and on Algebra as the Science of Pure Time. 409 
i and k being positive integers, such that 
itm, kbm, i+k>m; (93.) 
and if we put for abridgment c 
Ja) +4,=a, VJ 6, + 6,2 =, (94.) 
and 
a’ pe ~ 
Bs Awe 95. 
Y = TxOx3x dx Ix2x3xak? (95.) 
we shall have, by principles lately explained, 
ores = y (96.) 
and therefore 
Cc; } + Ys Orde => C2 > Ys Cr dg yrs (97.) 
if then the entire sum (90.) of all these couples (¢,, c.) be put under the form 
= (4, e)=(2%, = ©), (98.) 
the letter = being used as a mark of summation, we shall have the corresponding li- 
mitations 
Za pdy, Bat — zy, : 
Ze bly, q+ —Zy, 
(99.) 
m (m+1) 
2 
= being the positive sum of the such terms as that marked (95.). This 
latter sum depends on the positive whole number m, and on the positive numbers 
a, (2; but whatever these two latter numbers may be, it is easy to show that by taking 
the former number sufficiently great, we can make the positive sum = y become 
smaller, that is nearer to 0, than any positive number é previously assigned, however 
small that number 6 may be. For if we use the symbols F,, (a), F,, (8)5 Fn (a+ 3), to 
denote positive numbers connected with the positive numbers a, 3, a+, by relations 
analogous to those marked (87.) and (88.), so that 
7 Ge a” 
F = —~+t— +... + —.~_,__—_ 
m(a)= 1+ 7+ 4+ + TXaxax um 
(100.) 
it is easy to prove, by (68.), that the product r,, (a) x r,,() exceeds the number 
