LENGTH OF TEETH-INSIDE GEARING-INVOLUTE TEETII. 441 the positive sign applying to rolling outside, or describing the face, and the negative sign to rolling inside, or describing the flank. Hence the velocity of the tracing-point at a given instant is For the following wheel, q, and q have to be interchanged, so that, if r, be the radius of that wheel, The equations 2 and 3 evidently give the means of finding the distance of sliding between a pair of teeth, in a different form from that given in Article 453; for that distance is 456. To Inside Gearing all the preceding principles apply, observing that the radius of the greater, or concave pitch surface, is to be considered as negative, and that in Article 453, the difference of the angular velocities is to be taken instead of their sum. 457. Involute Teeth for Circular Wheels, being the first of the three kinds mentioned in Article 447, are of the form of the involute of a circle, of a radius less than the pitch circle in a ratio which may be expressed by the sine of a certain angle, and may be traced by the pole of a logarithmic spiral rolling on the pitch circle, the angle made by that spiral at each point with its own radius vector being the complement of the given angle . But this mode of describing involutes of circles, being more complex than the ordinary method, is mentioned merely to show that they fall under the general description of curves described by rolling. In fig. 198, let C1, C, be the centres of two circular wheels, whose pitch circles are B1, B2. Through the pitch point I draw the intended line of action P, P, making the angle CIP with 1 perpendicular to P, P, with which two perpendiculars as radii, describe circles (called base circles) D1, D B ינ Suppose the base circles to be a pair of circular pulleys, connected by means of a cord whose course from pulley to pulley is PIP As the line of connection of those pulleys is the same with that of the proposed teeth, they will rotate with the required velocity-ratio. Now suppose a tracing-point T to be fixed to the cord, so as to be carried along the path of contact P, I P„. That point will trace, on a plane rotating along with the wheel 1, part of the involute of the base circle D1, and on a plane rotating along with the wheel 2, part of the involute of the base circle D., and the two curves so traced will always touch each other in the required point of contact T, and will therefore fulfil the condition required by Article 451. B2 D2 C2 Fig. 198. All involute teeth of the same pitch work smoothly together. To find the length of the path of contact on either side of the pitch point I, it is to be observed that the distance between the fronts of two successive teeth as measured along P, I P2, is less than the pitch in the ratio sin : 1, and consequently that if distances not less than the pitch × sin be marked off either way from I towards P, and P, respectively, as the extremities of the path of contact, and if the addendum circles be described through the points so found, there will always be at least two pairs of teeth in action at once. In practice, it is usual to make the path of contact somewhat longer, viz., about 24 times the pitch; and with this length of path and the value of which is usual in practice, viz, 75, the addendum is about of the pitch. The teeth of a rack, to work correctly with wheels having involute teeth, should have plane surfaces, perpendicular to the line of connection, and consequently making, with the direction of motion of the rack, angles equal to the before-mentioned angle 6. 458. Sliding of Involute Teeth.-The distance through which a pair of involute teeth slide on each other, is found by observing that the distance from the point of contact of the teeth to the pitch point is given by the equation which reduces equation 3 of Article 455 to the following : This distance may also be expressed in terms of the extreme distances of the point of contact from the pitch point. denoted by t1, to; then Let these be 2 sin (2A.) For inside gearing, the difference of the reciprocals of the radii of the wheels is to be taken instead of their sum. The preceding formulæ, which are exact for involute teeth, are approximately correct for all teeth, if ◊ be taken to represent the mean value of the angle CIP between the line of centres and the line of action. The usual value of being 75°, sin @ = 31 nearly. 459. The Addendum of Involute Teeth, that is, their projection beyond the pitch circle, is found by considering, that for one of the wheels in fig. 198, such as the wheel 1, the real radius, or radius of the addendum circle, is the hypothenuse of a right-angled triangle, of which one side is the radius of the base circle CP, and the other is PI + the portion of the path of contact beyond I. Now CP=r, ⋅ sin ; PI=r1. cos. Let to be the portion of the path of contact above mentioned (=qe sin ), and d, the addendum of the wheel 1; then (r1+d,)2= r; • sin3 + (r, cos +t) ;...(1) and for the wheel 2 the suffixes 1 and 2 are to be interchanged. The usual value of sin is about and that of cos about 31 32' 4' The same formulæ apply to teeth of any figure, if be taken to represent the extreme value of the angle CI P. 460. The Smallest Pinion with Involute Teeth of a given pitch p, has its size fixed by the consideration that the path of contact of the flanks of its teeth, which must not be less than p· sin é, cannot be greater than the distance along the line of action from the pitch point to the base circle, I Prcos. Hence the least radius is ..(1.) which, for= 7510, gives for the radius r = 3.867 p, and for the circumference of the pitch circle, p x 3.867 × 2 = = 24.3 p; to which the next greater integer multiple of p is 25 p; and therefore twenty-five, as formerly stated, in Article 447, is the least number of involute teeth to be employed in a pinion. 461. Epicycloidal Teeth. For tracing the figures of teeth, the most convenient rolling curve is the circle. The path of contact which a point in its circumference traces is identical with the circle itself; the flanks of the teeth are internal, and their faces external epicycloids, for wheels; and both flanks and faces are cycloids for a rack. Wheels of the same pitch, with epicycloidal teeth traced by the same rolling circle, all work correctly with each other, whatsoever may be the numbers of their teeth; and they are said to belong to the same set. For a pitch circle of twice the radius of the rolling or describing circle (as it is called), the internal epicycloid is a straight line, being in fact a diameter of the pitch circle; so that the flanks of the teeth for such a pitch circle are planes radiating from the axis. For a smaller pitch circle, the flanks would be convex, and incurved or under-cut, which would be inconvenient; therefore the smallest wheel of a set should have its pitch circle of twice the radius of the describing circle, so that the flanks may be either straight or concave. In fig. 199, let B be part of the pitch circle of a wheel, C C the B C Fig. 199. R R line of centres, I the pitch-point, R the internal, and R' the equal external describing circles, so placed as to touch the pitch circle and each other at I; let DID' be the path of contact, consisting of the path of в approach D I, and the path of recess I D'. In order that there may always be at least two pairs of teeth in action, each of those arcs should be equal to the pitch. The angle é, on passing the line of centres, is 90°; the least value of that angle is =CID=<CID. It appears from experience that the least value of should be about 60°; therefore the arcs DI = I D' should each be one-sixth of a cir cumference; therefore the circumference of the describing circle should be six times the pitch. It follows that the smallest pinion of a set, in which pinion the flanks are straight, should have twelve teeth, as has already been stated in Article 447. 462. The Addendum for Epicycloidal Teeth is found from the formula already given in Article 459, equation 1, by putting for the angle CID, and for t, the chord I D' 2 r。 cose, r。 being the radius of the rolling circle. (r1+d1) = risin2 For the usual value of 4, 60°, sin2 Hence = 462 A. The Sliding of Epicycloidal Teeth is deduced from equation 3 of Article 455, by observing, that the radius vector of the point of contact is and that the extreme values of 7 are the arcs of approach and 463. Approximate Epicycloidal Teeth.-Willis has shown how to approximate to the figure of an epicycloidal tooth by means of two circular arcs, one concave for the flank, the other convex, for the face, and each having for its radius, the mean radius of curvature of the epicycloidal arc. Willis's formulæ are deduced in his own work from certain propositions respecting the transmission of motion by linkwork. In the present treatise they will be deduced from the values already given for the radii of curvature of |