of Trigonometrical Quantities from one another. 161 



if we find y' r, logy" = log r + y', log y'" = log r + y", &c. 



Then y', y", y'", &c. will be a series of successive approxi- 

 mations to y or A log sin x. 



Also if z' s, log z" r: log * z', log z"' log * z", &c. 

 then shall z', z", z'", &c. be a series of approximations to the 

 value of z, 



.1 in .iv,iu, i'h \L Jnonrmiii ^fL -vAuia '.>// 



12. Our formulae, although only approximations, are yet more 

 accurate than is necessary with the ordinary tables. Others 

 somewhat more simple, and sufficiently correct, may be deduced 

 from them, as follows : H >} 



The two formulae (D) (E) may evidently be written thus, 



cos (x + A x) A , 

 A loe sin x cos* x A log tan x ; 



* rns ,T 



COAX 



sin (x + A x) , 



A log cos x sin 2 x - A log tan x . 



sin oc 



cos (x + A oc\ 

 Near the beginning of the quadrant, ; - ' = 1 nearly, 



COS 00 



and towards the end Sm . "V = 1 nearly. Therefore, in 



Sill 00 



these cases, 



A log tan x = cos 8 x A log tan x nearly ; ) .p. 



A log cos x =. sin a? A log tan x nearly. ) 



13. That we may estimate the degree of approximation, let 

 us multiply the series for A log tan x by cos 2 at, and by sin 2 x ; 

 the results are, 



cos x A cos* x sin 2 x (A a?) 2 . ) 



~~ + &c - 



cos'* A log tan * = 



sin x cos*a? sin 2 d? (A x)* 



A.r --- - -- 4- &c. 



sin 



By comparing these formulge with the series for A log sin x, and 

 A log cos x (art. 1.) we shall find 

 VOL. x. P. i. x 



