that the principle just stated applies to the overcoming of all sorts of resistance, and not to the lifting of weights only.

The weight of the moving piece itself in a mechanical power may either be wholly supported at the bearing, if the piece is balanced; or if not, it is to be regarded as divided into two parallel components, one supported directly at the bearing, and the other being included in the effort or in the resistance, as the case may be.

The relation between the effort and the resistance in any mechanical power may be deduced from the principles of statics; viz. In the case of the LEVER (including the wheel and axle), from the balance of couples of equal and opposite moments; in the case of the INCLINED PLANE (including the wedge and the screw), from the parallelogram of forces; and in the case of the pulley, from the composition of parallel forces. The principle of virtual velocities, however, is more convenient in calculation.

The total load in a mechanical power is the resultant of the effort, the resistance, the lateral components of the forces acting at the driving and working points, and the weight directly carried at the bearings; and it is equal and directly opposed to the re-action of the bearings or supports of the machine.

By the purchase of a mechanical power is to be understood the ratio borne by the resistance to the effort, which is equal to the ratio borne by the velocity of the driving point to that of the working point. This term has already been explained in connection with the pulley, in Article 201, pages 215, 216.

The following are the results of the principle of virtual velocities, as applied to determine the purchase in the several mechanical powers:

I. LEVER. The effort and resistance are to each other in the inverse ratio of the perpendicular distances of their lines of action from the axis of rotation or fulcrum; so that the purchase is the ratio which the perpendicular distance of the effort from the axis bears to the perpendicular distance of the resistance from the axis.

Under the head of the lever may be comprehended all turning or rocking primary pieces in mechanism which are connected with their drivers and followers by linkwork.

II. WHEEL AND AXLE.-The purchase is the same as in the case of the lever; and the perpendicular distances of the lines of action of the effort and of the resistance from the axis are the radii of the pitch-circles of the wheel and of the axle respectively.

Under the head of the wheel and axle may be comprehended all turning or rocking primary pieces in mechanism which are connected with their drivers and followers by means of rolling contact, of teeth, or of bands. By the "wheel" is to be understood

the pitch-cylinder of that part of the piece which is driven; and by the "axle," the pitch-cylinder of that part of the piece which drives.

III. INCLINED PLANE, and IV. WEDGE.-Here the purchase, or ratio of the resistance to the effort, is the ratio borne by the whole velocity of the sliding body (represented by BC in fig. 165, page 233, and Cc in fig. 166, page 234) to that component of the velocity (represented by B D in fig. 165, page 233, and Ce in fig. 166, page 234) which is directly opposed to the resistance: it being understood that the effort is exerted in the direction of motion of the sliding body.

The term inclined plane may be used when the resistance to the motion of a body that slides along a guiding surface consists of its own weight, or of a force applied to a point in it by means of a link; and the term wedge, when that resistance consists of a pressure applied to a plane surface of the moving body, oblique "to its direction of motion.

V. SCREW.-Let the resistance (R) to the motion of a screw be a force acting along its axis, and directly opposed to its advance; and let the effort (P) which drives the screw be applied to a point rigidly attached to the screw, and at the distance r from the axis, and be exerted in the direction of motion of that point. Then, while the screw makes one revolution, the working point advances against the resistance through a distance equal to the pitch (p); and at the same time the driving point moves in its helical path through the distance (4 22+p2); therefore the purchase of the screw, neglecting friction, is expressed as follows:

VI. PULLEY. (See Articles 200 and 201, pages 214 to 216.)— In the pulley without friction, the purchase is the ratio borne by the resistance which opposes the advance of the running block to the effort exerted on the hauling part of the rope; and it is expressed by the number of plies of rope by which the running block is connected with the fixed block.

VII. The HYDRAULIC PRESS, when friction is neglected, may be included amongst the mechanical powers, agreeably to the definition of them given at the beginning of this Article. By the resistance is to be understood the force which opposes the outward motion of the press-plunger, A, fig. 159, page 224; and by the effort, the force which drives inward the pump-plunger, A'. The intensity of the pressure exerted between each of the two plungers

and the fluid is the same; therefore the amount of the pressure exerted between each plunger and the fluid is proportional to the area of that plunger; so that the purchase of the hydraulic press is expressed as follows:

R A transverse area of press-plunger

P A'

=

transverse area of pump-plunger

and this is the reciprocal of the ratio of the velocities of those plungers, as already shown in Article 209, page 223.

The purchase of a train of mechanical powers is the product of the purchases of the several elementary parts of that train.

