will not roll either in a straight line or in a curve without great difficulty, because if they roll in a straight line, the wheels on one side being larger in diameter than those on the other, either the larger or the smaller ones must slip on the path in which they roll. If they roll on a curve, then each pair of wheels has a tendency to roll in a curve independent of the other, and therefore the wheels must slip laterally if both pairs roll on the same track. Thus, suppose two pairs of wheels, _a_, _a′_, and _b_, _b′_, fig. 157, to be made conical and attached to a frame so that their axles are parallel to each other. Now each pair of such wheels will have a tendency to roll in circular paths, _a′ i_, _a h_, and _b′ k_, _b j_, the centres of which are at _m_ and _n_, or at the apices of the cones of which their peripheries form a part. If they are made to roll in circular paths, _c d_, _e f_, described from a point _g_, then each pair of wheels must slip laterally over the space between the paths _a′ i_, _a h_, in which they would naturally roll and that in which they are made to roll. Thus the wheel _a_ would slide laterally the distance between the curve _a h_ and _a f_, and _a′_ that between _a′ i_ and _c d_; _b_ would slide from _b j_ to _b f_ and _b′_ from _b′ k_ to _b′ d_. It will thus be seen that in order that two pairs of wheels may roll with equal ease in a straight line and in curves, the wheels in the one case must be of equal diameters and the axles parallel, and in the other case the wheels must be of unequal diameters and their axles be _radial_[69] to the curve. This is equally true of any number of pairs of wheels. If we have three, four, or any number of axles, with wheels all attached to the same frame, if their axles are parallel, and the wheels of the same diameter, they will roll in a straight line; but if their wheels are conical and their axles radial, they will roll in a curve. [68] The periphery is the outside surface on which the wheel rolls. This part of a wheel is usually called the “_tread_.” [69] That is, that their centre lines incline towards each other, and if extended far enough would meet at the centre of the curve. [Illustration: _Fig. 157._ _Scale_ _³⁄₁₆ In. = 1 Foot._] For the preceding reasons it is therefore sufficiently obvious that if a locomotive is to run on both straight and curved tracks, on the former the wheels should be of the same diameter and the axles parallel, and on the latter the wheels should be conical and the axles radial. [Illustration: _Fig. 158._ _Fig. 159._ Scale ³⁄₈ in. = 1 foot.] QUESTION 255. _How are the wheels made so that in curves they will act as though they were of the conical form described and on a straight track all be of the same diameters?_