peating the process by which FL is drawn, or by the following process:-About C, with the radius C N P F, draw a circle; FLN will be a straight tangent to that circle; and so also will all the tangents to the flanks at their inner ends. Therefore, from the inner ends of all the flanks, both front and back, draw straight lines touching the circle C N, and so placed that the straight lines from the front and back flanks of the same tooth shall not cross each other; these lines will show the proper positions for the side parts of the clearing curves. When the flank-circle coincides with the base-circle (as in the smallest pinion of a given pitch), the side parts of the clearing curves coincide with the radii drawn from the centre C to the inner ends of the flanks.

132. Involute Teeth for Racks.-The following is the process of designing the teeth of a straight rack which is to gear with an involute-toothed wheel of a given pitch and a given obliquity:-In fig. 87, let A B be the pitch-line of the rack, and let A III' be the pitch. Lay off A IE

the given angle of obliquity, and from A let A fall A E perpendicular to I E; then I E will be the normal pitch; further, if the path of contact is

E

H

Fig. 87.

to consist of two halves, each equal to half the normal pitch, I E will be one of those halves; then in EI produced make IF I E, and I F will be the other half of the path of contact. Through E, parallel to A B, draw E G G'; this will be the addendum line; through F, parallel to B A, draw F H; this will be the flank-line, marking the inner ends of the acting surfaces of the teeth. Perpendicular to A B draw I K, equal to the greatest addendum in the set of wheels of the given pitch and obliquity with an allowance for clearance added, as in Rule III. of Article 131, page 123; through K, parallel to A B, draw a straight line; this will be the root-line, with which the bottoms of all the hollows between the teeth are to coincide.

The traces of the fronts of the teeth are straight lines perpendicular to E F, and the fronts themselves are planes perpendicular to EF. The backs of the teeth are planes inclined at the same angle to A B in the contrary direction.

133. Peculiar Properties of Involute Teeth.-Involute teeth have some peculiar properties not possessed by teeth of other figures.

I. Sets of involute teeth have a definite and constant normal pitch; being, as already explained, the distance between the fronts of successive teeth, measured on the path of contact, or on the circumference of the base-circle; and all wheels and racks with involute teeth of the same normal pitch gear correctly with each other.

II. The length of the line of centres, or perpendicular distance between the axes, of a pair of wheels with involute teeth of the same normal pitch, or the perpendicular distance from the axis of a wheel with involute teeth to the addendum-line of a rack with which it gears, may be altered; and so long as the wheels, or wheel and rack, are sufficiently near together to make the path of contact longer than the normal pitch, and sufficiently far asunder for the crests of each set of teeth to clear the hollows between the teeth of the other set, the wheels, or the wheel and rack, will continue to work correctly together, and to preserve their velocity-ratio; although, in the case of a pair of wheels, the pitch-lines, the pitch as measured on the pitch-lines, and the obliquity, will all be altered when the length of the line of centres is altered. In other words, the velocity-ratio of a pair of wheels with involute teeth of the same normal pitch is the reciprocal of the ratio of the radii of their

base-circles, and depends on this ratio alone; and the velocity-ratio of a wheel and rack with involute teeth of the same normal pitch

depends solely on the radius of the base-circle of the wheel and on the angle of obliquity of the line of connection.

Another way of stating this property of involute teeth is, that the pitch-lines of wheels and racks with such teeth are arbitrary to an extent limited only by the necessity of having a path of contact of a certain length.

One practical result of this is (as Mr. Willis first pointed out), that the back-lash of involute teeth is variable at will, being capable of being increased or diminished by moving the wheels, or the wheel and rack, farther from or nearer to each other, and may thus be adjusted so as to be no greater than is absolutely necessary in order to prevent jamming of the teeth-a property not possessed by teeth of any other figure.

III. Given (in fig. 88), the centres, C, C, the base-circles, D D, D' D', and the addendum-circles, A A, A' A', of a pair of spurwheels with involute teeth of a given normal pitch, to find the line of connection, the pitch-point, the pitch-circles, the pitch on the pitchcircles, and the path of contact.

Draw a common tangent, P P', to the two base-circles in such a position as to run from the driver to the follower in the direction of motion. That common tangent will be the line of connection: the point I, where it cuts the line of centres, will be the pitch-point: two circles, B B and B' B', described about C and C respectively, and touching each other in I, will be the pitch-circles: the pitch on the pitch-circles will be greater than the normal pitch in the CI C'I

ratio

=

CP C' P

and the part E E' of the line of connection which lies between the two addendum-circles will be the path of contact. IV. Given (in fig. 89), the centre, C, the base-circle, D D, and the addendum-circle, A A, of a spur-wheel with involute teeth of a given

normal pitch; also the pitch-line, B' B', and the addendum-line, A' A', of a rack which is to have involute teeth of the same normal pitch; to

find the pitch-point, the pitch-circle of the wheel, the line of connection, the pitch, as measured on the pitch-lines, the path of contact, and the position of the fronts of the teeth of the rack.

