OF ALGEBRAIC QUANTITIES. 351 



For this may be proved (by Treatise, Art. 128.) nearly in the same manner as 

 the preceding article. 



36. Let a be a quantity inclined to unity at an angle greater than ~ and 

 less than c ; 



TT-.^ ,= H^ + i (^)' + i C-^)^ + «-■}■ 



For this may be proved nearly in the same manner as Art. 34. 



37. Let a be a quantity inclined to unity at an angle greater than -^ and 

 less than c, and let a — 1 be in length less than unity ; 



Then a! =z a - \ — \ {a — \f -\- \ {a - \f — &c. 

 — 1 



For this may be proved nearly in the same manner as Art. 34. 



38. Let a be a quantity inclined to unity at an angle less than -7-, and let 

 m be any quantity whatever ; 



Then (ay = 1 + (A + jo c J"^^) m + {^L±P±^^Ziy J + &c. 



where A = 2 {j^ + x (^)^ + &c.}, 



or a — 1 — ^ (a ~ 1) + &c., if a — 1 be in length less than unity. 

 For a' = A 



.-. a' = A + p c ^/"^^ 



= (byArt.33.) 1 + {K+pcJ~^^) m + i^+P^''^-'^^ ^' +&c. 



3 c 



39. Let a be a quantity inclined to unity at an angle greater than — and 

 less than c, and let m be any quantity whatever ; 



Then /ay = 1 + (A + f+J . c V:^!) m + ^^+^j/'^ ^'m+ &c., 



where A = 2 {^ + ^ (j^)' + &c.}, 



or a — 1 — i (a — 1)^ + &c., if a — 1 be in length less than unity. 



