Observations on Inclined Planes. 113 



cannot be used to advantage, will be seen, if we ascertain the depth 

 of flange necessary on a curved rail way. Put r = radius of the in- 

 terior wheel, r-\-x= radius of the exterior wheel including the-flange, 

 R = radius of interior rail, R-\-d= radius of the exterior, d = dis- 



dv 



tance between the rails, then R t K+dy.r : r-{-x, hence a; = — = 



R 



depth of the flange. Let R=500 feet, d—A.b, r=1.5, then x=l 



of an inch nearly. If R=1000 feet, oc=--j\ of an inch for the depth 



of the flange. From these examples it will be seen that the flange 



must be too small for the ordinary purpose of keeping the car on the 



rails on a straight road. There is also a disadvantage in the use of 



the conical wheel. On a straight road there must necessarily be a 



small allowance of breadth between the rails, in order to give play to 



the wheels ; hence a car will constantly change its position in the 



small space allowed it, in passing from one side to the other, and 



thus alternately raise and depress the sides of the car. This will 



perhaps produce more friction on a straight road, than the wheels 



will obviate on a curve. 



When the conical wheel is used, the play given to the wheels on 



a curve should strictly correspond to the slope of the rim in such a 



manner as to produce the intended effect. 



Let R = radius of the exterior rail. 



r = inner or greater radius of the wheel. 



r'= outer or less radius of the wheel. 



d = distance between the flanges inclusive. 



h = breadth of rim of the wheel. 



J? = play allowed the wheels while passing a curve. Then 

 to find p when the other quantities are given, we have 



bdr 



^'~R{r-r')-br 

 And to know the form of the wheels when the necessary play is al- 

 lowed, we have from the last equation 



pR 



In this equation, the inner radius r of the wheel, and the breadth b, 

 are supposed to be known, whence r' is found, and hence the proper 

 conical shape may be given to the. wheels. But after all, the conical 

 wheel does not remove altogether the friction, unless the axles are 

 made to take a position perpendicular to the rails. The method of 

 effecting this to advantage is yet a desideratum. 

 Vol. XXIII— No. 1. 15 