From C let fall C I perpendicular to B' B'; then I will be the pitch-point; and a circle, B B, of the radius C I, will be the pitchcircle of the wheel. From I draw I P, touching the base-circle D D; I P will be the line of connection. The pitch, as measured on the pitch-lines, will be greater than the normal pitch, in the CI

ratio of the radius of the pitch-circle to that of the base-circle. C P

The path of contact will be the part E E' of the line of connection, which is contained between the addendum-line of the rack, A' A', and the addendum-circle of the wheel, A A. The fronts of the teeth of the rack are to be planes perpendicular to I P, or, in other words, parallel to P C.

V. By the application of the preceding principles, two or more wheels of different numbers of teeth, turning about one axis, can be made to gear correctly with one wheel or with one rack; or two or more parallel racks, with different obliquities of action, may be made to gear correctly with one wheel, the normal pitches in each case being the same; and thus differential movements of various sorts may be obtained. This is not possible with teeth of any other form.

The obliquity of the action of involute teeth is by many considered an objection to their use; and that is the reason why, notwithstanding their simplicity and their other advantages, they are not so often used as other forms. In anticipation of the subject of the dynamics of machinery it may be stated, that the principal effect of the obliquity of the action of involute teeth is to increase the pressure exerted between the acting surfaces of the teeth, and also the pressure exerted between the axles of the wheels and their bearings, nearly in the ratio in which the radius of the pitch-circle of each wheel is greater than the radius of the base-circle, and that a corresponding increase of friction is produced by that increase of pressure. In the example of Article 131, that ratio is 65: 63.

134. Teeth for a Given Path of Contact.-In the three preceding Articles the forms of the teeth are found by assuming a figure for the path of contact-viz., the straight line. Any other convenient figure may be assumed for the path of contact, and the corresponding forms of the teeth found, by determining what curves a point moving along the assumed path of contact will trace on two discs, rotating round the centres of the wheels with angular velocities, which bear that relation to the component velocity of the tracing-point along the line of connection which is given by the principles of Article 127, page 118. This method of finding the forms of the teeth of wheels is the subject of an interesting treatise by Mr. Edward Sang.

All wheels having teeth of the same pitch, traced from the same path of contact, work correctly together, and are said to belong to the same set.

135. Teeth Traced by Rolling Curves. (A. M., 452.)—From the principles of Articles 122 and 123, pages 114, 115, it appears that at every instant the position of the point of contact, T, of the acting surfaces of a pair of teeth (fig. 82, page 115), and the corresponding position of the pitch-point I in the pitch-lines of the wheels to which those teeth belong, are so related, that the line, I T, which joins them, is normal to the surface of each of the teeth at the point T. Now this is the relation which exists between the tracingpoint T, and the instantaneous axis or line of contact I, in a rolling curve of such a figure, that, being rolled upon the pitch-line, its tracing-point T traces the outline of a tooth. (As to rolling curves and rolled curves, see Articles 72, 74, 75, 77, 78, 79, pages 51 to 62.)

In order that a pair of teeth may work correctly together, it is necessary and sufficient that the instantaneous normals from the pitch-point to the acting surfaces of the two teeth should coincide at each instant; and this condition is fulfilled if the outlines of the two teeth be traced by the motion of the same tracing-point, in rolling the same rolling curve on the same side of the pitch-lines of the respective wheels.

The flank of a tooth is traced while the rolling curve rolls inside of the pitch-line; the face, while it rolls outside.

B

Ꭱ

B

E

To illustrate this more fully, the following explanation is quoted from the Article "Mechanics (Applied)," in the Encyclopædia Brit annica (see fig. 90):-"If any curve, R, be rolled on the inside of the pitch-line, B B, of a wheel, the instantaneous axis of the rolling curve at any instant will be at the point I, where it touches the pitch-line for the moment; and consequently the line A T, traced by a tracingpoint T, fixed to the rolling

R

A.

E

Fig. 90.

curve, will be everywhere perpendicular to the straight line TI; so that the traced curve A T will be suitable for the flank of a tooth, in which T is the point of contact corresponding to the position I of the pitch-point. If the same rolling curve R, with the same tracing-point T, be rolled on the outside of any other pitch-line, it will trace the face of a tooth suitable to work with the flank AT.

"In like manner, if either the same or any other rolling curve

K