ELLIPTIC AND HYPERBOLIC SECTORS. 433 



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



y-y^ y-y^ 



(o 



*3 ~ * ■*^2 ~ *1 



Now in both curves, 



x' + of = ^3 + cy\ , 



,2 I „ ..1 „2 , - .,2 



^; + cy\ = «; + cy\ ; 

 wherefore, by transposition and resolution into factors, 



(z^- X) (^3 + x) = ciy - ^3) iy + y^), 

 (ar^ - x) {z,^ + x,)= c (^, - y^) (y, + y^) : 



and hence, 



y-y^ *3+^ 



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

 mula (a), may be expressed thus, 



«3-* *2-*. (1) 



(2) 



y.+y^ y +yz 



«2 + ^, ^3 + 



X 



^£ZL=Slll2A, (3) 



^2-^i _g( y+y3) ^ ^4j 



From these, again, there are obtained 



{x^-x){y^-y^) = {x^-x;){y-y^), .... (6) 



(^3 + ^)(i/i+3^2)=(*2 + *l)(^+J^3)' • • • • (6) 



(,x^-z)ix,^ + x) =c{y^+y^{y-y^, . . . (7) 

 (x^ + x)(x^-x^)=ciy^-y^)iy+y^), . . , (8) 



By performing the multiplications here indicated, and adding and subtracting 

 the results, we find 



i^tV^—x, V.> =x y — xv , ) 

 From (5) and (6) 4 '^' '^' '^ ^" l (9) 



(^33'2-^2i'3 =^,y-^y,;) 



!^3 ^, + ^y. y, = X..X + <^y y,> \ 

 ' ' ' ' ' y (10) 

 «3^2 + '^^2^3=^!^ + <^3'yi;j 



