336

MISCELLANEOUS

TRACT 38.

of a double oscillation, or of passing from A and arriving atA again, will be quadruple the time of passing over the ra-dius AC, or = 2 x 3-1416 v/ — l h 24' 29 ".

PROBLEM XIV.

Tofind the Time of a Pendulum vibrating in the Arc of dCycloid.

Let s be the point of suspension,

sa = the arc sb or sc the length of the pendulum,ca = ab = sb or sc the semi-cjmloid,ad = ds the diameter of its generating circle, to whichfke, hig are perpendiculars.

To any point g draw the tangent gp, also draw go paralleland pa perpendicular to ad. Then pg is parallel to thechord ai by the nature of the curve. And, by the natureof forces, the force of gravity: force in direct, gp : : gp :gq : : ai : ah : : ad : ai ; in like manner, the force of grav.: force in curve at e : : ad : ak; that is, the accelerativeforce in the curve, is as the corresponding chord ai or ak ofthe circle, or as the arc ag or ae of the cycloid, since ag isalwavs = 2ai. So that the process and conclusions for thevelocity and time of describing any arc in this case, will bethe same as in the last problem, the nature of the forces be-ing the same, viz, as the distance to be passed over to thelowest point A.

From which it follows, that the time of a semi-vibration,in all arcs, ag, ae, See, is the same constant quantity