174 
APOLLONIUS OF PERGA 
2 A OPT: (GL) = V{GN. NT): GN 
= PN.^ -.GN (L 37, 39) 
LJd 
= PN.GT:CT.CN.~ 
CA 
= 2 A CPT: GA . GB; 
therefore (GL) = GA . GB. 
The remaining propositions of the Book trace the variations 
of different functions of the conjugate diameters, distinguishing 
the maximum values, &c. The functions treated are the 
following: 
p', the parameter of the ordinates to PP' in the hyperbola, 
according as A A' is (1) not less than^), the parameter corre 
sponding to A A', (2) less than p but not less than \ p, (3) less 
than \p (VII. 33-5). 
PP'^p', as compared with AA'-^p in the hyperbola (VII. 36) 
or the ellipse (VII. 37). 
pp'+p' 
„ AA'+p in the hyperbola (VII. 
38-40) or the ellipse (VII. 41). 
PP' .p' 
„ AA'. p in the hyperbola (VII. 42) 
or the ellipse (VII. 43). 
pp' 2 +p' 2 
„ A A' 2 +p 2 in the hyperbola, accord 
ing as (1) A A'is not less than 
p, or (2) AA'<p,but A A' 2 not 
less than \ {AA' * p) 2 , or (3) 
AA /2 <i (A A'~p) 2 (VII. 44-6). 
PP' 2 +P' 2 
„ AA' 2 +p 2 in the ellipse, according 
* as AA' 2 is not greater, or is 
greater, than {AA' +^?) 2 (VII. 
47, 48). 
PP' 2 -/ 2 
„ A A' 2 ~p 2 in the hyperbola, accord 
ing; as A A' > or < p (VII. 
49, 50). 
pp'2 ^ } /2 
„ A A' 2 ~ p 2 or BB' 2 ^h 2 the ellipse, 
according as PP' > or < p f 
(VII. 51).