128 MR. DAVIES GILBERT ON THE PROGRESSIVE IMPROVEMENTS 
resistance, in others a half, and in some few instances one-third : evidently 
becoming less as the wheels were increased in size ; and always greatly di- 
minished by the introduction of an additional axis for multiplying the angular 
velocity. In similar machines recently constructed on the best principles, 
friction is said not to exceed a tenth. 
With the hope of reducing the amount of this great impediment to the 
useful application of motive force, I have been led to consider what would be 
the most proper form of teeth or cogs, and by how many intermediate steps a 
given increase of angular velocity might be most advantageously effected. 
It is quite clear, with respect to cogs or teeth, that to give them theoretical 
perfection, they should be so formed as to communicate an equable velocity 
from the driving to the driven wheel ; and at the same time to roll upon them- 
selves free from any sliding, the cause of friction and of abrasion. 
Either of these properties may be separately obtained ; but the two are 
utterly incompatible, except at the limit where teeth disappear and the wheels 
themselves are in contact. 
For producing equable velocities. 
The involute from a circle, receives continually an accession of length to its 
radius of curvature equal to the development of the generative arc. Conse- 
quently, if involutes are formed in opposite directions from two circles of 
magnitudes inversely proportionate to the angular velocities of their respec- 
tive wheels, the extremities of the two radii of curvature will always remain 
in contact, forming together a tangent to the evolute circles crossing the line 
of the centres. The angular velocities must therefore be uniform, while the 
surfaces will slide on each other, and create a friction proportionate to the 
difference of length between the radii of curvature. 
For avoiding friction between the cogs or teeth. 
As logarithmic spirals preserve always a constant angle between the ordi- 
nate and the curve, if two similar logarithmic spirals act against each other, 
they must continue to roll without friction, the ordinates remaining in contact 
on the line of the centres, but causing angular velocities in the inverse pro- 
portion of those ordinates. 
It is plain therefore that the two properties are incompatible, since the loci 
of contact between the curves producing them are in different lines. But since 
the involutes may be generated from circles indefinitely small, without refer- 
