412 
MINUTES OF PROCEEDINGS OF 
vertical section, DEGII, of any service wheel, DE being the part of the 
tire which is resting on the ground. The figure DEGH is, in fact, a prin¬ 
cipal section of the frustrum of a cone the apex of which is at C. AB 
represents the axis of the wheel. 
When the wheel is compelled to roll in a straight line on a rough plane, 
it is evident that there can only be one line in the surface of the tire which 
rolls truly. Let the point F represent the section of the rolling line. 
The wheel may be considered to consist of an indefinitely great number of 
parallel laminal discs, perpendicular to the axis; and as all such discs pass 
over equal distances in equal intervals of time, it is evident that the discs 
between E and F revolve more rapidly than is necessary for them to cover, 
by rolling simply, that portion of the road covered by the rolling disc; and 
the discs between D and F, on the other hand, roll at a slower speed than 
is sufficient to cover by rolling the distance traversed by the rolling disc. 
Hence friction in opposite directions is set up on each side of the true 
rolling line, the tendency on one side being to increase, and on the other to 
retard the velocity of rotation; and the moments about the axis due to 
retardation must be equal to those due to acceleration. 
In finding the position of the rolling line, F, on the surface, DE, of the 
tire, we may suppose the weight to be borne equally by every part of the 
touching surface along DE. 
Tig. 2 represents any laminal disc (W) lying between OF and AD 
Fig. 2. 
(Fig. 1), the circumference of which is smaller than that of the rolling 
disc. 
Let it be supposed that the rolling disc, in revolving through an angle (p, 
rolls a distance ac (Fig. 2). W must cover this distance; but it can only 
roll a distance ab , equal to that portion of its circumference which sub¬ 
tends (p. Therefore the distance be, which is the difference between ac and 
ab, must rub the surface ad in passing it; and the frictional action generated 
in passing over ac could only be maintained in its full force for a distance be. 
Hence, calling any distance ac equal to s, and be equal to s 1} 
— = that co-efficient of the weight borne by any disc which at each 
s instant is referable to friction. 
