be supposed to vary as the square of the velocity V, then will the work expended on this resistance vary as V2. S, or as V. T, since S-V. T. If then U, and U, represent the work done at the moving and working points during the time T, then does the modulus (equation 114) assume, in this case, the formn

U1=A. U2+B. V.T+C. V'. T . . . . . (115).

THE MODULUS OF A MACHINE MOVING WITH AN ACCELERATED OR RETARDED MOTION.

150. In the two last articles the work U,, done upon the moving point or points of the machine, has been supposed to be just that necessary to overcome the useful and prejudicial resistances opposed to the motion of the machine, either continually or periodically; so that all the work may be expended upon these resistances, and none accumulated in the moving parts of the machine as the work proceeds, or else that the accumulated work may return to the same amount from period to period. Let us now suppose this equality to cease, and the work U, done by the moving power to exceed that necessary to overcome the useful and prejudicial resistances; and to distinguish the work represented by U, in the one case from that in the other, let us suppose the former (that which is in excess of the resistances) to be represented by U1; also let U, be the useful work of the machine, done through a given space S., and which is supposed the same whatever may be the velocity of the motion of the machine whilst that space is being described; moreover, let S, be the space described by the moving point, whilst the space S, is being described by the working point.

Now since U, is the work which must be done at the moving point just to overcome the resistances opposed to the motion of that point, and U' is the work actually done upon that point by the power, therefore U-U, is the excess of the work done by the power over that expended on the resistances, and is therefore equal to the work accumulated in the machine (Art. 130.); that is, to one half of the increase of the vis viva through the space S, (Art. 129.); so that. if, represent the velocity of any element of the machine (whose weight is w) when the work U' began to be done, and, its velocity when that work has been com pleted, then (Art. 129.),

1 2g

.. U'=A. U2+B. S,+w(v,v,') . . . . . (116).

. .
.....

If instead of the work U' done by the power exceeding that U, expended on the resistances it had been less than it, then, instead of work being accumulated continually through the space S,, it would continually have been lost, and we should. have had the relation (Art. 129.),

The equation (116) applies therefore to the case of a retarded motion of the machine as well as to that of an accelerated motion, and is the general expression for the modulus of a machine moving with a variable motion. Whilst the co-efficients A and B of the modulus are dependent wholly upon the friction and other direct resistances to the motion of the machine, the last term of it is wholly independent of all these resistances, its amount being determined solely by the velocities of the various moving ele ments of the machines and their respective weights.

THE VELOCITY OF A MACHINE MOVING WITH A VARIABLE MOTION.

151. The velocities of the different parts or elements of every machine are evidently connected with one another by certain invariable relations, capable of being expressed by algebraical formulæ, so that, although these relations are different for different machines, they are the same for ail circumstances of the motion of the same machine. In a great number of machines this relation is expressed by a constant ratio. Let the constant ratio of the velocity v, of any element to that V, of the moving point in such a.

Sub

machine be represented by λ, so that v,V,, and .et v, and V, be any other values of v, and V,; then v,^V ̧. stituting these values of v, and v, in equation (116), we have

1

U'—A . U,+B. S‚ + — (V,' — V1')±w22.
-(V,'— V‚')±wλ' . . . . . (117);

2g

in which expression wλ represents the sum of the weights of all the moving elements of the machine, each being multiplied by the square of the ratio of its velocity to that of the point where the machine receives the operation of its moving power. For the same machine this co-efficient w is therefore a constant quantity. For different machines it is different. It is wholly independent of the useful or prejudicial resistances opposed to the motion of the machine, and has its value determined solely by the weights and dimensions of the moving masses, and the manner in which they are connected with one another in the machine. Transforming this equation and reducing, we have

by which equation the velocity V, of the moving point of the machine is determined, after a given amount of work U has been done upon it by the moving power, and a given amount U, expended on the useful resistances; the velocity of the moving point, when this work began to be done being given and represented by V1.

It is evident that the motion of the machine is more equable as the quantity represented by Zwλ is greater. This quantity, which is the same for the same machine and different for different machines, and which distinguishes machines from one another in respect to the steadiness of their motion, independently of all considerations arising out of the nature of the resistances useful or prejudicial opposed to it, may with propriety be called the co-EFFICIENT OF EQUABLE MOTION.* The actual motion of the machine is more equable as this co-efficient and as the co-efficients A and B (supposed positive) are greater.

* The co-efficient of equable motion is here, for the first time, introduced into the consideration of the theory of machines.

TO DETERMINE THE CO-EFFICIENTS OF THE MODULUS OF A MACHINE.

152. Let that relation first be determined between the moving pressure P, upon the machine and its working pressure P,, which obtains in the state bordering upon motion by the preponderance of P,. This relation will, in all cases where the constant resistances to the motion of the machine independently of P, are small as compared with P, be found to be represented by formulæ of which the following is the general type or form:

P‚=P, . §,+*, . . . . . (119);

where, and, represent certain functions of the friction and other prejudicial resistances in the machine, of which the latter disappears when the resistances vanish and the former does not; so that if (0) and (0) represent the values of these functions when the prejudicial resistances vanish, then =0 and 40= a given finite quantity dependent for its amount on the composition of the machine. Let P represent that value of the pressure P, which would be in equilibrium with the given pressure P,, if there were no prejudicial resistances opposed to the motion of the machine. Then, by the last equation, P, P, . 4,0.

1

But by the principle of virtual velocities (Art. 127.), if we suppose the motion of the machine to be uniform, so that P, and P, are constantly in equilibrium upon it, and if we represent by S, any space described by the point of application of P, or the projection of that space on the direction of P, (Art. 52.), and by S, the corresponding space or projection of the space described by P, then P. SP, . S. Therefore, dividing this equation by the last, we have

which is the modulus of the machine, so that the constant

The above equation has been proved for any value of S1, provided the values of P, and P, be constant, and the inotion of the machine uniform; it evidently obtains, therefore, for an exceedingly small value of S,, when the motion. of the machine is variable.

GENERAL CONDITION OF THE STATE BORDERING UPON MOTION IN A BODY ACTED UPON BY PRESSURES IN THE SAME PLANE. AND MOVEABLE ABOUT A CYLINDRICAL AXIS.

153. If any number of pressures P1, P2, P3, &c. applied in the same plane to a body moveable about a cylindrical aris, be in the state bordering upon motion, then is the direction of the resistance of the axis inclined to its radius. at the point where it intersects the circumference, at an angle equal to the limiting angle of resistance.

For let R represent the resultant of P, P, &c. Then, since these forces are supposed to be upon the R point of causing the axis of the body to turn upon its bearings, their resultant would, if made to replace them, be also on the point of causing the axis to turn on its bearings. Hence it folL lows that the direction of this resultant R cannot be through the centre C of the axis; for if it were, then the axis would be pressed by it in the direction of a radius, that is, perpendicularly upon its bearings, and could not be made to turn upon them by that pressure, or to be upon the point of turning upon them. The direction of R must then be on one side of C, so as to press the axis upon its bearings in a direction RL, inclined to the normal CL (at the point L, where it intersects the circumference of the axis) at a certain angle RLC. Moreover, it is evident (Art. 141.), that since this force R pressing the axis upon its bearings at L is upon the point of causing it to slip upon them, this inclination RLC of R to the perpendicular CL is equal to the limiting angle of