720 Mk. STOKES'S DISCUSSION OF A DIFFERENTIAL EQUATION 



be the more liable to break under the action of a heavy body moving swiftly over it. The eflPect 

 of the inertia may possibly be thought worthy of experimental investigation. 



20. The mass of a rail on a railroad must be so small compared with that of an engine, or 

 rather with a quarter of the mass of an engine, if we suppose the engine to be a four-wheeled one, 

 and the weight to be equally distributed between the four wheels, that the preceding investigation 

 must be nearly applicable till the wheel is very near the end of the rail on which it was moving, 

 except in so far as relates to regarding the wheel as a heavy point. Consider the motion of the 

 fore wheels, and for simplicity suppose the hind wheels moving on a rigid horizontal plane. Then 

 the fore wheels can only ascend or descend by the turning of the whole engine round the hind axle, 

 or else the line of contact of the hind wheels with the rails, which comes to nearly the same thing. 

 Let M be the mass of the whole engine, I the horizontal distance between the fore and hind axles, 

 h the horizontal distance of the centre of gravity from the latter axle, k the radius of gyration 

 about the hind axle, x, y the coordinates of the centre of one of the fore wheels, and let the rest of 

 the notation be as in Art. 1. Then to determine the motion of this wheel we shall have 



de \il {2cx-x'Y 



M 

 whereas to determine the motion of a single particle whose mass is — we should have had 



M d^y _M Cy 



4, dt' 4 (2ca; - ar'y 



Now h must be nearly equal to - , and k' must be a little greater than 1/*, say equal to 1 P, so 

 that the two equations are very nearly tlie same. 



Hence, /3 being the quantity defined by equation (5), where S denotes the central statical 



Mg 

 deflection due to a weight — - , it appears that the rail ought to be made so strong, or else so 



4 



short, as to render jS a good deal larger than ^. In practice, however, a rail does not rest merely 



on the chairs, but is supported throughout its whole length by ballast rammed underneath. 



21. In the case of a long bridge, /3 would probably be large in practice. When ji is so large 



/3V 

 that the coefficient — - — — , or ir d^e"^^* nearly, in <b(a:) may be neglected, the motion of 



the body is sensibly symmetrical with respect to the centre of the bridge, and consequently T, as 

 well as y, is a maximum when a; = 1. For this value of a: we have 4 (x — or) = 1, and therefore 

 !s = T = y. Putting Ci for the (i + l)"" term of the series (9), so that d = Ji2~', we have 



for x = ^ 



r=C„+ C, + Co+ {39) 



'here Cp = ^ , , C, 



and generally, 



2 + li' ' 6+/i' 



(i + l)(i + 2) 



whence T is easily calculated. Thus for /3 = 5 we have tt/S^ e"^^* = .031 nearly, which is not large, 

 and we get from the series (39) 7"= 1.27 nearly. For /3 = 10, the approximate value of the 



