loss expressed in degrees will be greater in proportion, the quantity of steam in the cylinder will also be greater; this point may, however, be more readily explained by an example. If steam be expanded from 30 lbs. pressure to 5 lbs. pressure, there is a loss of temperature of 48°. The number of units of heat abstracted from the metal will of course be proportional to this quantity, and the quantity of heat which the cool metal will absorb from the hot steam will also be proportional to the same quantity. If we expand steam at 60 lbs. down to 10 lbs., the loss of heat will be 67°, and the quantity abstracted from the hot steam at the next stroke will be proportional to this; but the quantity of metal has remained constant, whereas the weight of the steam is doubled: hence, to heat the metal 48°, steam at 30 lbs. pressure will have to yield 11 units of heat, whereas to heat the metal 67°, steam at a pressure of 60 lbs. per square inch would only have to lose 7 units. These considerations tend to show that the experiments give results which are reliable only under the circumstances under which they are conducted, and that steam of a higher pressure or engines differently constructed will give different results.

Some well-conducted experiments on the relations of heat to steam and mechanical work are now very much wanted, and it appears to us that these experiments should be, performed with apparatus of very accurate construction, admitting of a great variety of pressures, and also allowing of variations in the general circumstances, in order that the quantity of heat lost by radiation and conduction may be estimated.

We may here remark upon the use of other gases besides steam to propel thermo-dynamic engines. The most important of these applications consists in the employment of atmospheric air, and the air or caloric engines appears in many respects to have the advantage over steam-engines: the principle of working is of course similar, that is to say, the air is expanded by heat in order to obtain pressure.

Engines propelled by ether have also been proposed, but we are not aware that they have been found practically useful. We would in concluding this chapter recommend our readers to examine C. W. Williams's theory of the evaporation of water; which we have refrained from discussing in these pages, as it is as yet not established, although there are many points of importance which may be decided without very great difficulty.

CHAPTER VIII.

ON THE PRINCIPLES OF MECHANICAL CONSTRUCTION.

WE will now give a brief account of mechanics as applied to the construction of machinery, commencing with an account of the means of concentrating power. We will take as an example the ordinary lever. It will first be necessary to consider the manner in which a force acts round a centre. Let us suppose a force of 10 lbs. to act perpendicularly on one end of a bar, of which the other end is carried upon a centre, then the revolving force upon that centre will be proportional to the intensity of the weight or force, and to its distance from the centre. Let the bar be 6 feet long, then the relative intensity of the revolving force may be represented by 60 ft.-lbs. This revolving force is called a moment. We may find an equivalent moment by using a weight of 6 lbs. and a 10 ft. bar, for the moment in this case will also be 60 ft.-lbs. The general rule to find the moment of any given force about any given point will be, multiply the intensity of the force by its distance from the point measured perpendicularly to the direction of the force. In Fig. 24 we illustrate the manner in which this distance is measured. A force w acts in the direction ab; it is required to find its moment about the point c; from c let fall a perpendicular upon a b, and call the length of this perpendicular x, then will the moment of the weight about c

= WX.

Fig. 24.

From the above remarks it appears that any two moments will be equal when the distances of the weights producing them from the centres to which they are referred, vary inversely as the weights. Suppose this condition to be fulfilled, and let the

moments act about the same centre, but in opposite directions; then will a condition of equilibrium be attained, and in this balance of moments it is that the principle of the lever consists.

Let us suppose that we have an ordinary bar supported at one third of its length upon a pin or gudgeon, about which it is free to revolve, the weight of the bar being at present neglected, and let the length of the bar be 9 ft., then, on one side of the centre, pin, or fulcrum, as it is termed, there will be a length of 6 ft., and on the other a length of 3 ft.; let a weight equal to 500 lbs. be attached to the shorter end, it is required to find the weight which must be attached to the longer end in order to balance this weight. The weights and their distances must vary inversely as each other; hence we may solve this question by proportion,

thus

6: 3 :: 500: 250.

250 lb. will therefore be the weight required. We may give as the general rule for solving similar questions the following. To find the weight which, attached to one arm of a given lever, will balance a known weight attached to the other arm, multiply the weight by the length of the arm supporting it and divide the product by the length of the other arm, the quotient will be the quantity required. Thus in the above case we have

500 × = 250.

This rule will apply to every kind of lever, care being taken to observe the conditions under which it acts; its principle, however, is the same whether the arms be in a straight line with each other, or whether they be parallel or contain an angle, and if the length of the arms remains constant, the same forces will maintain equilibrium. Various forms of levers are shown, Fig. 25, but the same length of arms is preserved in every case. We may here observe that the proportions between the weights and arms refer to relative quantities, and not to absolute; thus a lever having arms 3 ft. and 6 ft. long will have the same value as one with arms 4 ft. and 8 ft. long, for the proportion of the arms is the same in both cases, as shown by the following equation—

=2=8

Let us now compare the work performed when the arms move about the fulcrum in the case of the lever mentioned above. Let

the long arm move through 1 ft, then the amount of work executed will be

250 × 1 = 250 ft.-lbs.

Let us now examine the amount of work executed at the same time at the other end of the lever. We must first find the space

Fig. 25.

through which the end of the short arm will move, whilst that of the long arm moves through 1 ft. The ends of the arms describe circles about the fulcrum; hence, in moving through the space mentioned above, a part of the circumference of a circle will be described, and the distance passed through will vary as the length of the arms which are the radii of the circular arcs; hence, the end of the short arm, which carries the 500 lb. weight, will move through half the space of the long arm, or through ft., the lengths of the arms being 6 ft. and 3 ft., and the amount of work performed at the extremity of the short arm will be

which is equal to that performed at the end of the long arm.

From the above observations, we find that by means of a lever we may raise a given weight by a force equivalent to a much smaller weight, but at the expense of time; hence, in this case power is not gained, but a force expended during a certain time is concentrated to overcome a greater force, the static forces being unequal, but the quantity of work done by them in a given time being equal.

We may now generalize the results of the investigation of the laws of the lever, in order to apply it to other machines for con

centrating power in the following manner:-In any machine let x represent the distance through which a given force is to be exerted, or through which an equivalent weight is to be lifted. Let w equal this weight or force; let y equal the distance through which the pressure required to raise it will move in the same time that w will move through x, w being equal to the pressure, then the amount of work to be executed will be

=wx,

the work done by the motive power will be

= wy

These two quantities must be equal, or rather, to produce motion, one must preponderate by an infinitely small quantity, otherwise the apparatus will remain in equilibrio; the balancing forces may be found from the following equations:

These equations will of course apply to simple or complicated machines, where an uniform resistance is overcome by an uniform force, x being the distance through which the point of resistance moves in a given time, and y the distance through which the point of application of the power moves in the same time.

The pulley and axle are evidently identical in their action with the two arms of a lever. The screw and inclined plane act differently, but the law given above will of course be applicable, the distances moved through being very easily found; thus, when a single threaded screw revolves once, any body whichis being raised by it passes through a distance equal to that between two threads of the screw measured from centre to centre.

We may now instance another means of concentrating power, viz., by hydraulic pressure. Let a and b, Fig. 26, represent two