

ELLIPTIC AND HYPERBOLIC SECTORS. 435 



ax=x^z-cy^y^-\ ^^ 



ax = xx^ + cyy^-\ ^ 



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 AOP = sector ACP' + sector ACP", 



Sector ACP' = sector ACP - sector ACP" : 

 Let us now consider the co-ordinates 



^, y ; «,, y, ; ^,, y„ 

 as/unctions of the sectors AGP, ACP', ACP", and let us put 



Sector ACP" = a, sector ACP^^S, sector ACP = 7. 



Also, let us express 



x^ by / (a), «, by / (13), xhy f (7), 



y^ by F (a), y^ by F (^), y by F (7) ; 



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

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



«./(y) = a./(/3 + «)=/(/3)/(a)-c. F(/3)F (a)) ^ 



«.F(7) = «.F(/3 + «)=F(/3)/(«)+/(^)F(«) j' 



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

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



These are entirely analogous to the well 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 ellipse 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 : further, 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 



