III.

On Rolling Curves. By HAMNETT HOLDITCH, M.A., Fellow of
Caius College, and of the Cambridge Philosophical Society.

[Read December 10, 1838.]

IN the fifth volume of the Acta Petropolitana, Euler referred to a class of curves which, when caused to turn round fixed centres, possessed the property of communicating motion to each other without friction; he deduced also their characteristic property, that the point of contact remains always in the straight line joining their centres: he has not however followed out the investigation so as to furnish actual forms of curves, neither has this been done by any other writer that I am aware of, and consequently no method exists by which such curves can be obtained. But as they are practically employed in a manner which I shall proceed to explain, and commonly found by a tentative process, it appeared worth while to search for forms and rules for their construction, independently of the analytical interest that may be supposed to attach to such investigation.

Let Anm, Bn,m, (Fig. 1.) be two curves capable of rolling together, and having their centres of rotation A and B fixed at a distance equal to the sum of their apsidal distances, Am being a long and Bn, a short apsidal distance, then if n Am be caused to turn round in the direction of the arrow, it will press against Bn,m, and communicate a rotation to it. This action will, however, cease when the point m has reached n ̧; for beyond that point the radii of mAn will diminish, and its circumference begin to recede from the other curve.

No continuous motion of B can therefore be derived from that of A, if they be continuous curves, unless their outlines be treated like the pitch lines of ordinary wheels, and be indented with small teeth at regular distances; these teeth, as in the usual forms, projecting nearly as far beyond the pitched line or circumference as they extend within it. If this be done, it will be found that the circumference of A will retain its hold on that of B in all positions, as well on the receding as on the advancing sides of the curve. A continuous uniform rotation of one curve will produce a rotation of the other, not uniform, but continually varying in its angular velocity, as the ratio of the radius of A to that of B; this becomes then a commodious contrivance for converting an equable angular velocity into an unequal one, and is sometimes so used by Mechanists. Fergusson's well-known Cometarium was constructed on this principle: it is to be found in use in some silk machinery, where it is introduced for the purpose of correcting the unequal action of the common excentric in laying the silk upon the bobbins; it has also been used by Messrs Bacon and Donkin, in their printing machinery. I am informed by Professor Willis, who drew my attention to the subject of these curves, and furnished me with the above practical information, that the copious collections of Messrs Lanz and Betancourt, and that of Borgnis, furnish no example of the application of rolling curves to the purposes of machinery; which may therefore be considered to have been unknown to them.

When two such curves roll on each other, let r be the distance of their point of contact from the centre of rotation of the first curve, rde and the angle made by with a fixed radius; then is the tandr gent of the angle the curve makes with r; and r, and 0, being correr,do sponding quantities in the second curve, is the tangent of the dr

angle it makes with r,, and as r and r, are in the same straight line, and the curves must have a common tangent at the point of contact, these two angles must be equal, and

Also, if c be the distance of the centres, r+r, = c, and

.. dr2 + r2 do2 = dr2
dr2 (1+

(1

r2d02
dre

r2do2
dr2

= dr2 ( 1 +
:(1 = dr2 + r2de2,

or the differentials of the lengths of those parts of the curves which have been in contact are equal, and

are equations which contain the analytical conditions of such curves.

We will first consider the case of two equal and similar curves rolling on each other. Since

de
dr

is some function of r,

rde dr

must also

be a function of r, let it = f(r); and as r, and e, belong to a point in a similar and equal curve,

r,do

dr = f(r); and r1 = c − r; :. ƒ(r) =ƒ(c − r),

the solution of which equation is f(r) = p(r, cr); any symmetric function of r and cr, and if any form be given to in the

rde

equation = p(r, cr), the integration of the latter will give the

dr

equation to a curve having the proposed property. If we suppose it to have greater and less apsidal distances a and b, which most curves which can practically be used, must possess; then, as in revolving the greater apsidal distance of one must come into contact with the less apsidal distance of the other, a + b = c;

Now (ar). (r − b) = (a + b) . r — r2 — ab = r. (c

symmetric function of r and cr; and as at apses

assume

rdo

=

X(r, c− r)

dr √(a− r). (r− b) function of r and c-r which is not divisible by √(a − r). (r — b), the curve will be confined between the apsidal distances; and supposing also, that contains only positive integral powers of r and c - r, this last equation can always be integrated in finite terms.

consists of even powers of a only, and therefore X will contain no negative powers of x, and will be of the form

and limiting the investigation, for the sake of simplicity, to the first

-

.. (a − r). (r — b) = (a + b) . r — p2 — ab = 2ar – p2 — a2 + ß2 = ß2 — (r − a)2;

k,+ kẞ cos P.do
a + ß cos &

a)2

2.do

do + kado - kß cos pdp.