220 EIGHTH REPORT — 1838. 



which being integrated gives 



M + M'~ M(/i-/)- «V^' 

 where V is the value of V corresponding to S = and T = 0, 

 and is therefore the initial velocity. 



Hitherto the train has been assumed to move with accelerated 

 motion down an inclined plane. If it ascend, having received 

 any initial velocity, V, the motion will be retarded, and the 

 equation will be 



{M {h -vf) + « V'^} gdi:= - (M + M) rf V. 

 Substituting as before, let 



..- «v' ..,v = */^5±Z).rf.. 



V a 



•* -M(/i+/) 

 Hence we have 



VUa {h +/) (1 + x'-)gdT = - (M + M') dx 

 •.• '/Ma (A + /) jrp dx 



which being integrated between the limits x and x', the value x' 

 corresponding to T = 0, we have 



^ jWMa (A + ,n ,p\ x' - X 



And substituting for x and x' their values, we find 



^ ^Ma(A+/) „^_ >/M«(A+/).(V^-V) .^, v 



*""' M + M' ^^~ M(A+/) + «VV' •• ^"'-^ 



The relation between V and S will be found as before : 



2agS _ ( Mik+f) +a\>^' \ 

 M + M'~ VM(A f /) + aVV' • • • V— ; 



If the train move down a plane, of which the gradient is such 

 that h < /, the motion will be retarded, and in that case the 

 equations may be put under the forms 



^-•^"1 M + M^ ^^J- M{f-h) + a\\' (-^-^ 



2as-S 



/ M(/-A) + aV^ \ 



M +M'~ VM(/-/0 



Such are, then, the equations of the motion of a train of wheeled 

 carriages which are submitted to the action of accelerating and 

 x-etarding forces, or retarding forces only which are independent 



