If the total resistance against which the train is maintained in motion with an instantaneous velocity of V feet per second is R, the rate at which energy is expended in moving the train is represented by the product **RV**, and this must be the rate at which energy is supplied to the train after deducting all losses due to transmission from the source of power.

If now the prism P be interposed as in the figure, the whole beam is not only refracted upward, but also spread out into the spectrum **RV**, the horizontal breadth of the band of colours being the same as that of the original image S.

Hence if all the energy supplied to the train is utilized at one axle there is the fundamental relation **RV** (I) Continuing the above arithmetical illustration, if the wheels to the axle of which the torque is applied are 4 ft.

= **RV** (2) where T 1, T2, T3, &c. are the torques on the axles whose respective angular velocities are wl,w2, W3, &c.

Multiplying through by w we obtain Tw = 2FwD = 2µWwD = **RV** (4) This is a fundamental energy equation for any form of locomotive in which there is only one driving-axle.

The relation between the b.h.p. and the torque on the driving-axle is 55 o B.H.P. =Tu., (9) It is usual with steam locomotives to regard the resistance R as including the frictional resistances between the cylinders and the driving-axle, so that the rate at which energy is expended in moving the train is expressed either by the product **RV**, or by the value of the indicated horse-power, the relation between them being 55 0 I.H.P. =**RV** (Io) or in terms of the torque 55 0 I.H.P.X€=**RVe**=TW (II) The individual factors of the product **RV** may have any value consistent with equation (to) and with certain practical conditions, so that for a given value of the I.H.P. R must decrease if V increases.

The intensity may be expressed by 12= (2+Cv) 2 +(2+Sv) 2 and the maxima and minima occur when dC dS (z+Cv)a`j+(2+Sv)dV=0, whence sin **rV** 2 +cos27rV 2 =G..

Xxm MU - **RV** - 0 7u g xxo A=T 1 + h [4 8u - rh 13u + g07 80 65u 4-`644K640oa7u+...

At-run `% **rV**/en?Bc4 n .

**Rv** ~

SeIndian **rv** ices.

Removing the summation signs in equation (52) in order to restrict its application to two points and dividing by the common time interval during which the respective small displacements ds and ds were made, it becomes Pdsfdt = Rds/dt, that is, Pv = **Rv**, which shows that the force ratio is the inverse of the velocity ratio.

**Rv** Z :

**Rv** ndcra.