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).