14 PHILOSOPHICAL TRANSACTIONS. [aNNO 1713. 



such a prodigy. Both the magistracy and clergy appointed supplications to be 

 made, and to carry in procession to the Capuan gate, which leads to the moun- 

 tain, the relics of St. Januarius, the tutelar saint of this city; where about the 

 first or second hour of the night, towards the north, where perhaps there was 

 not so large a quantity of ashes, a star or two were seen, and the azure face of 

 the heavens began to appear; and afterwards the darkness, which had obscured 

 the day, gradually to diminish in the night; and the ashes, by the shifting of 

 the wind, to be driven into the sea. The following day continued somewhat 

 dark, by reason of the remains of the ashes interspersed in the air. Vesuvius, 

 having thus covered the fields with ashes, and belched out its grit for several 

 days, so that its black torrent had almost reached the neighbouring sea, at 

 length, in about 15 days, it ceased. 



Of the Motion of a Tense String, By Brook Taylor, Esq. F. R. S. N° 337, 

 art. 4, p. 26. Translated from the Latin. 



Lemma 1. — Let adfb, and AAipB, fig. 13, pi. 1, be two curves, so related 

 that, drawing any ordinates cAd, E<pF, it is every where cA : cd :: Eip : ep. Then 

 theordinates being diminished ad infinitum, so as the curves may coincide with 

 the axis ab; I say that the ultimate ratio of the curvature in A to the curvature 

 at D is as cA to cd. 



Demonstration. — Draw the ordinate cM very near to cd, and to d and A draw 

 the tangents at and a9, meeting the ordinate cd in t and 9. Then, because 

 c^: erf:: cA : CD, by hypothesis, the tangents produced will meet one another 

 and the axis in the same point p. Hence by the similar triangles cdp, c^p, and 

 CAP, c6p, it will be c9 : ct :: cA : cd (:: cS : cd by hypothesis) :: ^ (= c9 — c$) 

 : dt {= ct — cd). But the curvatures in A and d, are as the angles of contact 

 9A(J and tod; and because (JA and dn coincide with cc, those angles are as their 

 subtenses ^ and dt, that is, by the analogy above, as cA and cd. Therefore, 

 &c. Q. E. D. 



Lemma 1. — At any instant of its vibration, let a tense cord, stretched between 

 the points a and b, take any form of curve aj&ttb, fig. 14. Then will the incre- 

 ment of the velocity of any point p, or the acceleration arising from the force 

 of tension in the string, be as the curvature of the string in the same point. 



Demonstration. — Conceive the string to consist of equal rigid particles, infi- 

 nitely small, as/)pand ptt, &c. ; and erect the perpendicular pr = the radius of 

 curvature at p, in which let the tangents pt and -rt meet at ty and their parallels 

 •ITS andj!)^ at s, also the chord pir in c. Then, by the principles of mechanics, 

 the absolute force, by which the two particles pv and ptt are urged towards r, 

 will be to the force of tension in the string, as st to pt; and half this force, by 



