Dr. Hirst on Equally Attracting Bodies. 319 



spectively to tn-srO and '^=-^- In fact, the substitution we 



made in (16) is equivalent to determining any point in the plane 

 by means of the angle 6 and the parameter m of the curve (17), 

 which passes through the same. Further, this parameter being 

 constant for all points of such a curve, we easily conclude that 

 all the equally attracting curves', when crossing the same, are 



inclined to the radii vectores at a constant angle - — ot. 



In the particular case under consideration, the auxiliary curves 

 (17) are logarithmic spirals, 



r= e'"^ (23) 



Wlj cos IS ^ ' 



each of which cuts every radius vector at a constant angle 

 TT 1 . 



g—'oyo, where cos i3-Q=—. This system of curves (23) com- 

 prises not only the limiting curve ot = 0, but also the given spiral 

 (19), which corresponds to the value ct = i!7q. Further, each 

 spiral (23) is crossed by all the equally attracting concave and 

 convex curves at constant angles, equal respectively to 



OT— n-Q, TT — (ct + i!7o); 



where, for distinctness, the positive or negative inclination in 

 question is estimated by the angle, less than two right angles, 

 through which the tangent of the curve must be turned, in a 

 positive or negative sense, around its point of contact, in order 

 that it may coincide with the tangent of the spiral. 



It will be immediately seen that the concave curves cut all the 

 spirals between the infinitely distant one and the given curve (19) 

 in such a manner that their inclination to the same is always 



positive, and diminishes continuously from ^ — -utq to 0, whereas 



they cut all the spirals between (19) and the limiting curve, so 

 as to make this inclination negative, and to increase negatively 

 from to vtq. In short, the concave curves as they approach the 

 pole from an infinite distance cut the intervening auxiliary spi- 

 rals ever more and more acutely, or rather they coincide with the 

 latter more and more in direction, and only reach the given spiral 

 (19) when it reaches the pole, after encircling the same an infi- 

 nite number of times. Subsequently these concave curves 

 reappear on the inner side of the given spiral, and continually 

 recede from the latter so as to cut the auxiliary spirals more and 

 more obtusely, until finally they terminate abruptly in the limit- 

 ing spiral ill such a manner as to be perpendicular to the radii 

 vectores. 



