...curves, both for historical interest and for the variety of its physical applications. This curve is generated by the motion of a point on the circumference of a circle which rolls along a right line. Let A be the point where the motion commences; then, in any position of the generating...

...curves, both for historical interest and for the variety of its physical applications. This curve is generated by the motion of a point on the circumference of a circle which rolls along a right line. Let A be the point where the motion commences ; then, in any position of the generating...

..._!?L*.4 = 3y2 P 9y* 3y2 (9y< Ca'y When y = 0, r = ao , and when y = ± oo , r = oo . 5. The cycloid, or curve generated by the motion of a point on the circumference of a circle, while the circle rolls on a straight line. Let the radius of the generating circle =: a. Place the...

...parabola y2 = a?x. [l+£ 6a*y When y = 0, r = ao , and when y = ± oo , r = « . 5. The cycloid, or curve generated by the motion of -a point on the circumference of a circle, while the circle rolls on a straight line. Let the radius of the generating circle = a. Place the origin...

...:i,/2 — ~ Q,,5* V y' + a»)* When y = 0, r = x , and when y = ± oo , r — oo . 5. The cycloid, or curve generated by the motion of a point on the circumference of a circle, while the circle rolls on a straight line. Let the radius of the generating circle = a. Place the origin...

...acted on by gravity for a small arc only. If a particle be dropped down a cycloid (the curve traced by a point on the circumference of a circle which rolls on a straight line), the time of oscillation will be the same wherever the starting point may be. On account of this remarkable...

...radius of curvature ; but the conclusion is verified by the known properties of cycloids and trochoids. A point on the circumference of a circle which rolls on a straight line has a cusp in its path at the point where it comes to the straight line. This cusp is a point of zero...

...Archimedes previously noticed (§ 62). 123. Epicycloids and Hypocycloids. These curves are the roulettes of a point on the circumference of a circle which rolls on the outside or inside of a fixed circle (§ 9G). Let 0 denote the centre and a the radius of the fixed...

...outlines made up of curves called epicycloidal and hypocycloidal curves. These curves are developed by the motion of a point on the circumference of a circle that is rolled on another circle. When the rolling circle ia on the outside of the other, the curve...

...in drawing gear teeth are the cycloidal and involute. Cycloid. The cycloid is a curve generated by a point on the circumference of a circle which rolls on a straight line tangent to the circle. The rolling circle is called the describing or generating circle and the point,...