may jump, that is, half the weight of the engine exclusive of the driving-wheels,-W=11 to 14 tons, W1=84 to 114 tons, w=n, g=32.19084 whence I have made the following calculations from formula (59.).

It appears, by formula (59.), that the displacement of the centre of gravity necessary to produce a jump at any given speed, is not dependent on the actual weight of the engine or the wheels, but on the ratio of their weights; and, from the above table, that when the weight of the engine and wheels is 63 times that of the driving-wheels, a displacement of 24 inches in the centre of gravity is enough to create a jump when the train is travelling at sixty miles an hour, or of two inches when it is travelling at seventy miles; this displacement varying inversely as the square of the velocity is less, other things being the same, as the square of the diameter of the wheel is less; for the radius of the wheel being 22n represented by a, the angular velocity is represented by w= and substituting this value, formula (59.) becomes

15a'

• It will be observed, that the cranks being placed on the axle at right angles to one another, when the centre of gravity on the one side is in a favourable po

If the weight W of the wheel be supposed to vary as the square of its diameter and be represented by pa2, this formula will become

still showing the displacement of the centre of gravity necessary to produce a jump to diminish with the diameter of the wheel. These conclusions are opposed to the use of light engines and small driving-wheels; and they show the necessity of a careful attention to the true balancing of the wheels of the carriages as well as the driving-wheels of the engine. It does not follow that every jump of the wheel would be high enough to lift the edge of the flange off the rail; the determination of the height of the jump involves an independent investigation. Every jump nevertheless creates an oscillation of the springs, which oscillation will not of necessity be completed when the jump returns; but as the jumps are made alternately on opposite sides of the engine, it is probable that they may, and that after a time they will, so synchronise with the times of the oscillations, as that the amplitude of each oscillation shall be increased by every jump, and a rocking motion be communicated to the engine attended with danger.

Whilst every jump does not necessarily cause the wheel to run off the rail, it nevertheless causes it to slip upon it, for before the wheel jumps it is clear that it must have ceased to have any hold upon the rail or any friction.

The Slip of the Wheel.

If ƒ be taken to represent the coefficient of friction between the surface of the wheel and that of the rail, the actual friction in any position of the wheel will be represented by Y, f. But the friction which it is necessary the rail should supply, in order that the rolling of the wheel may be maintained, is X. It is a condition therefore necessary to the wheel not slipping that

If, therefore, taking the maximum value of in any revolution, we

X
Y1

find that fexceeds it, it is certain that the wheel cannot have slipped in that revolution; whilst if, on the other hand, ƒ falls short of it, it must

sition for jumping, it is in an unfavourable position on the other side, so that it can only jump on one side at once, and the efforts on the two sides alternate.

have slipped.

The positions between which the slipping will take place continually, are determined by solving, in respect to cos. 1, the equation

The application of these principles to the slip of the carriage-wheel is rendered less difficult by the fact, that the value of h is always in that case so small, as compared with the values of k and a, that

lected in formulæ (34.) and (35.), as compared with unity. tions then become

h

a

may be neg

Those equa

Now if ẞ>1, there will be some value of 0 for which+cos. 6=0, and

В

therefore 1+ẞ cos. 6=0; and since for this value of e,

Of course, the slipping, in the case of the driving-wheels of a locomotive,

is diminished by the fact, that whilst one wheel is not biting upon the rail the other is.

it follows that it corresponds to a maximum value of u, and

But if ẞ<1, then there is some value of cos. for which ẞ+cos. 6=0, and therefore for which u=infinity, which value corresponds therefore in this case to the maximum of

and in the other case it will be represented by the formula

In the first case, i. e. when ß < 1, the wheel will slip every time that it revolves, whatever may be the value of f. In the second case, or when B> 1, it will slip if ƒ do not exceed the number represented by formula (68.). The conditions (65.) are obviously the same with those (59.) which determine whether there be a jump or not, which agrees with an observation in the preceding article, to the effect, that as the wheel must cease to bite upon the rail before it can jump, it must always slip before it can jump. When the conditions of slipping obtain, one of the wheels always biting when the other is slipping, and the slips of the two wheels alternating, it is evident that the engine will be impelled forwards, at certain periods of each revolution, by one wheel only, and at others, by the other wheel only; and that this is true irrespective of the action of the two pistons on the crank, and would be true if the steam were thrown off. Such alternate propulsions on the two sides of the train cannot but

communicate alternate oscillations to the buffer-springs, the intervals between which will not be the same as those between the propulsions; but they may so synchronise with a series of propulsions as that the amplitude of each oscillation may be increased by them until the train attains that fish-tail motion with which railway travellers are familiar. It is obvious that the results shown here to follow from a displacement of the centres of gravity of the driving-wheels, cannot fail also to be produced by the alternate action of the connecting rods at the most favourable driving points of the crank and at the dead points *, and that the operation of these two causes may tend to neutralise or may exaggerate one another. It is not the object of this paper to discuss the question under this point of view.

NOTE F.

ON THE DESCENT UPON AN INCLINED PLANE OF A BODY SUBJECT TO VARIATIONS OF TEMPERATURE, AND ON THE MOTION OF GLACIERS.

If we conceive two bodies of the same form and dimensions (cubes for instance), and of the same material, to be placed upon a uniform horizontal plane and connected by a substance which alternately extends and contracts itself, as does a metallic rod when subjected to variations of temperature, it is evident that by the extension of the intervening rod each will be made to recede from the other by the same distance, and, by its contraction, to approach it by the same distance. But if they be placed on an inclined plane (one being lower than the other) then when by the increased temperature of the rod its tendency to extend becomes sufficient to push the lower of the two bodies downwards, it will not have become sufficient to push the higher upwards. The effect of its extension will therefore be to cause the lower of the two bodies to descend whilst the higher remains at rest. The converse of this will result from contraction; for when the contractile force becomes sufficient to pull the upper body down the plane it will not have become sufficient to pull the lower up it. Thus, in the contraction of the substance which intervenes between the two bodies, the lower will remain at rest whilst the upper descends. As often, then, as the expansion and contraction is repeated the two bodies will descend the plane until, step by step, they reach the bottom.

* A slip of the wheel may thus be, and probably is, produced at each revolution.