218; EIGHTH REPORT — 1838. 



whole resistance of a train moving with a velocity V, we shall 

 have 



R = rtV2 + B. 



Now, since B is proportional to the load, if M express the load 

 in tons, and /the resistance for a load of one ton with an inde- 

 finitely slow motion, we shall have 



B = M/, 



and therefore 



R = « V2 + /M. 



The coefficient a being the constant number which, being 

 multiplied by the square of the velocity, gives that portion of 

 the resistance which varies with the velocity, will depend on 

 the form and magnitude of the train, on the number, form, and 

 magnitude of the wheels, and in general on any circumstances 

 by which the resistance of the air to the moving parts of the 

 train may be affected. But it should be observed, also, that 

 there is nothing in the mere mathematical formula which limits 

 the term a V^ to represent the effect of the air ; that term in 

 fact represents any effect which would be attended with a resist- 

 ance proportional to the square of the velocity. 



If any means were devised by which the total resistance of 

 the same train at two different velocities could be found, the 

 value of the coefficient a might then be determined ; for let R 

 and R' be the two resistances of the same train at the velocities 

 V and V, then we have 



R =«V2 + M/ 

 R' = aV'2 + M/ 



' ^ V^ — V'*' 



Hence it appeal's that the difference between the two observed 

 resistances, divided by the difference of the squares of the cor- 

 responding velocities, woukl be the value of a. 



But as the estimation of the resistance of trains by any direct 

 means is attended with difficult}^, it may be useful to seek in 

 the circumstances of accelerated and retarded motion on inclined 

 planes which are straight, other means for the solution of this 

 problem. 



If R, as already explained, express the ratio of the retarding 

 force produced by the whole resistance to the retarding force 

 of gravity, expressed as usual by g, then the velocity which 

 gravity would destroy in the time d T being g d T, the velocity 

 which the resistance would detroy in the same time will be 

 KgdH. 



