Sales a Q pie : 
the positive factor eel be made <6, chat is, as near as we please to 0, 
: 1 1 1 
and also the other factor, as being < A ileus + 7, and therefore SESE y aps if 
m+1>na. Pursuing this train of reasoning, we find that as m Spe te greater 
and greater without end, the couple F,, (a,, a;) tends to a determinate /imit-couple, 
which depends on the couple (%, a,), and may be denoted by the symbol r,(a,, a), 
or simply F(a, a), 
F (M4, Q2)=F_ (dh, G2) =" Fm (Ay, Az) 5 C7) 
and similarly, that for any determinate number a, whether positive or not, the 
number F,,(a) tends to a determinate limit-number, which depends on the number a, 
and may be denoted thus, 
F (a) =F, (a) = L Fn(a). (118.) 
It is easy also to prove, by (112.), that this function, or dependent number, ¥ (a); 
must always satisfy the conditions 
F(a) x¥ (8) =F (a +8), (119.) 
VOL. XVII. 4D 
