306 Dr. Hirst on Equally Attracting Bodies. 



I. On equally attracting Curves. 



To determine any point in space, we shall employ polar coor- 

 dinates, the attracted point being always considered as the pole ; 

 and when a system of curves are referred to this point as pole, 

 we shall, for the sake of brevity, call all points of such curves 

 situated upon the same radius vector corresponding points ; the 

 arcs of these curves intercepted between the same two vectors we 

 shall term corresponding arcs or elements, according as the same 

 are of finite or infinitesimal length ; and corresponding elements 

 produced indefinitely, correspo?iding tangents. This being un- 

 derstood, the problem with which alone we shall at present 

 occupy ourselves may be thus enunciated : ' To find all the curves 

 whose elements attract the pole in the same manner as the corre- 

 sponding elements of a given curve. 



At first let the given attracting curve be in the plane of the 

 attracted point, so that the equation of the former may be 



then, assuming the density of the attracting matter to be every- 

 where the same, the attraction of an element of the given curve 

 upon a point situated at the pole will be proportional to 



ds dO / „ dr'^ 



■ d9 / 



r^+^^=d9\/u^+u'% 

 where m= — , and u'=-^ = 2 ' jB- If jo be the variable per 



dO' 



pendicular from the pole upon the tangent, it is well known that 



p : r=rd0 : ds, 

 so that each of the above three expressions is equivalent to 



dO 



P' 

 The attraction of the corresponding element of any other curve 



being 



dd 



Pi' 

 we easily conclude that the correspo7iding elements of two or more 

 curves, and hence the curves themselves, will attract the pole equally, 

 provided their corresponding tangents are equidistant from the pole. 

 The curves or their corresponding elements may here, of course, 

 attract the pole in the same direction or in opposite directions ; 

 practically no difficulty will arise in distinguishing between the 

 two cases ; and if we allow the term ' equally attracting ' to em- 



