54 Letter from the Rev. Dr. Lardner to Peter Barlow, Esq. 
noticed my inferences respecting gradients, I find that you say 
that, ? 
“‘ As the question wholly turns upon velocity, it is of course 
impossible to exhibit the effect experimentally.” 
Now although I do not perceive this to be at all a matter 
of course, but on the contrary have found it very easy to 
reduce questions depending on velocity to experiment, yet I 
beg to observe that the present question does not depend 
either wholly or at all upon velocity. Whatever be the speed 
of the load upon the inclined plane, provided only it be main- 
tained uniform, my theory of gradients (if it deserve to be so 
called) will still hold good. It will take the same expenditure 
of mechanical force to move a load on such inclined planes as 
I have described, and the mean between the ascending and 
descending forces will be the force along a level plane. You 
must surely be so well acquainted with the laws of friction that 
it is needless for me to remind you that that resistance is alto- 
gether independent of the velocity. And I would also beg to 
observe that the case is one totally distinct from the consi- 
deration of accelerating forces. In page 91 you say: 
“This point cannot be met experimentally, and J am there- 
fore obliged here to depend only on demonstration. The case 
certainly involves no difficulty of conception to those acquainted 
with theoretical mechanics, &c.” 
I admit that it does not; but I apprehend the conception 
which those acquainted with theoretical mechanics form of it 
will be altogether different from that at which you appear to 
have arrived, and I therefore regret that you seem to have 
forgotten your expressed intention of giving a demonstration 
of your own peculiar view of the matter. In the next page 
(92) you mention the intention as one which you dad, but seem 
to have immediately abandoned it. 
It will be very gratifying to me, and I am sure it will be 
useful to all who are practically engaged in those extensive 
enterprises for the formation of lines of communication through 
the country, if you will show how these views of mine are at 
variance with the established principles of mechanics. Al- 
though I am not aware that any one has hitherto pointed out 
the property which I have explained in reference to inclined 
planes of less inclination than the angle of repose, yet, so far 
as I am informed, there is no difference of opinion whatever 
as to the legitimacy of the method of estimating the tractive 
force both in ascending and descending these planes. The 
same formulz that I have used, viz. L (¢ + sins), have been 
in substance universally adopted in estimating the mechanical 
force necessary to work railroads. You will find that many 
