on the Theory of Gradients in Railways. 99 
within itself an acting power which exactly balances the fric- 
tion at all velocities; then I consider that the descent of this 
body wiil take place under precisely the same circumstances 
as the former, and therefore that the space described in the 
time ¢ will be as before27 = ¢v + g sin <.é, and that in both 
these cases the velocity will be uniformly accelerated. 
But if the internal acting power within the body will only 
balance the friction when the velocity of the body is v, there 
will be three circumstances on which the velocity of descent 
will depend, viz. 
First, the original velocity = v. 
Secondly, the accelerating force = g sin «. 
Thirdly, the retarding force arising out of the excess of 
friction beyond that of the internal force employed to over- 
come it. 
In this form the solution of the problem falls under the 
case of variable forces, and is so involved as not to admit of 
solution without employing differentials of the second order. 
Now, I consider a locomotive engine to be a body constituted 
as above supposed, that is, liable to friction, but containing 
within itself an acting power capable of overcoming the fric- 
tion, so that where gravity does not act, the motion of the 
body continues uniform; and if on reaching a descending 
plane, the internal force was still such as to balance the fric- 
tion due to the increased velocity, then its descent would be 
governed by the same laws as supposed in the second case, that 
is, the velocity after any time ¢ would bev’ = v + 2tg sin ¢, 
and the space described would be 7 = tv + g sine.@. 
The actual law of locomotive machines is not yet well un- 
derstood ; some engineers are of opinion that an increased ve- 
locity, by throwing the steam faster into the funnel, causes an 
increased draught, which produces a proportionally greater 
quantity of steam, in which case the above would be an ex- 
act expression for the space described; and if it is not so, and 
we still reject the consideration of retardation, then at all 
events the above formula will mark a limit in the problem, 
and the velocity and space thus obtained will be the greatest 
that can be acquired in a given time. This, therefore, is the 
most favourable view of the question for Dr. Lardner in com- 
paring his velocities with mine. Let us see, then, what the re- 
sults are on this supposition. We have seen that, 7 being any 
length of plane, gsine® + vi = 1, 
1 Sale si ae 
or Ps 
: vt = =! sans 
g sine gsine gsin’e¢ 
h denoting the height of the plane. 
