RAILWAY CONSTANTS. 20/ 



S by eliminating T. Hence we have 



2Mg{h-f)dS -2 (M + M') V(^V = 0. 



2 M^ {h -/) S = (M + M') (V^ - V'2) (6.) 



2 (M + M') S = M^ (A -/) T2 + 2 V (M + M') T . (7.) 



It is evident that from the formulae (5.), (6.), and (7.)j the value 

 of / may be found if the initial velocity V' of the train and the 

 time of passing the posts by which the plane is staked out be 

 observed. 



If the train be allowed to move from a state of rest by gravity 

 alone^ the formulae will be simplified by the condition V' = 0. 

 They then become 



Mg{h-f)T = {M + M')Y .... (8.) 



2Mglh -f) S = (M + M') V^ . . . . (9.) 

 2 (M + M') S = M^- {h -f) T2 . . . . (10.) 



In the preceding formulae the load is considered as descend- 

 ing the gradient. If it ascend^ gravity will become a retarding 

 force, and the sign of h must be changed ; also the sign oi dY 

 will become negative. The formulae (5.), (6.), and (7.)? will then 

 become 



M^{/+/OT = (M + M')(V'-V) ..... (11.) 

 2M^(/+ A) S = (M + M') (V'2- V2) .... (12.) 

 2 (M + M') 8=-Mg{f+ h) T^ + 2V'(M + M')T (13.) 



If in this case the load having the initial velocity V be allowed 

 to run until it stop, we shall have V = •.* (11.), and (12.) be- 

 come 



Mg{f+h)T = {M + M')Y> . . . . (14.) 



2M^(/ + A)S = (M + M')V'2 . . . . (16.) 



In the case of retarded motion in descending a gradient less 

 steep than the angle of friction, h in these formulae must be 

 taken negatively. 



If, therefore, the train move down an inclined plane from a 

 state of rest, we shall have (9.) 



instead of 



Y^ = 2g{h-f)S, 



according to M. de Pambour. The value of V^, therefore, ob- 

 tainetl by him (neglecting all resistances which depend on the 

 velocity) is greater than the truth in the ratio of M + M' 

 toM. 



