52 Letter from the Rev. Dr. Lardner ¢o Peter Barlow, Esq. 
conspires with the drawing power. ‘The effective resistance, 
therefore, to the drawing power will be ¢sin s, referred to the 
weight of the load as the unit. The total expenditure of me- 
chanical force, therefore, necessary to cause the required de- 
scent will be L (¢ — sine). Now if we wish to determine the 
whole rnechanical power expended in ascending and descend- 
ing the plane, it is only necessary to add L (¢ + sin «) to L (¢— 
sine): the sum is2 L¢. Now it is obvious that this would be 
the amount of mechanical force expended in drawing the same 
load backwards and forwards on the level plane of the length L. 
If the angle of inclination of the plane were the angle of re- 
pose, then the tendency of the weight down the plane by its 
gravity would be precisely equal to the friction or sine’ =¢: 
there would in fact be no resistance to the motion down the 
plane, and consequently any velocity imparted to the load 
down the plane would be continued uniform without any 
drawing power to the bottom, supposing of course the plane 
to be free from the inequalities which would alter the amount 
of friction. 
To ascend such a plane, on the other hand, would require a 
drawing force of twice the amount necessary for a level, since 
t+ sine =2¢; and we accordingly arrive at the same con- 
¢lusion,—that in ascending and descending planes whose in- 
clinations do not exceed <’, the total expenditure of mechanical 
power is the same as on the level, the only difference being 
that on the level it is expended by one continued uniform 
éxertion, and that on the inclinations it is greater in the as- 
cent and jess in the descent, the mean being the amount upon 
the level. 
T explained fully, both before Parliament and at the British 
Association, that this reasoning would not extend to greater 
elevations thax ¢', for that in these cases the power saved in 
the descent wonld be less than the excess expended in the 
ascent, and that, consequently, such gradients would always 
occasion a loss of mechanical power. 
Now really this conclusion is so plain a result of first prin- 
ciples that I have been utterly at a loss to discover in what can 
originate our difference of opinion about it. It struck me, 
therefore, that this discrepancy must have its origin, not in 
the above reasouing, but in some difference of conception re- 
specting the very foundations of mechanical science. It oc~ 
curred to me, therefore, to look over your Reports, to. see 
whether the same difference as to first principles would lead you 
to conclusions upon other points different from those at which I 
should have arrived: in this I was not disappointed, for I found 
in another case, in which the force of gravity is considered, and 
