NOTE ON THE INTERSECTION OF TWO CURVES, 

 WHOSE EQUATIONS ARE GIVEN IN POLAR CO- 

 ORDINATES, WITH AN ILLUSTRATIVE EXAMPLE. 



By Prof. Lawrence Crawford, M.A., D.Sc, F.R.S.E. 



(With two text figures.) 



I. The point with coordinates (r. 0) may also be written as 

 the point ( — r, i8o° -{- 6)- From this it follows that the inter- 

 sections of two curves / {r, 0) = o, F {r, 0) =: o are not 

 necessarih' completely given by finding the points on the two 

 curves for which the r and the are the same, the points (r^, 0^) 

 on the first and (r.^, 6*2) ^^ the second, for which ^3 = — r-^, 0^ = 

 180° -r- O- must also be considered. I have not seen this men- 

 tioned, though it is implied in the standard question, to find the 



/ 

 equations of two common chords of — =^ i -\- e cos {0 — 7)^ 



r 

 L 



_ = I + ^' cos (^ — S).* 

 r 



An illustrative example is the intersection of two 

 -conies with a common focus and axes perpendicular,. 



/ /' 



whose equations may be written — := i -j- ^ cos 0, — = 



r r 



I -j- c' sin 0. The points for which i\ t=z ;%, q^ = ^^ are given 

 by /' (i -j- ^ cos 0) = I (i -{- e' sin 0), i.e., e'l cos -j — e'l sin 

 = 1 — /'. This equation can only give two values of between 

 0° and 360°, and therefore only two points of intersection of the 

 conies. The other points of intersection are those for which r., 



I 

 = — 'fi, 60 = 180° -f" 01^, and are given by — = i -}- e cos ^1, 



I' 



— = I -f e' sin 02, rn= — r„ 0. = 180° -f $^, 

 r, 



/' 

 . • . = I — e' sin 0^ 



ri 

 . - . ^1 is a root of /' ( i + e cos ^) = — I (i — ^' sin ^) , 

 i.e., el' cos — e'l sin ^ = — / — /'. 

 No root of this equation can be also a root of the other equation, 

 or differ from a root of that equation by 180°. 



It may be noted that the points for which rj = n, 0^ = 

 0o lie on 

 l—V 

 — ■=^ e cos — e' sin 0, one chord through two points of 



Clement-Jones, Introduction to Algebraical Geometry, p. 379. 



