Let C1, C2 (fig. 96) be the poles of a pair of rolling curves; T1, T2 any pair of points of contact; U1, U2 any other pair of points of contact. Then, for every possible pair of points of contact, the two following equations must be simultaneously fulfilled:—
Sum of radii, C1U1 + C2U2 = C1T1 + C2T2 = constant;
arc, T2U2 = T1U1.
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A condition equivalent to the above, and necessarily connected with it, is, that at each pair of points of contact the inclinations of the curves to their radii-vectores shall be equal and contrary; or, denoting by r1, r2 the radii-vectores at any given pair of points of contact, and s the length of the equal arcs measured from a certain fixed pair of points of contact—
dr2/ds = −dr1/ds;
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which is the differential equation of a pair of rolling curves whose poles are at a constant distance apart.
For full details as to rolling curves, see Willis’s work, already mentioned, and Clerk Maxwell’s paper on Rolling Curves, Trans. Roy. Soc. Edin., 1849.
A rack, to work with a circular wheel, must be straight. To work with a wheel of any other figure, its section must be a rolling curve, subject to the condition that the perpendicular distance from the pole or centre of the wheel to a straight line parallel to the direction of the motion of the rack shall be constant. Let r1 be the radius-vector of a point of contact on the wheel, x2 the ordinate from the straight line before mentioned to the corresponding point of contact on the rack. Then
dx2/ds = −dr1/ds