of Trigonometrical Quantities from one another. 159 



Therefore, because A log tan x = A log sin x A log cos x, 



8 x sin 2 x (A a?) 2 , cos 4 a- + sin 4 x (A a-) s ) 



~H 2 Q ' 3 '' 3 H 



This last series being multiplied by 



cos x cos (a? + A a?) = cos* x cos x sin x A a- cos 2 a |V ^^- + &c. 



and the same series also by 



(A a 1 ) 2 



sin x sin (a? + A a?) = sin* x + cos x sin x A a: sin 2 a' , ' + &c. 



the results, continued as far as three terms, will be 



(cos a: 1 (A a?) 2 2 cos 4 x sin 4 a? (A a?) 3 ) 



cos a- cos (.r + A x) A log tan x =. m \ -. -A a? ^- v ' H -^ h c, \ ; 



(sin a? sm 2 cos a? sin 3 a? 



f sin x 1 (Aa:) 2 2 sin 4 a? cos* x (A a 1 ) 3 



sin a- sin (a? + A a?) A log tan x =. m\ A x 4- - =- - -\ 5 = rr" + c. 



(cosa? cos 2 a- 2 cos 3 a? sin a? 6 



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

 A log cos x, it will appear that 



A log sin x = cos x cos (x + A a:) A log tan x + m (2 cot x + tan x) , &c. (10) 

 A log cos x = sin a? sin (x + A a?) A log tan .r + m (2 tan a- + cot a-) 5^-, &c. (11) 



Hence we may infer the following properties of formulas (D) (E) : 



(1.) The two formula? are least correct towards the extremities 

 of the quadrant. 



(2.) The first is most correct when tan # = v/f , that is, when 

 x =r 54 1 : nearly, and the second when tan x = \/%, in which 

 case x 35^ nearly, because, in these cases, the functions 

 2 cot x -f- tan x, and 2 tan x -j- cot x are the least possible. 



(3.) When x is a small angle, the error of the expression for 

 A log sin x is nearly double that for A log cos x. The reverse 

 is true when x is nearly a right angle. 



9. Supposing the increment A x to be one minute, the er- 

 ror of the formula for A log sin x, when x 2, will be 



