316 Dr. Hirst on Equally Attracting Bodies. 



of jo and r^ the one will increase whilst the other decreases^ and 

 vice versa ; so that whilst the one curve through tn, like the given 

 curve, is concave towards the pole, the other will be convex. As 

 the point m recedes to a greater distance, the curves passing 

 through the same cut the radius vector more and more acutely, 

 until ultimately they both coincide with it in direction. The 

 asymptotes of the two curves passing through the infinitely di- 

 stant point of the line OM are of course the two tangents of the 

 circle P/x which are parallel to OM. Had the point m been 

 situated between /i and the pole, then as it would have been im- 

 possible to draw tangents from it to the circle P/z., the curves 

 could not have been constructed. In fact the equation (14) 

 necessitates the inequality ^< < o> or r>j!^. Had m coincided 

 with the point /i, the two curves could only have been constructed 

 on one side of the radius vectoi*, to which they would have been 

 perpendicular. In short, generally, two equally attracting curves 

 terminate abruptly in every such point [i, and are there at right 

 angles to the radius vector. The locus of the point /i, which we 

 shall call the limiting curve, is represented by the equation 



r=p-=:z — (15) 



In general, therefore, we may conclude that through every point 

 m of the angular space M, Mg, except those within the space 

 bounded by the limiting curve (15), pass two,and only two,curves 

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

 responding elements of the given curve. Every two such curves 

 are equally inclined to their common radius vector; and as far as 

 convexity or concavity towards the pole is concerned, the curva« 

 ture of the one is similar, of the other dissimilar, to that at corre- 

 sponding points of the given curve. Thus the whole series of 

 curves which fill the angular space Mj M^, with the exception 

 of that portion of it bounded by the limiting curve into which 

 none of them enter, may be considered as composed of two 

 groups, the individuals of which are distinguished from each 

 other by the nature of their curvature towards the pole, and are 

 represented by the particular integrals of the equations 



« '= - ^ ^''-u^ ,'\ .^g. 



Besides the curves contained in these two groups, there are 

 innumerable others, both continuous and discontinuous, whose 

 elements attract the pole in the required manner. All such, 

 however, are necessarily mere combinations of diiferent portions 

 of the curves of the two groups already described. We may 

 remark, lastly, that if the given curve has a point of inflection. 



