442 PHILOSOPHICAL TRANSACTIONS. [aNNO 1708. 



it is m = 2, and m — • 1 = 1 . But the velocity of a body in an ellipsis, is to 

 the velocity of a body moving in a circle at the same distance, in a less ratio 

 than that of y/l to 1. And the velocity in an hyperbola is to the velocity in a 

 circle, in a greater ratio than that of -v/ '2 to I . — If the body move in the nauti- 

 cal spiral, its velocity is every where equal to the velocity of a body describing 

 a circle at the same distance : for in this case it is m = 3, and m — l 

 = 2. 



Problem. — Supposing that the centripetal force (the absolute quantity of 

 which is known) is reciprocally as the square of the distance, and a body is 

 projected in a given direction, with a given velocity ; to find the curve in which 

 the body will move. 



'[' Let the body be projected in the right line ab, fig. Q, with the given velo- 

 city c. And since the absolute quantity of the centripetal force is known, 

 there will thence be given the velocity with which a body can describe a circle 

 at the distance sa, by the same force : for it is equal to that acquired by a body 

 in falling through 4- sa, while urged constantly by the same force. Let that 

 velocity be c. Erect ak perpendicular to ab, in which take ar a fourth pro- 



portional to c*, c% — ; then ar will be the radius of curvature at a. Draw 



■rh perpendicular to as, andnK perpendicular to ar; then drawing sk, it will give 

 the position of the axis. Make the angle fak equal to the angle sak. Then 

 f FA be parallel to sk, the figure in which the body moves will be a parabola. 

 But if it meet the axis sk in f ; and if the points s and f fall on the same side 

 of the point k, the figure will be an hyperbola; but if s and f fall on contrary 

 sides, the figure will be an ellipsis. Hence, with the foci s, f, and the axis = 

 sA + FA, the section may be described, in which the body wjll move. 



Concerning the New Philippine Islands. By the Jesuits Father Paul Clain and 

 Father Le Gobien. N^ 317, p. ISQ. 

 The Pintados are large islands separated from each other by arms of the sea, 

 where ebbing and flowing renders their navigation very difficult and dangerous. 

 At the town of Guivam, in the isle of Samal, or Ibabao, the last and most 

 southern of the eastern Pintados, were ig Palaos, or inhabitants of certain new 

 discovered islands; the easterly winds, that blow on these seas from December 

 to May, having driven them 300 leagues from their own islands, to this town, 

 where they arrived in two small vessels, called paraos: of which adventure we 

 received this following account. 



: They embarked to the number of 35 persons, to pass over to one of the 

 neighbouring islands ; when there arose a very strong wind, that forced them 



