Dr. Hirst on Equally Attracting Bodies. 313 



transformations, become 



_ 2(ft-c) 1 



'"ITT^I (12) 



_ 2{a + c) I 

 P^ 1-cos^J 



a + ccos^ i . . 

 \^^) 



^ c + a cos ^ 



each of which represents a conic section having a focus coinciding 

 with the pole, and its principal axis with the prime radius. The 

 curves (12) are parabolse whose vertices A and B, fig. 2, are on 

 opposite sides of the pole F, at the distances a—c and a + c re- 

 spectively ; they cut each other perpendicularly in the points d 

 and e. Of the two equally attracting curves (13), the first is an 

 ellipse whose major and minor axes AB and DE are respectively 

 equal to 2« and 2b ; the second is a hyperbola, its transverse 

 axis ABi being equal to 2c, and its conjugate axis to 2b. In 

 short, the relation between these two curves is such that, besides 

 having a focus F and vertex A in common, the principal axis, AB 

 or AB„ of the one is equal to the focal distance, FFg or FFj, of 

 the other. Inasmuch as r and r^ have like or unlike signs ac- 

 cording as cos ^ ^ , it is easily seen that the correspond- 

 ing elements of the semi- ellipse DAE and semi-hyperbola 

 D, A El, having a common vertex A, attract the poJe in the same 

 direction, whereas those of the semi-curves D B E and D'j Bj E'„ 

 which have no point in common, attract the pole equally, but in 

 opposite directions. The equally attracting curves and their in- 

 termediate curve the parabola dKe, have a common vertex A, 

 and a common tangent HHj in the same, which latter being the 

 locus of the foot of the perpendicular upon a tangent of the para- 

 bola, is also the locus of the intersection point of a pair of cor- 

 responding tangents of the ellipse and hyperbola. We may con- 

 ceive the ellipse to be turned 180° first around the prime radius 

 as an axis, and afterwards around the line through the pole per- 

 pendicular to the prime radius ; by so doing — which is equiva- 

 lent to changing r into — r in (13) — the vertices B and Bj will 

 be made to coincide, as shown by the dotted ellipse A' D' B E' in 

 the figure, and we shall again have a pair of equally attracting 

 curves. The corresponding elements of the semi-curves D' B, E' 

 and D'l B, E'j will now attract F in the same direction, and those 

 of the semi-curves D' A' E' and DjAEj in opposite directions. 



