VOL. XXVIII.] PHILOSOPHICAL TRANSACTIONS. 15 



which one particle j&p is urged, will be to the tension of the string, as ct to tp, 

 that is, because of the similar triangles ctp, tpR, as tp or pp to Rt or pr. There- 

 fore, because of the force of tension being given, the absolute accelerating 

 force will be as — . But the acceleration generated is in a ratio composed of 

 the ratios of the absolute force directly and of the matter moved inversely; and 

 the matter moved being as the particle vp; therefore the acceleration is as 

 — , that is, as the curvature at p. For the curvature is reciprocally as the radius 



PR 



of the osculatory circle, a. e. d. 



Prob. 1. To determine the Motion of a Tense String. — In this and the fol- 

 lowing problems, I suppose the string to move through a very small space from 

 the axis of motion; and that the increment of tension from the increase of 

 the length, as also the obliquity of the radii of curvature, may be safely neg- 

 lected. 



Tlierefore let the string be stretched between the points a and b, fig. 15; 

 and by a bow let the point z be drawn to the distance cz from the axis ab. 

 Then, taking away the bow, because of the flexure in the point c alone, that 

 will first begin to move, by lemma 2. But as soon as the string is bent in the 

 nearest points <p and d, these points will also begin to move; and then e and e; 

 and so on. Also because of the great flexure in c, that point will at first move 

 very swiftly ; and thence the curvature being increased in the next points d, e, 

 &c. these will be accelerated very swiftly, and at the same time the curvature in 

 c being diminished, that point in its turn will be accelerated more slowly. And 

 in general, those points which are slower being accelerated the more, and those 

 that are quicker, less accelerated, it will be brought about at length, that the 

 forces being duly tempered to each other, all the motions will conspire toge - 

 ther, and all the points will at the same time approach the axis, going and 

 returning alternately ad infinitum. 



Now for this purpose, the string must assume the form of a curve acdeb,^ 

 the curvature of which, in any point e, is as its distance e» from the axis; the 

 velocities of the points c, d, e, &c. being also in the ratio of the distances from 

 the axis, cz, d9, En, &c. For in this case the spaces ex, DcT, es, &c. described 

 in the same infinitely small time, will be to each other as the velocities, that is, 

 as the spaces to be run through cz, d^, &c. Therefore the remaining spaces, 

 xz, S^f £ti, &c. will be to each other in the same ratio. Also, by lemma 2, the 

 accelerations will be to each other in the same ratio. So that, the ratio of the 

 velocities always continuing the same as the ratio of the spaces to be described,, 

 all the points will arrive at the axis together, and all at once depart from it j 

 therefore the curve acdeb is rightly determined, o. e. d. 



