ELLIPTIC AND HYPERBOLIC SECTORS. 435 



ax=xx^ + cyy^-y g, 



\ c. 



y. ) 



ax=xx^^-cyy^ 

 «-y., = x^y -X 



5. Retaining the hypothesis that PA, one of the parallel chords, passes 

 through A, the vertex of the diameter, let the semidiameters CP, CP', CP" be 

 drawn. Then, by a known property of the curves, the sectors ACP", P'CP will 

 be equal, so that 



Sector ACP = sector ACP' + sector ACP", 



Sector ACP' = sector ACP - sector ACP" : 

 Let us now consider the co-ordinates 



^. y ; ^,. y, ; •«2, y„ 

 a&funrMons of the sectors AGP, ACP', ACP", and let us put 



Sector ACP"=a, sector ACP'=/3, sector ACP=7. 

 Also, let us express 



X, by / (a), ^, by / (/3), xhy f (7). 



y,byF(a), 3/,byF(^), yhyV^y): 



where the letters / and F are to be regarded as characteristics of the functions. 

 Then, applying this notation to equations A', B', we have 



«-/(7)=«-/0 + «)=/('3)/(«)-c F(/3)F(«)| ^ 



(3) F («)1 



a. F(7)=a . F(/3 + «)=F (/3)/(a) +/(^) F 



a./(/3) = a./(7-a)=/(7)/(a) + c.F(7)F(a) ) 



a.F(/3) = a.F(y-«) = F(7)/(a)-/(y)F(a) f *^- 



These are entirely analogous to the weU known formula for the cosine and 

 sine of the sum, and of the difference of two angles ; indeed the latter are com- 

 prehended in the former, and to produce them, it is only necessary to conceive 

 the eUipse to change into a circle ; in which case the symbol c, independently of 

 the sign prefixed to it in the formula, must be regarded as positive, and its value 

 = 1 : fmther, we must then exchange the functional mark/ for cos., the abbrevia- 

 tion of cosine, and F for sin. that of sine. 



6. The reasoning throughout this memoir is perfectly general, and inde- 

 pendent of the sign of the modulus c ; and, in imitation of that employed to esta- 

 blish the calculus of angles, it may be carried on to a great extent. Indeed, pro- 



VOL. XVI. PART II. 3 T 



