^2 REPORT — 1856. 



Accordingly, in the trigonometry of the parabola, 



^ V sec(<p-^f) + l 

 If 



sinj, _ sin(f-x)^ (kk) 



tan;// sin(x— ;^)' 



it is easily shown that tan cp, tan x^ and tan \p are in harmonic progression. 



Hence it follows in parabolic trigonometry, that if 



tan(/) _ tan(9^-rx) (^-k) 



tan;|/ tan(x-r;//)' 



sin (p, sin x, and sin \// are in harmonic progression. 



Let w be conjugate to \p and to, while w, as before, is conjugate to f 

 and x- Then we shall have 



tan w = tan (f-'-X'^^)' 

 or 



tan (^ -^ X "^ */') = *^^" ^^'^ X ^^^ 4' + tan x sec 4' sec 



+ tani//sec0seex+ tan (^tan xtan;]/ (n;) 



and 



sec (0-'-x-'-^)=^sec0sec x seci^+ see^ tan x tan ;// 



+ secxtan )//tan + sec )//tan tan X (p) 



. , , , ix sin rf)+ sin Y+ sin \I/+ sin (ftsin Y sin J/ . ^ 



Sin((p-^X-^V)= ^ r^ ■ ' , . ^ r^ — r-^, • . ((T) 



1 +sinxsinv/'+sin ;f/sm0+sin0sinx 



whence in the trigonometry of the circle, 



sin (<p + x+ */')— S'" ''OS X ^'^^ 4' + ^''" X '^o® 'I' ^*^-'' 



+ sin >//co5 0cosx~s>n0sinxsin ;// (p) 



cos (^ + X + 4') — COS cos xcos \|/ — COS0 sin x^in \p 



— cos X sin i// sin 0— cos i// sin sin X (i") 



tan U + y + -J,)-^ tan (/.+ tan x+ tan ;^- tan <fi tan x tan ;// 

 ' 1 — tanxtan;^— tan i^tan^— tan^tanx 



(«) 



We have here a remarkable illustration of that fertile principle of duality 

 which may be developed to sucli an extent in every department of pure ma- 

 thematical science. 



The angle ^-^0 may be called the diqjlicate of the angle 0, the angle 

 (p-^f-^cf) the triplicate, and the angle (f-'-ip to n terms) the n-plicate of the 

 angle (p. 



The reader will observe that in this paper the signs -■- and -r connect the 

 angular magnitudes of tiie parabola, while numerical quantities are connected 

 by -J- and — . Thus in the circle, we have f + x and a + h indifferently, while 

 in the parabola we must use the notation ^-'-x or ^-rx, but a + b or a—h, 

 as in the circle. 



