814 Dr. Hirst on Equally Attracting Bodies. 



Hence we may conclude generally, if an ellipse and hyperbola 

 have a focus and vertex in common, and if the principal axis of the 

 one be equal to the focal distance of the other, then the common 

 ^ocus ivill be attracted equally and in the same direction by all cor- 

 responding arcs of the semi-curves which have a common vertex, 

 and equally, but in opposite directions, by all corresponding arcs of 

 the remaining semi-cui'ves. 



The axis-major of the ellipse remaining the same, if we allow 

 its foci to approach one another the curve will of course approxi- 

 mate more and more to a circle as its limit ; at the same time 

 the distance between the foci of the hyperbola will remain con- 

 stant whilst its transverse axis will diminish indefinitely ; in 

 other words, both its branches, one of which always touches the 

 ellipse, will ultimately coincide with each other and with the tan- 

 gent of the circle. As a particular case of the present theorem, 

 therefore, we are conducted again to the well-known one before 

 alluded to as due to Joachimsthal. On the other hand, by allow- 

 ing the axis-major as well as the focal distance of the ellipse to 

 increase indefinitely, those of the hjrperbola will do the same, and 

 both curves will ultimately coincide with one another and their 

 intermediate curve, the parabola. 



We must rest satisfied with a mere indication of one or two 

 other particular cases of the equations (10) and (11). When 



71 = 1, C= , Cj= -^^ , we have 



„ v^v^o ^ „ sm 



p = 2a , p^ = 2a 



sm « 

 a sin 26 



sin(^-f«) ' ' sin(^-«) ' 

 The two first are circles passing through the pole whose radii 



are respectively and -^ , and whose centres are in the 



^ '' cos u sin a 



prime radius, and in the perpendicular to the same through the 

 pole; they have, moreover, a common chord, whose length is 

 2a, inclined to the prime radius at the angle «. The equally 

 attracting curves in this case are of the third degree, and differ 

 from each other only in being differently, although symmetri- 

 cally, situated with respect to the prime radius. This curve of 

 the third degree has a double point at the pole, and an asym- 

 ptote inclined to the prime radius at the angle a. When n = 2, 

 the orthogonal auxiliary curves are both cardioides ; when n = ^, 

 lemniscata ; when n= — |, equilateral hyperbolce; and so on. 

 The pairs of equally attracting curves in each case, though easily 

 traced, do not promise sufficient interest to warrant our entering 

 here into a closer examination of their properties. 



