OF ALGEBRAIC QUANTITIES. 
351 
For this may be proved (by Treatise, Art. 128.) nearly in the same manner as 
the preceding article. 
3 c 
36. Let a be a quantity inclined to unity at an angle greater than — and 
less than c ; 
Then d = 2 + J (j^) 3 + * 1 )' + &c.}. 
For this may be proved nearly in the same manner as Art. 34. 
3 c 
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' = a — 1 — \ {a — l) 2 + ^ {a — l) 3 — &c. 
- 1 
For this may be proved nearly in the same manner as Art. 34. 
__ c 
38. Let a be a quantity inclined to unity at an angle less than and let 
m be any quantity whatever ; 
Then (a\ m = 1 + (A + p c J~^l) m + (t— ^ '' - m + &c. 
where A = 2 + § (~) + &c.}, 
or a — 1 — i (a — l) 2 + &c., if « — 1 be in length less than unity. 
For a! = A 
/. a! = A + p c — 1 
p 
((.)-)' = (A + p c J — 1) m, 
(ay = (yy A+pc 
= (by Art. 33.) 1 + (A + ^ c ^ — 1) m + ~ m + &c. 
3c 
39. Let a be a quantity inclined to unity at an angle greater than 4 
less than c, and let m be any quantity whatever ; 
Then ( a y = 1 + (A + V + 1 • c J~^) m + 
where A = 2 {^=4 + i (iTl) + &c '}’ 
and 
(A + p + 1 c V — 1)' 
1.2 
m + &c., 
or a _ i _ a ( a — i) 2 _j_ & c ., if a — 1 be in length less than unity. 
