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ROYAL SOCIETY OF CANADA 



That is (tan d -f tan aY + (1 + tan 6 tan a)' = (1 + tan' d -\- 

 tan' ^) (1 -f" tan' a), which is an equation of the second degree in 

 tan 6 and tan <^. It simpHfies down to the form 



tan' (f) = 4 sin' a (tan 6 cot a). 



As before, this becomes for finite values of x^ y and a where R tends 

 to infinity, the equation to the parabola met with in analytical plane 

 geometry of the form y^ = 4 ûk». 



XII, The curve that corresponds 

 to the ellipse may be dealt with in the 

 same manner. 



Let the curve be as shewn. Suppose 

 we call the angular length OA = a. 

 Suppose here PF = e . F M where '^ < 1. 

 Let F be the focus. 

 Then 



AF = e . AD 

 A'F = e {AD + 2 a) 

 That is 



2a = e{2a^2 AD) 



A O FADo^ 



e AD 



a {\ - e) = a - OF 



That is 



OF = a 

 Then, too. 



(1 





OD = a-\- 

 As before 



(1 + tan< PF) . "+;,^°''+r'^^^t"°-^ = sec'PF. 



^ ' (l -\- tan tan a . ey 



Next we shall get PM the perpendicular from P on the circle 



a 

 tan a = tan — . 



By the same reasoning as before — here — replaces {— a) — we 

 should get 



And 

 sin' PI' 



sin'-^ (1 - tan cot ~y 

 «i^^ ^'^ = (1 + tan' ^ + tan' /) = ""' ^^>'- 



(1 -f- tan^ -f t an'0) sec'a^ - (1 + tan ^ tan a e)' 

 (1 -|- tan' d + tan' 0) sec' a e. 



A'' say; 



