We shall omit the discussion of the former case, and proceed to the latter. Let P, represent the pressure parallel to its axis which is to be overcome by the action of the screw. Now it is evident that the pressure thus produced upon the thread of the screw is the same as though the whole central portion of it within the thread were removed, or as though the whole pressure P, were applied to a ring whose thickness is As or 2D. Now the area of this ring is represented by {(R+D)' -(R-D), or by 4RD. So that the pressure of P„, upon every square unit of it, is represented by P2 4RD Ar Let ar represent the exceedingly small thickness of such a ring whose radius is r, and which may therefore be conceived to represent the termination of the exceedingly thin cylindrical surface passing through the point p; the area of this ring is then represented by 2-rar, and therefore the pressure upon P1. 2-rar it by 2 " 4RD or by Prar Now this is evidently the pressure sustained by that elementary portion of the thread which passes through p, whose thickness is Ar, and which may be conceived to be generated by the enwrapping of a thin plane, whose inclination is 4, upon a cylinder whose radius is; whence it follows (by equation 311) that the elementary pressure AP,, which must be applied to the arm of the screw to overcome this portion of the resistance P thus applied parallel to the axis upon an element of the thread, is represented by · AP1 = (RD) () { sin. (+) cos., 0 +tan., }; whence, passing to the limit and integrating, we have cos. (++) tan. +tan. ❤, 1-tan. 9, tan. 9,—tan. 1 (tan. 9, +tan. .) tan. +tan. 1 (1—tan. 9, tan. 9,){1—tan. › tan. (9, +9.)} =tan. 9, +tan. +tan. (,,) tan. 2. Neglecting dimensions of tan. 9, and tan., above the first*, P. R+D .. P1 =2RD {(tan. 9, +tan. 1+tan. (9, +9,) tan. "),*+ 21 R-D Φι Substituting in this expression for tan. its value (equation 313), it becomes R+D P1 =2RDS {2 tan. 9, +Rr tan. I + R2 tan.” I tan. (9, +4,)+ R-D pr tan. 9,} dr. Integrating and reducing, P1 = P.R( 2 PR tan. 1+ (1+) tan.9, +(f) tan.9, + a l tan. 'I tan. (9, +9,) } . . . . . (315); whence we obtain by (equation 121) for the modulus, U=U,{1+{(1++) tar Φι tan.,+ tan. 'I tan. (,+)} cot. I }. . . . . (316). 256. Whence it follows that the best inclination of the thread, in respect to the economy of power in the use of the square screw, is that which satisfies the equation The inclination of thread of a square screw rarely exceeds 7°, so that the term tan. 'I tan. (,+,) rarely exceeds 015 tan. (9,+9,), and may therefore be neglected, as compared The integration is readily effected without this omission; and it might be desirable so to effect it where the theory of wooden screws is under discussion, the limiting angle of resistance being, in respect to such screws, considerable. The length and complication of the resulting expression has caused the omis sion of it in the text. with the other terms of the expression; as also may the term +(2) tan. 1, since the depth 2D of a square screw being usually made equal to about 4th of the diameter, this term does not commonly exceed tan. . Omitting these terms, observing that L-2R tan. I, and eliminating tan. I, 257. Let P., P., P., P, represent the pressures applied by the piston rod, the crank rod, the air pump rod, and the cold water pump rod, to the beam of a steam engine; and suppose the directions of all these pressures to be vertical.* Let the rods, by which the pressures P,, P,, P„, P, are applied to the beam, be moveable upon solid axes or gudgeons, whose centres are a, d, b, e, situated in the same straight line passing through the centre C of the solid axis of the beam. Let P1, P2, P., P. represent the radii of these gudgeons, p the radius of the axis of the beam, and 9,, 2, .,,, the limiting angles of resistance of these axes respectively. Then, it the beam be supposed in the state bordering upon motion * A supposition which in no case deviates greatly from the truth, and any error in which may be neglected, inasmuch as it can only influence the results about to be obtained in as far as they have reference to the friction of the beam; so that any error in the result must be of two dimensions, at least, in respect to the coefficient of friction and the small angle by which any pressure deviates from a vertical direction. by the preponderance of P,, each gudgeon or axis being upon the point of turning on its bearings, the directions of the pressures P1, P„, P„, P, R, will not be through the centres of their corresponding axes, but separated from them by perpendicular distances severally represented by p, sin. 1, fa sin., P, sin. O, P, sin. ., and p sin. , which distances, being perpendicular to the directions of the pressures, are all measured horizontally. Moreover, it is evident that the direction of the pressure P, is on that side of the centre a of its axis which is nearest to the centre of the beam, since the influence of the friction of the axis a is to diminish the effect of that pressure to turn the beam. And for a like reason it is evident that the directions of the pressures P,, P,, P, are farther from the centre of the beam than the centres of their several axes, since the effect of the friction is, in respect to each of these pressures, to increase the resistance which it opposes to the rotation of the beam; moreover, that the resistance R upon the axis of the beam has its direction upon the same side of the centre C as P1, since it is equal and opposite to the resultant pressure upon the beam, and that resultant would, by itself, turn the beam in the same direction as P, turns it. Let now a, Ca, a,=Cd, a,=Cb, a, Ce. Draw the hori zontal line of fCg, and let the angle aCf 8. Let, moreover, W be taken to represent the weight of the beam, supposed to act through the centre of its axis. Then since P,, P., P, P, W, R are pressures in equilibrium, we have, by the principle of the equality of moments, taking o as the point from which the moments are measured, P, . of = P, . og+ P1. oh +P.. ok + W. oC. 2 Now of Cf-Co=a, cos. -p, sin. ❤,—p sin. ❤, og=Cg+ Co=a, cos. +p, sin. +p sin. o, oh-Ch-Co=a, cos. + P, sin. -p sin. 9, ok=Ck+Co=a, cos. +p, sin. 9, +p sin. 9. Ps .. Pa, cos. -(p, sin. ❤, +p sin. ❤)} P, {a, cos. +(p, sin. 9, +p sin. )} + Pa, cos. +(p, sin. 9,-p sin. ) + Pa, cos.+(p, sin. ❤,+p sin. ❤)} +Wp sin. = (319). Multiplying this equation by 4, observing that a represents the space described by the point of application of P, so that Pa represents the work U, of P; and similarly that Pa, represents the work U, of P,, P,a,, that U, of P, and P, that U, of P., also that a, represents the space S, described by the extremity of the piston rod very nearly; which is the modulus of the beam. Its form will be greatly simplified if we assume cos. =1, since is small,* suppose the coefficient of friction at each axis to be the same, so that 9,=,=9 ̧=91, and divide by the coefficient of U,, omitting terms above the first dimension in የ1 sin. 7, &c.; whence we obtain by reduction. 258. The best position of the axis of the beam. Let a be taken to represent the length of the beam, and the distance aC of the centre of its axis from the extremity to which the driving pressure is applied. *In practice the angle 6 never exceeds 200, so that cos. never differs from unity by more than 060307. The error, resulting from which difference, in the friction, estimated as above, must in all cases be inconsiderable. |