The object of producing a purchase expressed by a number greater than unity is, to enable a resistance to be overcome by means of an effort smaller than itself, but acting through a greater distance; and the use of such a purchase is found chiefly in machines driven by muscular power, because of the effort being limited in amount.

SECTION IV.-Of Dynamometers.

340. Dynamometers are instruments for measuring and recording the energy exerted and work performed by machines. They may be classed as follows:

I. Instruments which merely indicate the force exerted between a driving body and a driven body, leaving the distance through which that force is exerted to be observed independently. The following are examples of this class :

a. The weight of a solid body may be so suspended as to balance the resistance, as in Scott Russell's experiments on the resistance of boats. (Edin. Trans., xiv.)

b. The weight of a column of liquid may be employed to balance and measure the effort required to drag a carriage or other body, as in Milne's mercurial dynamometer.

c. The available energy of a prime mover may be wholly expended in overcoming friction, which is measured by a weight, as in Prony's dynamometer (described further on).

d. A spring balance may be interposed between a prime mover and a body whose resistance it overcomes.

II. Instruments which record at once the force, motion, and work of a machine, by drawing a line, straight or curved, as the case may be, whose abscissæ represent the distances moved through, its ordinates the resistances overcome, and its area the work performed (as in fig. 241, page 346).

A dynamometer of this class consists essentially of two principal parts: a spring whose deflection indicates the force exerted between a driving body and a driven body; and a band of paper, or a card, moving at right angles to the direction of deflection of the spring

with a velocity bearing a known constant proportion to the velocity with which the resistance is overcome. The spring carries a pen or pencil, which marks on the paper or card the required line. The following are examples of this class of instruments:

a. Morin's Traction Dynamometer.

b. Morin's and Hirn's Rotatory Dynamometers.

c. The Steam Engine Indicator.

III. Instruments called Integrating Dynamometers, which record the work performed, but not the resistance and motion separately.

BF F

нал

E WE

Fig. 249.

K

341. Prony's Friction Dynamometer measures the useful work performed by a prime mover, by causing the whole of that work to be expended in overcoming the friction of a brake. In fig. 249, A represents a cylindrical drum, driven by the prime 'mover. The block D, attached to the lever B C, and the smaller blocks with which the chain E is shod, form a brake which embraces the drum, and which is tightened by means of the screws F, F, until its friction is sufficient to cause the drum to rotate at an uniform speed. The end C of the lever carries a scale G, in which weights are placed to an amount just sufficient to balance the friction, and keep the lever horizontal. The lever ought to be

so loaded at B that when there are no weights in the scale, it shall be balanced upon the axis. The lever is prevented from deviating to any inconvenient extent from a horizontal position by means of safety-stops or guards, H, K.

The weight of the load in the scale which balances the friction being multiplied into the horizontal distance of the point of suspension C from the axis, gives the moment of friction, which being multiplied into the angular velocity of the drum, gives the rate of useful work or effective power of the prime mover.

As the whole of that power is expended in overcoming the friction between the drum and the brake, the heat produced is in general considerable; and a stream of water must be directed on the rubbing surfaces to abstract that heat.

The friction dynamometer is simple and easily made; but it is ill adapted to measure a variable effort; and it requires that when the power of a prime mover is measured, its ordinary work should be interrupted, which is inconvenient and sometimes impracticable.

342. Morin's Traction Dynamometer. The descriptions of this and some other dynamometers invented by General Morin are abridged from his works, entitled Sur quelques Appareils dynamométriques and Notions fondamentales de Mécanique.

Fig. 250 is a plan and fig. 250 a an elevation of a dynamometer

for recording by a diagram the work of dragging a load horizontally. a a, b b are a pair of steel springs, through which the tractive force is transmitted, and which serve by their deflection to measure

Fig. 250.

that force.

Fig. 250 A.

They are connected together at the ends by the steel links f,f. The effort of the prime mover is applied, through the link r, to the gland d, which is fixed on the middle of the foremost spring; the equal and opposite resistance of the vehicle is applied to the gland c, which is fixed on the middle of the aftermost spring. When no tractive force is exerted, the inward faces of the springs are straight and parallel; when a force is exerted, the springs are bent, and are drawn apart, through a distance proportional to the force. The springs are protected against being bent so far as to injure them by means of the safety bridles i, i, with their bolts e, e. Those bridles are carried by the after-gland, and their bolts serve to stop the foremost spring when it is drawn forward as far as is consistent with the preservation of elasticity and strength.

The frame of the apparatus for giving motion to the paper band is carried by the after-gland. The principal parts of that apparatus are the following:

1, store drum on which the paper band is rolled, before the commencement of the experiment, and off which it is drawn as the experiment proceeds;