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--2= radius of the exterior wheel including the flange, 
R = radius of interior rail, R--d= radius of the exterior, d = dis- 
tance between the rails, then R: R+d::r: r+a, hence on t= 
depth of the flange. Let R=500 feet, d=4.5, r=1.5, then a= 
of an inchnearly. If R=1000 feet, e=,', 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 = ss 
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. 
7 = outer or less radius of the wheel. 
d = distance between the flanges inclusive. 
b = breadth of rim of the wheel. 
p = play allowed the wheels while passing a curve. Then 
to find p when the other quantities are given, we have 
See z bdr 
: PS R(r—r)—or 
And to know the form of the wheels when the necessary pia is al- 
lowed, we have from the last equation 
taudek (d+-p)br per. 
pR 
_ In this equation, the inner radius r of the wheel, and the breadth 3; 
are supposed to be known, whence 7” 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 
this to advantage is yet a desideratum. pa ieee 
Vou. XXIII.—No. 1. 5 
