Observations on Inclined Planes. 109 



Put w and t^''=the absolute weights of the descending and ascend- 

 ing traias respectively, v and v'=^ihe\v weights in the direction of the 

 plane, ^"=the accelerating force by which the united trains move, 

 a:=the friction of all the moving parts, compared with g, F=a: ex- 

 pressed in pounds, F = the friction of rope, sheaves, and rope-roll in 



10 lo' g 



pounds. Thenv=-, v'=^ — , g'—-, we have also the proportion 



s:'{v — v'\ ffiw — w') 



V ', V - v' . ' sr t 2" , and hence sr =^ : — , — =: -; : — -x. 



•^ ° ' ■= v-{-v n{w-{-v/) 



Again, when friction is removed, we always have t^=~j}, but since 



2s 



g" is diminished by friction, we have ^2=^;; , and hence a: = 



g -X 



2s . <§'('^^ "~ ■^') 25 



g" — 7z, and by substitution x=—; — ; — j\ — rr- x being found, we 

 ° ^3' J n{w-\-w') t^ => ' 



have^ : x: '.w-^v/ : F'=^ =the whole resistance in pounds, 



x{w-\-w') 

 and since F=F'— /, F=^- — — /. Now putting a and <?=the 



sum and difference of the weights, and substituting for x its value, 



d 2as 



we have F = - r^ — f D. 



n gt^ •' 



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 

 ascending 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, and 

 the equation becomes 



