310 Dr. Hirst on Equally Attracting Bodies. 



tan ^((7— <Po) = — 7 — i — - — ■ w i - i — — c — 



Uu\ -\-U(ll' 



U Mj 



_ tan(g-0) + tan(^-<j),) 

 ~ 1 — tan (^— ^) . tan (^— ^,) 



= tan[2^-((^ + 0,)]. 

 This equation is evidently fulfilled when 



2{e-<i>^)=2e-[ci>+4>,)+k'jr, 



where k is any positive or negative whole number, or zero. Hence 



2(</'o+^|)=</>+<^i; 



from which we conclude that the one JDerpendicular, produced if 

 necessary, either bisects the angle made by the other two, or the 

 supplement of that angle ; and the same conclusion applies of 

 course to the tangents to which these lines are perpendicular. 



Since the three tangents and the line joining their common 

 intersection with the pole constitute a harmonic pencil, one line 

 of which bisects the angle made by two others, the fourth, which 

 is the conjugate of the first, must also be perpendicular to the 

 same ; in other words, the locus of the intersection of a pair of 

 corresponding tangents of two curves whose corresponding elements 

 attract the pole equally, is also the locus of the foot of the perpen- 

 dicular from the pole upon the tangent of the intermediate curve. 



Any number of pairs of equally attracting curves may be now 

 determined. For having assumed any curve p as intermediate 

 curve, the system of curves which cut the same orthogonally may 

 be found by a simple quadrature. If any individual of this system 

 be selected as the curve pj, then the corresponding elements of 

 the two cui'ves whose equations are 



^ P Pi 



n~P p/ 

 will attract the pole equally. 



Amongst the numerous examples which might be cited, the 

 following are not without interest. Let the arbitrary interme- 



