130 
MR. DAVIES GILBERT ON THE PROGRESSIVE IMPROVEMENTS 
Let the diameters of the two wheels be as 1 : d, then will the amount of 
pressure on the axis be d + 1 . Assuming the axis and wheels next in suc- 
cession to be similar in all respects, the pressure on them will be less than on 
the former in proportion to the increased angular velocity, but the prime 
mover having the disadvantage of leverage in the same proportion, the retard- 
ing effect of friction will be precisely equal in both ; whence it is obvious, 
that the same ratio should continue throughout all the series, or that the mul- 
tiplication of angular velocity should proceed in a geometrical proportion. 
Let then the whole increase of angular velocity be represented by a, and let 
the number of axes employed be x. Then in the usual mode of applying the 
wheels the friction on each axis will be a 1 -f- 1 . And the friction of the whole 
\_ 
of the axes will be (a 1 + 1) x. To find when this is a maximum, let A = the 
y 
natural logarithms of a. Then will the fluxion of (a x -f- 1) x be 
— A . a x . x~ - 1 x -f- a x - x -f- x when this is put = 0 
A 
By approximating it will be found that when 
i 
a = 120 x or the number of axes = 3.745, and ( a x -f l).a? or the fric- 
tion 17-9 instead of 121 if x were unity and the whole 
angular velocity communicated at once, about one seventh 
part. 
a — 100 x — 3.6 friction 15.64 about 10 parts in 64 of the whole, as 
above. 
a — 40 x = 2.88 friction 13.25, about one third part of 41. 
When a = 3.59 the minimum of friction falls on a single axis. 
a = 12.85 the minimum of friction is on two axes. 
a = 46.3 the minimum of friction is on three axes. 
a = 166.4 the minimum falls on four axes. 
In practice it is obvious that x must represent a whole number. 
Let a = 64 x may be either two or three. 
