1826.} the Use of continued Fractions, ^c. 421 



_ t 



tan, t = -j— J_ m, ., 



5 + -r- 4M3 



9 4- &c. 



Example II.— Log. (a + 1) = a - -^a^ + -^ a' — -5-a* + 

 &c. or a - -|- . a + ^^ . a - "g^g'.V - ^ + ^^- becomes 



5 H -r- -5 a 



6 H 



7 + &c. 



Example III. — Given s~' = 1 T "^ 2 2T3 "*" 2. 3.4 "' 



&c. or 1 _ 4- + fi-^ - i-:^^ + &c. This, therefore, be- 



I 



— X 



7 — &c. 

 _ 1 



~ T" + —- X 



1 — X 



2 + — X 



a — - — X 



2 + — a; 



3 - -— X 



7 — &c. 



Example IV. — Segment to versed sines v, V, or 



3 I SV" ~ TV ' 3~V "*" 7 V.9 V • 3~V" "" 7 V.9V,11 V * 



5V ^ 



8cc.) 



2v ^ \ V 



(l+sV+Ldi' 2.0. 





All these formulse converge much more rapidly than the series 

 they i-epresent, and afford a ready mode of" approximating their 

 values, when the arbitraries t, a, x, v, 6cc. are given in small 



integers or fractions. 



{To le continued.) 



