Dr. Hirst on Equally Attracting Bodies. 315 



To find any number of pairs of curves whose coiTesponding 

 arcs attract the pole equally is, therefore, a problem whose solu- 

 tion may be considered complete ; but the one originally pro- 

 posed — to find all the curves whose elements attract the pole in 

 the same manner as the elements corresponding to them of a 

 given cui've — is not only more general, but its complete solution 

 involves far greater difficulties. At present we must content 

 ourselves with glancing at a few of the more salient properties, 

 and at a few particular cases of such curves, each of which fulfils 

 the differential equation 



u^ + u>'=~=<o^ (14) 



where o) represents a given function of 0, 



Let Mj Mg (fig. 3) be a portion of the given attracting curve 

 intercepted by the lines OMj, OM2 through the pole ; and for 

 the sake of a distinct conception, let us suppose that within the 

 limits under consideration the given curve is everywhere conti- 

 nuous, concave towai'ds the pole, and free from all singular 

 points, such as cusps, apses, multiple points, &c. ; or still more 

 definitely, let us suppose that, with respect to the given curve, 



r and -j^ are continuous positive functions of 6, and that r and/> 



increase simultaneously vidth 0, so that the perpendicular OP 

 upon the tangent to M will always fall on the same side of 

 (below) the radius vector. Around as a centre, and with the 

 radius OP, describe a circle cutting the radius vector OM in the 

 point fj,. Any point m beyond /u, being taken in this radius vec- 

 tor, let the two tangents mp and wzjo, to the circle P/x. be drawn, 

 cutting the immediately following radius vector OM' in the 

 points m" and m' ; then mm" and mm' will cleai-ly be the ele- 

 ments corresponding to MM' of the two equally attracting curves 

 passing through m. Similarly, if P'//,' be the circle around 

 touching the tangent at M', and through the points m", m' two 

 tangents to the same be drawn, — the former below, the latter 

 above the radius vector, — the elements adjacent to mm" and mm' 

 on the respective curves may be found ; and by proceeding in 

 this manner we may easily conceive the equally attracting curves 

 OT, m m^ and m'j i.i m'c^ passing through m completely constructed. 

 If r, and 'i\ be the variable radii vectores of these two curves, it is 

 easily seen that the one increases whilst the other diminishes 



with the increase of 6, so that -r^ and -f§, and hence also «', and 



do at) 



u'g have always opposite signs for the same value of 9. Further, 



the perpendiculars on the corresponding tangents of both curves 



being equal, iip and r, increase or decrease simultaneously, then 



