ELLIPTIC AND HYPERBOLIC SECTORS. 433 



Hence, in both curves, having regard to the signs of the co-ordinates 



y-y^ y^-y■l . 

 = , (« 



x-x x~x^ 



Now in both ciures, 



3^ + cy- =i^^-V cy\, 



x'^ + cy\=x'^ + cy\; 



wherefore, by transposition and resolution into factors, 



(^3- x){x^+ x) = c{y -y,)(.y + y,), 

 (x„ - X) (:i, + T,) = c (j/, - y.^) (^, + yj : 



and hence, 



y-y, ^,+^ 



z^-x, _ c^y^+y^) 



y-y^ ^^+*', 



By comparing these with formula (a), we obtain three others, which, with for- 

 mula (a), may be expressed thus, 



y-y^ y-yl 



y,+y,.j/ +yz ^2) 



x.^ + x^ x^ + x 



^Zi=,^iZl±^, (3) 



y-y^ ^2+=», 



^.-^,_c(y+.y3) (4) 



y^-y, =«3+=^ ' ' 



From these, again, there are obtained 



k\-i:')iy-y^-^«-^^'<y-y^^ .... (5) 



(«, + «)(i',+y2)=(«2 + «,)(i'+i'3)> • • • • (6) 

 {x-x)(x^ + x^=ciy^+y^iy-y^, ... (7) 

 {x^ + x'){x-x^=ciy-y^(_y^-y^, . . . (8) 



By performing the multiplications here indicated, and adding and subtracting 

 the results, we find 



Fron.(5)a.d(6)i'-^'-^-^^=^^^^-^^^'l (9) 



{''^y.-^^y, =^,.^-*.5',; J 



( x^x,+cy,y, — xj; + cyy,, ) 

 From (7) and (8) ^ ' '' ' ')- (10) 



