330 
MR. WARREN ON GEOMETRICAL REPRESENTATION 
£ 
34. Let a be a quantity inclined to unity at an angle less than where c 
is the circumference of the circle ; 
then o' = 2 + £ ( ~ -j) + £ (^+4) + &c ’} * 
ct GC ' 
For let a = b ^1^7, where b is a positive quantity, and — positive and less 
than 
then a! = b' + a — 1 ; 
0 0 
ct 
Now, since a = b ^1\t, we have (by Treatise, Art. 135.) 
, \* i t i / i (B + a \/ — \ fx 2 
/«\ = 1 + (B + a J — 1) X 4 77 f &C. 
1.2 
where x is any possible quantity, 
muI B = 2 + * (|7i) S + * (£1)‘ + &c .} 
= possible hyperbolic logarithm of b 
= b'; 
O 
Also (by Treatise, Art. 132.) 
y*y = i + a x + 1 2 -I - ^ cc -j 
where A = 2 1|=| + § (£-=4) + J (£7l) + &c -} ’ 
.•. equating the coefficients, 
A = 13 -f- ci — 1 
= b' -f- a — 1 
••• = 2 {^n + i (f+r) + i (stt) + &c -} • 
£ 
35. Let a be a quantity inclined to unity at an angle less than -j, and let 
a — 1 be in length less than unity ; 
Then a! = a — 1 — § (a — 1)' + ( a — I) 11 — &c. 
