and on Algebra as the Science of Pure Time. 387 
ratio ¢ = b can be found, whatever incommensurable number the exponent a may 
be, is easily proved from the circumstance that none of the conditions (336.) are in- 
compatible with one another if ) > 1, and that none of the conditions (337.) are in- 
compatible with each other in the contrary case, by reason of the constant increase or 
constant decrease of the fractional power Bin for constantly increasing values of the 
fractional exponent *. And that only one such positive ratio ¢ = b can be found, 
or that no two different positive ratios c, c’, can both satisfy all these conditions may 
be proved for the case » > 1 by the following process, which can without difficulty be 
adapted to the other case. 
The fractional powers of comprised in the series 
1. 3) dh im tim 
bn, bm, bm, ... bm, bom, (339.) 
° ° 
increase (when )>1) by increasing differences, of which the last is 
im l+m 
m 
Ob™ + bw =b (01 + bn); ($40.) 
this last difference, therefore, and by still stronger reason each of the others which 
precede it, will be less than = if 
l>kb (841.) 
and 
O1+bn<7: (342.) 
. and this last condition will be satisfied, if 
| m>1(01 +b), (343.) 
J and m (like 7 and &) denoting any positive whole numbers ; for then we shall have 
i So, (344.) 
and by still stronger reason 
(Lx 7)" >B, 1+ 7> Bry (345.) 
observing that 
(1 +4) ">1+4%, ifm>1, (346.) 
VOL. XVII. 4A 
