Bote] 
XVII. Letter from Peter Barlow, Esg., F.R.S., to the Rev. 
D. Lardner, LL.D., F.R.S., on the Theory of Gradients in 
Railways. 
Dear Sir, 
AS you have addressed a letter to me in the last London 
and Edinburgh Philosophical Magazine, I feel myself 
bound by courtesy to reply to it through the same Journal, 
not, however, with a view of entering into any controversy on 
the question. You have in your letter very clearly stated your 
mode of solution, I will endeavour to explain also the grounds 
of my objection; the readers of the Lond. and Edinb. Philo- 
sophical Magazine will then be able to form their own judge- 
ment. 
First, ¢ being the fraction expressing the ratio of the fric- 
tion to the load, and ¢ the angle of the plane’s inclination, you 
take ¢ to denote the resistance on the horizontal plane, ¢ 
+ sine to denote the same on the ascending plane, and 
¢ — sin e for the same on the descending plane. ‘Then, assum- 
ing what may not, perhaps, be quite true, (but to which I do 
not here make any objection,) 2. e. that in locomotive engines 
the power generated and expended is the same in the same time, 
you arrive at the conclusion, that in cases of uniform velocity 
the resistance into the velocity is constant: taking, therefore, 
V to denote the velocity on the horizontal plane, and v the ve- 
locity on the descending plane (which you also assume to be 
uniform), you arrive at the equation 
(¢— sine) v = 7V. 
wv iV 
iy #Semies 
And by this formula your tables of velocities are computed for 
the Great Western and Basing lines; but where the formula 
gives more than 40 miles an hour the results of the computa~ 
tion are not stated. 
Thus, for example, assuming, as you do, 25 miles to be the 
Whence 
, , 1 
velocity on a horizontal plane, and ¢ = 250° the formula be- 
comes 
_ ‘I 
(:004—sin e)" 
I have gone over all the numbers in yqur four tables, and, 
except very trifling numerical errors, I find your results con- 
sistent with this formula as far as you have given the com- 
puted velocities; but those which you have not stated (by 
v= 
