ON THE TRIGONOMETRY OP THE PARABOLA. Jl 



Since sec(^-L^)=sec^0 + tan^^, and tan(^-'-^)=2tan0sec0, 



sec (^ -^ <^) + tan (<l>-^<t>)= (sec <j) + tan (py. 

 Again, as 



sec (^ -i- -*- ^) = sec (^ -•- ^) sec + tan (^ -*- ^) tan ^, 

 and 



tan (^ -L ^ -L ^) = tan (0 -L ^) sec ^ + sec (0 -L ^) tan 0, 



it follows that 



sec(<j>-^(j)-^(p)+tan(({,-^<p-^(j))=(sec(j>+t&n(py, 



and so on to any number of angles. Hence 



sec((j)-^(p-^(p...to n(j>) + tan ((l>-^(p-^(p ... to w^)=(sec0 + tan<^)". (6) 

 Introduce into the last expression the imaginary transformation 



tan0= V — lsin0, 

 and we get Demoivre's imaginary theorem for the circle, 



cos/i^+z^/— 1 sin«(^={cos0+ V'— 1 sin^}". 

 This is a particular case of the more general theorem 

 sec(a-J-/3-Ly-LS-J- &c.) + tan(a-L/3-^y-J-S-i- &c.) 



=(sec a+ tan a)(sec /3+ tan /3)(sec y + tan y)(sec S+ tan S) &c.* 



In the circle, 



i±jggi- , / l+sin2^ , . 



l-tan(/. V l-sin20 ^V 



Accordingly, in the parabola. 



1 + >v/3i sin ^ _ /H- V-l^tan(0-J-^) ^ ^ ^ , . 

 1 — -/lllsin^ ^ 1 — V^tan(0^0)" 



2 sin 20 — sin 



tan" 6 = „ ■ ^^ I — r- 



'^ 2sm20+sm 



tan (0-^0) 

 In the circle, 



2sin20— sin40^ /"hM 



• . ,_ 2tan(0-^0)-tan (0-^0-^0-1-0) ,^^v 



^'° ^^ 2 tan (0-L0)-f tan (0-1- 0-^0^0) '''^^^ 



hence in the parabola, 



In the circle, 



cos 20= cos* — sin*0, (cc) 



hence in the parabola, 



sec (0 -1-0)= sec* 0— tan* (yy) 



In the circle, 



X Q J. <> sin (0-f-v) sin (0— y) ,,,^. 



tan^0— tan^y = — yLl-M. ^Z — az . .... (dd) 



^ ^ cos^ cos^ X 



therefore in parabolic trigonometry, 



c 



sec" sec" x 

 In the circle, 



sln^0-sin^X = ^^^^^^^^l^^^/^^ 0^) 



taui>=./lE^^ (ee) 



V l-hcos20 ^ ^ 



* Hence cos(a-|-/3-|-y-i-5-|-&c.)-|-A/^sin(a-|-;8-f-7-f-5-|-&c.) 

 =(cos«+A/^siua)(cos/3-l- V'^sinj8)(cosy-|- V^ siny)(cosJ-i-V—l8itt^)&c« 



