VOL. XXXIX.] PHILOSOPHICAL TRANSACTIONS. 4^ 



order to draw a tangent to the locus of p ; join dc, ds and cs, and constitute 

 the ; ngle dqr, equal to cqb, so that gr lie the contrary way from gd that qb 

 lies from gc, and let qr meet dc in r. Constitute also the angle dnt, equal 

 SNA, with the like precaution, and let nt meet ds in t. Join kt, and produce 

 it till it meet cS in h ; then join ph, and make the angle cpl equal to sph, so 

 that PL and ph may lie contrary ways from cp and sp; then pl shall be a tan- 

 gent at p, to the locus described by p, the intersection of cq and sn. 



Mr. Maclaurin has also applied this doctrine to the description of lines through 

 given points. But he supposes he has said enough at present on this subject ; 

 and concludes, after observing that in the abovementioned treatise, he has 

 given an easy theorem, for calculating the resistance of the medium, when a 

 given curve is described with a given centripetal force in a resisting medium, 

 which he here repeats, because it has been misrepresented in a foreign Journal. 



Let V express the centripetal force with which the body that is supposed to 

 describe the curve, is acted on the medium ; let v express the centripetal force 

 with which the same curve could be described in a void ; suppose z = -, then 

 the resistance shall be proportional to the fluxion of z multiplied by the fluxion 

 of the curve ; supposing the area, described by a ray, drawn from the body to 

 the centre of the forces, to flow uniformly. Let this theorem be compared 

 with what the celebrated ma^iematician mentioned by that Journalist has given 

 on the same subject, and it will easily appear what judgment is to be made of 

 his assertion ; and since several persons, and particularly the gentleman men- 

 tioned above in this paper, testify that Mr. Maclaurin communicated to them 

 this theorem, before any thing was published on this subject by the learned 

 mathematician he nauies, his observation on this occasion must appear the 

 more groundless. 



From this theorem, the author draws this very general corollary; that if the 

 curve is such as could be described in a void by a centripetal force, varying ac- 

 cording to any power of the distance, then the density of the medium in any 

 place, is reciprocally proportional to the tangent of the curve at that place, 

 bounded at one extremity by the point of contact, and, at the other, by its 

 intersection with a perpendicular raised at the centre of the forces to the ray 

 drawn from that centre to the point of contact. Let al be the curve described 

 by a force directed to the point s (fig. 14) ; let lt touch the curve at l, and 

 raise st perpendicular to sl, meeting lt in t ; then the density in l shall be 

 inversely as lt, if the resistance be supposed to observe the compound propor- 

 tion of the density, and of the square of the velocity. 



Besides what is here observed, he proposes to illustrate and improve several 



VOL. veil. H 



