Gradients on Railways. 245 
only come to Mr. Barlow’s conclusion, that it is altogether 
erroneous, both in theory and practice,” when the assumed 
maintenance of uniform motion is objectionable, as it most 
certainly is, in the case of the deflections of rails. Contenting 
myself at present, then, with the opinion that the contending 
parties thus view the question of power expended, on differ- 
ent suppositions as to the way in which it is applied, I shall just 
take a very brief sketch of the question of velocity, when the 
motion is not assumed to continue the same through planes of 
different inclinations. 
Dr. Lardner supposes that in cases of uniform velocity, the 
resistance into the velocity is constant, and on this assumption 
deduces the equations stated by Mr. Barlow in page 97, viz. 
: tV 
(¢— sine) v=tV 0 ane 
This assumed principle is, in my opinion, decidedly inaccurate, 
more especially when it is contemplated that the carriages will 
pass along with the uniform velocity so expressed. For uniform 
motion can only be continued when the moving force continues 
equalto the resistance; and assuming with Dr. Lardner that the 
amount of friction is independent of the velocity, the speed will 
in such a case be quite indeterminate ; or, in other words, the 
power so applied will sustain uniformly any velocity that may 
have been previously communicated. It the friction were really 
independent of the velocity, while a moving force which exactly 
balances the resistance would maintain uniformly any pre- 
viously imparted motion, a moving force which exceeded the 
resistance would transmit the carriages with a velocity con- 
tinually accelerated, in conformity with what has been said by 
Mr. Barlow: but as the portion of resistance arising from the 
atmosphere at least, increases with the velocity, it is evident 
that the resistance will gradually augment till it balances the 
moving force, and so a uniform motion will eventually succeed. 
If the carriages be so acted upon as to retain a uniform velo- 
city v along a level plane, and with such velocity and moving 
power they arrive at the upper end of, and proceed down, an 
clined plane, the investigation given by Mr. Barlow, pages 
98 —100, will be strictly accurate on two suppositions, viz. 1. 
That the friction is independent of the velocity and inclination 
of the plane; 2. That the action of the moving power is not 
diminished by the increase of velocity. The former supposi- 
tion is sanctioned by Dr. Lardner ; the latter, as Mr. Barlow 
justly observes, if not true, will have the effect of giving the 
velocity and space passed over, rather in excess of the truth, and 
therefore the more favourable for a comparison with Dr. Lard- 
