[ Dr. Hirst on Equally Attracting Bodies. 309 



that point at the same angle as the latter is cut by the curvCj 

 must necessarily coincide with the tangent. Similarly, the equa- 

 tions of the corresponding tangents to the curves r, and p are 



U-MiCos(^-(^)+M'isin(^~(/))=0, ... (8) 



V-v cos {d-(f)) + v\ sin (^—0) =0, 



the latter of which may, by (3), be written thus : 



[U-MCOs(^-</>)+M'sin(^-<^)] "\ 



'-0)]=O;J 



(9) 



+ [U— Ml COS {6 — (j}) +u\ sin {d- 



and as the same values of U and cf) which siniRltaneously satisfy 

 the equations (7) and (8) also satisfy (9), it is evident that every 

 three corresponding tangents of the curves ?; r^, and p meet in 

 a point. 



It will be at once perceived that the foregoing relation between 

 every three corresponding tangents is true for any two curves 

 and their intermediate curve; by introducing the condition (1) 

 or (2), however, it may easily be proved that the tangent of the 

 intermediate curve not only passes through the intersection point 

 of the corresponding tangents of two equally attracting curves, 

 but also bisects the angle made by the latter. For if 0, 0i, ^q 

 be the respective angles between the prime radius and the per- 

 pendiculars let fall from the pole upon any three corresponding 

 tangents of the equally attracting curves and their intermediate 

 curve, then 



tan(^-<^) = --, tan (^ -</),) = -—, tan (^-</)o) = - ^, 



and 



_2- g ^^ + ^'i 



tan 3(6^ — d)^ = js = TT, — TTa • 



By means of equations (1) and (2) this expression is susceptible 

 of the following successive transformations : — 



tanSft'— 6o)= ri~i 



1 + 



M + Mj u' — m'j 



2{uu' — Mim'i) 



•i#* 1*2 4j'*.^ -J/'*. 



Uu' — U^u'i MW' — M,m', 



