Observations on Inclined Planes. 109 
Put w and w’=the absolute weights of the descending and ascend- 
ing trains respectively, v and v’=their weights in the direction of the 
plane, g’’=the accelerating force by which the united trains move, 
x=the friction of all the moving parts, compared with g, F=a ex- 
pressed in pounds, — friction of rope, sheaves, and rope-roll in - 
w/ “ 
pounds. Then v==>, v= 7, 8’ =*, we have also the proportion 
Py ‘ea ere / 77 g’'(v— wv) g(w— w’) . 
oho bu no ite te and hence g’/= "wre baad 
Again, when friction is removed, we always have ¢?= as but since 
g”’ is dimintebed by friction, we have t?= , and hence z= 
2 Z ‘ 
s w 9 
Ba and by substitution cee =e = x being found, we 
a(w+w’ 
have gt wi :w--w + FY re) =the whole resistance in pounds, 
and siricd PnP’ ff Fuck 2 flaw ply wie 
sum and difference of the weights, and substituting for « its value, 
2as 
we cers F="— ett Fee D. 
In this value of F, the resistance arising from the inertia of the 
sheaves and rope-roll, is included ; and in the application of the for- 
mula, half the weight of the rope is to be considered as constituting 
a part of the weight of each train; or a, the sum of the weights, is 
equal to the whole weight of the rope added to the weights of the 
ng and descending trains. Mr. Wood obtains the friction 
without the inertia, by introducing into his equations the inertia as 
equal to one half the weight of the sheaves and rope-roll. The ac- 
curacy is doubtful, as the inertia depends much on the form of the 
_ wheels. Where it is proposed to ascertain the exact amount of rub- 
bing friction, it would certainly be necessary to obtain as nearly as 
possible the true amount of inertia. But in ordinary practice, the 
error would not, perhaps, be important, if the resistance of inertia 
and friction were estimated together. 
Resuming equation D, it will be seen that if the ascending weight 
on the plane becomes indefinitely small, d becomes equal to a, 2 
the equation becomes a 
