414 
MINUTES OF PROCEEDINGS OF 
Consequently, the elementary moment 
and in the limit, 
ujjr=s)W-V)*» 
/x Wn 
bl (a — b) 
Therefore the whole moment from b to a 
(f — by) dy . 
- 4 ^ 
-! 5 f 
Similarly, tlie whole moment from A to 0 
_ y.W (l - ») r , _ „ f from y = b 
~ ~ U (b - c) J, { y 3,1 1 to y = c 
= —& 
and by the conditions these moments are equal; therefore 
n {2 a 2 — ab — h 3 } = (l — n) {5 3 + — 2c 3 }. 
If for b } its value, a — ^ ^ ^ 
^ , be put in this equation, and it be as 
far as possible simplified, we get 
+ ac — 2c 3 ) 
a 2 -c 3 ; 
l 
” = 3 
__ Z ® + 2c 
‘ 3 ’ a + 6 
, , 2 2& + c 
and l — ti — - * ——- 
3 <35 -f~ C 
Thus the position of the true rolling line is defined in terms of the 
breadth of the tire ( l ), and the large and small radii of the wheel, 
viz. a and c . 
To ascertain now the pressure exerted upon the pipe-box by the grind of 
the wheel. 
We have already found the tangential friction upon any elementary 
disc ST (larger than the rolling disc) to be equal to 
(y-i) 
The moment of this friction about an axis OF in the plane of the rolling 
disc is the product of the force of friction and the perpendicular upon its 
direction from the axis. This latter equals OS = x. 
