OF ALGEBRAIC QUANTITIES. 
349 
For, first, let a + (3 be less than c, 
then (by Art. 29.) a! b' = f , 
p q p + q 
.•. m a! -f - m V — m f , 
p q p + q 
••((:)")' + (©')'- («.)')■ 
■■■ ”■>(;)'■ & 
Next, let a + |8 be not less than c 
then (by Art. 29.) a! + V — f 
p q p+q + 1 
( \ 171 / 7 \ ^ / /* \ Wl 
?) •© = U + o- 
32. 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 — f, and let m be any quan- 
tity whatever ; 
/a\ m 
then vf/ = / / \ m , if a be not less than (3, 
QY v-J 
= / f \ m , if a be less than (3. 
\p- q- 1/ 
For this may be proved nearly in the same manner as the preceding article. 
33. Let m be any quantity whatever, and E the base of the hyperbolic 
0 
logarithms ; 
then (Ey i = 1 + m + ^ + TTvTs + &c - 
For let m — p + q — 1 , where p and q are possible quantities. 
then 
(?)"“(?) 
p + q V - 1 
er-o>- 
= ( 1 + p + ^2 + &°-) • ( 1 + q «y — i 
= 1 + (P + 9 ■/=!) + Y±JLSEJT 
— 1 + m + J~2 ^ cc • 
1 .2 
&C.^ 
+ &c. 
1.2 
