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: 2 P1 =P2. Þ2+Þ2 . . . . . (119); where, and 4, 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 ," and 4, represent the values of these functions when the prejudicial resistances vanish, then Φ. (=0 and Þ ̧”= 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 P2, if there were no prejudicial resistances opposed to the motion of the machine. Then, by the last equation, P1=P2. Þ ̧. 1 1 (0) 1 But by the principle of virtual velocities (Art. 127.), if we suppose the motion of the machine to be uniform, so that P1 and P, are constantly in equilibrium upon it, and if we represent by S, any space described by the point of application of P1, or the projection of that space on the direction of P1 (Art. 52.), and by S, the corresponding space or projection of the space described by P,, then P. S1 = P2. S2. Therefore, dividing this equation by the last, we have which is the modulus of the machine, so that the constant A in 2 The above equation has been proved for any value of S1, provided the values of P, and P, be constant, and the motion of the machine uniform; it evidently obtains, therefore, for an exceedingly small value of S1, 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 axis, 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. m L R For let R represent the resultant of P, P2, &c. Then, since these forces are supposed to be upon the point of causing the axis of the body to turn upon its bearings, their resultent would, if made to replace them, be also on the point of causing the axis to turn on its bearings. Hence it follows 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 perpendicular 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 resistance of the axis and its bearings.* Now the resistance of the axis is evidently equal and opposite to the resultant R of all the forces P1, P2, &c. impressed upon the body. This resistance acts, therefore, in the direction LR, and is inclined to CL at an angle equal to the limiting angle of resistance. Therefore, &c. RA B THE WHEEL AND AXLE. 154. The pressures P, and P, applied vertically by means of parallel cords to a wheel and axle are in the state bordering upon motion by the preponderance of P1, it is required to determine a relation between P, and P2. The direction LR of the resistance of the axis is on that side of the centre which is towards P1, and is inclined to the perpendicular CL at the point L, where it intersects the axis at an angle CLR equal to the limiting angle of resistance. Let this angle be represented by ø, and the radius CL of the axis by p; also the radius CA of the wheel by a,, and that CB of the axle by a,; and let W be the weight of the wheel and axle, whose centre of gravity is supposed to be C. Now, the pressures P1, P2, the weight W of the wheel and axle, and the resistance R of the axis, are pressures in equilibrium. Therefore, by the principle of the equality of moments (Art. 7.), neglecting the rigidity of the cord, and observing that the weight W may be supposed to act through C, we have, If, instead of P, preponderating, it had been on the point *The side of C on which RL falls is manifestly determined by the direction-towards which the motion is about to take place. In this case it is supposed about to take place to the right of C. If it had been to the left, the direction of R would have been on the opposite side of C. of yielding, or P, had been in the act of preponderating, then R would have fallen on the other side of C, and we should have obtained the relation P, . CA=P2 . CB-R. Cm; so that, generally, P, . CA=P2. CB±R. Cm; the sign + being taken according as P, is in the superior or inferior state bordering upon motion. Now CA=a,, CB=a,, Cm- CL sin. CLR=p sin. 4, and R =P1+P2± W, the sign + being taken according as the weight W of the wheel and axle acts in the same direction with the pressures P, and P2, or in the opposite direction; that is, according as the pressures P, and P, act vertically downwards (as shown in the figure) or upwards; 2 ... P1a1=P ̧a2+(P1+P2±W) p sin. 4, 22 1 2 ... P1(a,—p sin. 4)=P2(a2+p sin. 4)±Wp sin. 4. 2 Now the effect (Art. 142.) of the rigidity of the cord BP2 is the same as though it increased the tension upon that cord D+E. P from P, to (P2+ · P.): allowing, therefore, for the rigidity of the cord, we have finally P1(q1—p sin.q) = (P2+ D+E. P2)(a, + p sin. ¢) ± Wp sin. ¢, 2 aa1-p sin. 2 which is the required relation between P, and P, in the state bordering upon motion. α sin. and sin. are in all cases exceedingly small; we may therefore omit, without materially affecting the result, all terms involving powers of these quantities above the first, we shall thus obtain by reduction DS P.=P, (+) { 1 + (+-)p sin.}+1+G++) sin... (123). P‚=P, E) a *+ Ρ 155. The modulus of uniform motion in the wheel and axle. =(1. if we take Þ1=(1+ Now observing that represents the value of 9, when the prejudicial resistances vanish (or when =0 and E=0), which is the modulus of the wheel and axle. Omitting terms involving dimensions of sin. 4, and sin. , and above the first, we have E a E U=U2 ρ S,D W U=U, 1+++) sin. of +1+++) sin. (125). 156. The modulus of variable motion in the wheel and axle. 2 If the relation of P, and P, be not that of either state bordering upon motion, then the motion will be continually accelerated or continually retarded, and work will continually accumulate in the moving parts of the machine, or the work (124), |