on the Theory of Gradients in Railways. 9 
ing the effect of gradients, raised in my evidence before the 
Great Western Railway Committee, and subsequently more 
fully developed by me at the meeting of the British Association 
in Dublin, it will of course be evident that I am the person 
here alluded to. But since your Report must needs fall into 
the hands of many persons who have neither seen my evidence 
before Parliament, nor heard the discussion in Dublin, I think 
it right to explain briefly what the conclusions are at which I 
arrived, and which you declare to be erroneous both in theory 
and practice. 
There are on railways certain inclined planes, forming so 
small an angle with the horizon, that a load placed upon them 
will not descend by its gravity, the friction being greater than 
the tendency down the plane by gravity. Let the angle of 
elevation of such a plane be «, and let the greatest angle of 
elevation which is compatible with this conclusion be ¢. I 
shall call this angle, «’, the angle of repose. Now let us sup- 
pose an inclined plane at the inclination «, the length of which 
expressed in feet is L: let a load be placed upon the plane, 
the amount of which we shall take as the unit of weight. Let 
t be the ratio of the friction to the pressure peculiar to the 
nature of the road, the carriages, &c. which is of course a con- 
stant quantity, so long as the carriages and the road continue the 
same. Now the pressure upon the plane will be expressed by 
cos‘; but as « must be a very small angle, we may, without 
sensible error, take cos ¢ = 1, and consider the whole weight 
as pressing upon the inclined plane. In fact, ¢ is an angle 
so smal] that its sine does not exceed 0°004, and « being still 
stnaller, it is clear that cos¢ is so nearly equal to the unit that 
we are justified in this assumption. 
To determine the tractive force which must be applied to 
the load to draw it up the inclined plane, it is only necessary 
to add together the forces necessary to overcome the friction 
and the gravity: now the friction is ¢, and the gravity sin ¢; 
therefore the force which resists the motion up the plane will 
be¢+ sine. The moving power, therefore, which will keep 
the load moving up the plane at a uniform speed will exert a 
pull upon it which shall be expressed by ¢# + sine. The unit 
being the weight of the load, it is clear that the total expendi- 
ture of mechanical power in drawing the load the entire length 
of the plane will be expressed by L (¢ + sin «). 
Now to estimate the mechanical force necessary to draw the 
load at a uniform speed down the plane, we have only to con- 
sider that the force which is opposed to the drawing power is 
the friction ¢, diminished by that component of the weight of 
the load which is directed down the plane, and which of course 
H 2 
