OF ALGEBRAIC QUANTITIES. 



'For, first, let a + /3 be less than c, 



then (by Art. 29.) a! -\- V = f , 



V 9 P + ? 



.'. m a' -\- 7n b' = m f , 



p q T-Vi 



349 



(©O'-^CoO'^Cc^.)")'' 



...,byArt.27.)(»)".(^)"=^J; 



Next, let a + (3 be not less than c 

 then (by Art. 29.) (^ + V = f 



P 9 P + ? + 1 



(\ wi y J v ^^ / j:' \ ''^ 



^'/ \?/ \P + 9+0 



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 = j3, a and j3 being each positive and less 

 than c the circumference of the circle, and let -j^ "=■ f, and let m be any quan- 

 tity whatever ; 



then ;,/ = / /" X"*, if os be not less than &, 



(*)"' \v-9) ^ 



= / / X"*, if a be less than (3. 



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 



logarithms ; 



then /E\"' = 1 + „» + ^ + j-^ + &c. 



For letm=/>+ q ^ — 1, where jt and q are possible quantities, 

 = {}^V +1^ + &c.) . (l + 9 y ^=^- 1^ - &c.) 



= 1 + (;^ + ^/ v/^) + ^^-^4^^ + &c. 



= 1 + m + j-|-2 + &c. 



