iG PHILOSOPHICAL TBANS ACTIONS. [aNNO 1713. 



Further, the two curves acdeb, ajtiJsb being compared together, by lemma 1, 

 the curvatures in d and i will be as the distances from the axis dS and <^S; there- 

 fore, by lemma 2, the acceleration of any given point in the string, will be as 

 its distance from the axis. Hence, by sect. 10, prop. 51, of Newton's Prin- 

 cipia, all the vibrations, both great and small, will be performed in the same 

 periodical time, and the motion of any point be similar to the oscillation of a 

 body vibrating in a cycloid, q. e. i. 



Corol. — Curvatures being reciprocally as the radii of the osculating circles ; 



therefore, let a denote a given line, then will its radius of curvature at e 

 , aa 



be = — . 



Prob. 2. Given the Length and fVeight of a String, with the Weight by which 

 it is stretched; to find the Time of one Vibration. — Let the string be stretched 

 between the points a and b by the force of the weight p ; also let the weight of 

 the string be n, and its length l. Let the string be put in the position AppcB, 

 and at the middle point c raise the perpendicular cs = the radius of curvature 

 at c, and meeting the axis ab in d, and taking a point p very near to c, draw 

 the perpendicular pc and the tangent pt. 



Therefore, as in lemma 2, it appears that the absolute force by which the 

 particle pc is accelerated, is to the force of the weight p, as ct to pt, that is, 

 pc to cs. But the weight p is to the weight of the particle j&c, in a ratio com- 

 pounded of the ratios of p to n, and of n to the weight of the particle j&c, or 

 of L topCf that is, as p X L to N X pc. Therefore, compounding these ratios, 

 the accelerating force is to the force of gravity, aspXLtoxXcs. Con- 

 stitute, therefore, a pendulum of the length cd: then, by sect. 10, prop. 62, 

 of Newton's Principia, the periodical time of the string, will be to the peri- 

 odical time of the pendulum, as y/n x cs to v'p x l. But, by the same 

 prop, the force of gravity being given, the lengths of pendulums are in the 

 duplicate ratio of the periodical times; hence 



y X cs X CD ^ ^^ (writing ^ for cs, by corol. to prob. l) 12L^^ yf\]\ bethelength 

 of a pendulum, whose vibrations are isochronous with the vibrations of the 

 string. 



To find the line a, put the absciss of the curve ae = z, its ordinate ep = x, 

 and the curve itself ap = v, also cd = b. Then, by corol. to prob. 1, the 

 radius of curvature at p will be , , 



— . But V being given, the radius of curvature is -^: hence — = ^ ; there- 

 fore aaz = vxi; and, taking the fluents, aaz = -^vxx — -^vbb + vaa, where 

 the given quantity — -l^vbb -f- vaa is added, to make z =z v in the middle point 



