308 Dr. Hirst on Equally Attracting Bodies. 



it will be well to examine more minutely its geometrical signifi- 

 cation. Let p and pj be the variable radii vectores whose reci- 

 procals are v and v■^ respectively; then c and Cj being variable 

 parameters, the equations (5) represent two systems of ortho- 

 gonal curves. For if t and t, be the angles through which a 

 radius vector must be turned, in a positive direction, in order to 

 become parallel to the corresponding tangents of a pair of curves 

 from the systems (5), then 



v\ 1 



COtT,= --=p^. 



These expressions, being each independent of the variable para- 

 meters c and C], show that the corresponding tangents of all 

 curves belonging to the same system are parallel to each other. 

 But according to (4), 



C0tTC0tT, = — 1, 



hence 



r-T, = ±-; 



that is, the corresponding tangents of any two curves belonging 

 to different systems are perpendicular to each other, consequently 

 each curve of the one system cuts every curve of the other 

 orthogonally. 



Again, by equation (3) we have/ for the same values of 6, 



p r r, 



so that p is the harmonic mean between r and r, ; in other 

 words, every line through the pole is divided harmonically by 

 the prime radius and the three curves, whose variable radii vec- 

 tores we have represented by r, r^ and p. On this account we 

 may refer to the last curve as the intermediate curve with respect 

 to the two equally attracting ones. 



The equation of the tangent to the curve r, at the point r, 9, 

 may be thus written : 



U-Mcos(^-(^)-FM'sin(^-<^)=0, ... (7) 



where (j) and U (the reciprocal of the radius vector) are the vari- 

 ables. For when <f> = 6) we have evidently 



^~''' V'd4>~u' 



so that the right line represented by (7), inasmuch as it not only 

 passes through the point r, 6, but also cuts the radius vector to 



