VOL. XLIf.J PHILOSOPHICAL TRANSACTIONS. ' 673 



from the centre, and yet never arise to a certain finite altitude. The construc- 

 tion of the cases wherein this obtains, is performed by logarithms or hyper- 

 bolic areas, the angles described about the centre being always proportional to 

 the hyperbolic sectors, while the distances from the centre are directly or in- 

 versely as the tangents of the hyperbola at its vertex. The circle is an asymp- 

 tote to the spiral ; and this can never be, unless the velocities requisite to carry 

 bodies in circles increase while the distances decrease, or decrease while the 

 distances increase, in a higher proportion than the velocity in the trajectory; 

 that is, unless the force be inversely as a higher power of the distance than the 

 cube. Next follow theorems for computing the time of descent in any arc of 

 a curve, for finding the resistance and density of the medium, when the trajec- 

 tory and centripetal force are given, and for defining the catenaria and line of 

 swiftest descent in any hypothesis of gravity. 



Then the usual rules are derived from the inverse method for computing the 

 area, the solid generated by it, the arc of the curve, and the surface described 

 by it, revolving about a given axis. The meridional parts in a sphere, and any 

 spheroid, are determined with the same accuracy, and almost equal facility. 

 The attraction of a spheroid at the equator, as well as at the poles, is deter- 

 mined in a more general manner than in the first book, or in a piece of the 

 author's, published at Paris in 17-10, which obtained a part of the prize pro- 

 posed by the Royal Academy of Sciences for that year. Several mechanical 

 problems are resolved, concerning the proportion the power ought to bear to 

 the weight, that the engine may produce the greatest efi^ect in a given time; 

 and concerning the most advantageous position of a plane which moves parallel 

 to itself, that a stream of air or water may impel it with the greatest force, 

 having regard to the velocity which the plane may have already acquired. On 

 this occasion, it is shown, that the wind ought to strike the sails of a wind-mill 

 in a greater angle than that of 54° 44', against what has been deduced from the 

 same principles by a learned author. The same theory is applied to the motion 

 of ships, abstracting from the lee-way, but having regard to the velocity of the 

 ship; and among other conclusions it appears, that the velocity of a vessel of 

 one sail may be greater with a side-wind, than when she sails directly before 

 the wind ; which, perhaps, may be the case of those seen by Captain Dampier 

 in the Ladrone Islands, that sailed at the rate of 1*2 miles in half an hour with 

 a side-wind. 



The remainder of this chapter is employed in reducing equations from second 

 to first fluxions; constructing the elastic curve by the rectification of the equi- 

 lateral hyperbola; determining the vibrations of musical chords; resolving pro- 

 blems concerning the maxima and minima, that are proposed with limitations, 



VOL. vui. 4 R 



