If the focus Q, is to be the origin and CTj = R, the radius for polar coordinates, and /S 

 the angle which R makes with the positive x-axis, i.e. jS is the angle QQiQ', then our plane 

 figure is as follows: 



.rr-/3- 



Q 



7, 



^ 



■>- X 



2c 



Figure 9. 



By the law of cosines in triangle Q2QQ1 



CTi' = 4c' + R' +4cR cos |8 



From the condition ctj - R = 2a, CTi = R + 2a, and this value of ct, placed in (56) gives 

 (R + 2a)' = 4c' + R' + 4cR cos /3 , which when expanded gives 



R 



c cos j8-a 

 For the alternative form of (57), we have the well known formula 



, ... (s -bo) (s - Co) 



tan >2A = , where 2s = a^ + bo + c^ 



s(s -a,,) 



Here a^ = a^, bp = R, Cp = 2c, A = 77 - ^, 



Hence: s = a+c+R, s-ao = c-a, s-bo = a+c,s-Co = a-c+R, 



(c - a) (R + c + a) 



whence tan' ^2/8 = 



(c + a) (R - c + a) 



which is an alternative form of (57). 



Now (54), (55), (57) and (59) could have been obtained directly from (32), (34), (42) and 



(50) by replacing correctly the trigonometric functions of lengths by corresponding lengths, i.e. 



tan a = sin a = a, cos a = 1, etc. We place them side by side for direct comparison in the 



following table which will also serve as a summary for both: 



(56) 



(57) 



(58) 



(59) 



36 



