348 MR. WARREN ON GEOMETRICAL REPRESENTATION 
but g 1 = possible hyperbolic logarithm of g, 
O 
h' = h, 
rr J t 
••• g' + h' = 
O 0 
a 1 + U = g h! + a + (3 + p + q . c . J — 1 ; 
p q 0 
m « 4 ~ P> 
but, since g h . ^1^ c —f, 
(S h\' + a + (3 — 1 =/ , j if a + (3 be less than c, 
— f, not less - - 
(S h \ + a + ^+ J P + ?* c -'v/~ 1== /' ,ifa + f3be less than c, 
\ 0 ) p+q 
not less - - 
= f 
p + q + 1 
r. a! -\- b' = f , if a + (3 be less than c, 
p q p + q 
= f , 
not less - - 
30. Cor.) Hence if a and b be any quantities whatever, and a be inclined 
to unity at an angle = a, and b at an angle = (3, a and /3 being each positive 
and less than c the circumference of the circle, and 4 - = f ; 
then a! — b' — f , if a be not less than (3 , 
p q p-q 
= f , if a be less than (3. 
p-q - 1 
31. Let a and b be any quantities whatever, and let a be inclined to unity 
at an angle = a and b, at an angle = (3, a and (3 being each positive and less 
than c the circumference of the circle, and let a b = J\ and let m be any 
quantity whatever ; 
then (aV 1 . (by = ^ f y, if a + (3 be less than c 
— ( f not less - - . 
V' + -7 + V 
