98 Letter from Peter Barlow, Esq., to Dr. Lardner, 
supposing the break to be employed) are so extraordinary 
that I think you can scarcely consider them as correct. 
For example, taking a very common gradient of 16 feet to 
a mile, or sine = nina ea find 
| 
_ (004 —355 
Such a gradient is very common in practice, and in practice 
in such a case the break is not applied. The computed and 
practical result ought, therefore, to agree, supposing the 
solution correct ; but such a velocity as one of 103 miles per 
hour has never yet, I believe, been obtained. 
Again, with a slope of 1 in 250, by no means an uncom- 
mon gradient, your formula gives the velocity of descent zn- 
finite: now such gradients are descended without the break, 
but, of course, not with an infinite velocity. For slopes 
greater than the last, of which also there are many, your ve- 
locity passes through infinity, and becomes negative, and the 
time of descent negative also, or less than no time. 
I have certainly stated in my Second Report to the Direc- 
tors of the London and Birmingham Railway Company, that 
a solution which leads to such extraordinary results must be 
‘erroneous both in theory and practice.” And my opinion is 
not altered. The error, I conceive, arises from combining the 
two dissimilar forces ¢ and sin <, and then treating the ques- 
tion of descent as one belonging to the case of uniform mo- 
tions; whereas (according to my view of the subject) it pro- 
perly belongs to the class of accelerated motions. 
Your solution, however, and my objection to it are thus 
placed before the readers of the London and Edinburgh Phi- 
losophical Magazine, and I leave the question in their hands. 
I remain, dear Sir, yours truly, 
Woolwich, Jan. 2, 1836. PETER Bartow. 
v Ti 103:09 miles per hour. 
After closing my letter, I have thought it might be satis- 
factory to some of the readers of the Phil. Mag. to see the 
view I take of this question, which is as follows. 
Suppose a body free from friction to arrive at, or to be pro- 
pelled from, the upper end of an inclined plane with a velo- 
city v, and let the angle of the plane = «, then the velocity 
acquired by that body in the time ¢ will be v + 2g. sine. ¢; 
and the space descended will be ¢v + g sin ¢.2°; (g denoting 
the space fallen through by gravity in one second, or g 
= 16,, feet.) 
Suppose now a body subject to friction to reach the same 
plane with the same velocity, but that this body. contains 
