of Trigonometrical Quantities from one another. 151 



By adding the corresponding sides of equations (1), (3), and sub- 

 tracting those of (2), (4), we find 



sin (2 .r + Ax) wAa? sin x m (A a?)* .. , 

 *lg> s * + cos * cos (a- + A*) W = co'^ " 6" 



sin (2 a: + A a 1 ) m A a? cos a? m(Aa?) 3 , fi v 



A lug sin x - smarsin( , r + A . r) g -^ 9 



And, again, by multiplying both sides of equation (5) by 

 cos x cos (x -f- A a?), and both sides of equation (6) by 

 sin x sin (<r + A a?), and adding the results, there is obtained 



x. s sin x cos (a, 1 + A x) -^ 



cos a- cos (x + A a?). A log cos x \ i cos ^ I m(Aa,)* 



V -< > 2 C. 



. , -> i ( ) cos x sin (a? + A x) 

 + sin ,r sin (x + A x) A log sin x 1 ( . , ^ 



sin' a? 



The expression which forms the coefficient of - 6 , is, by 

 the calculus of sines, equivalent to 



sin A x + cos 2 x sin (2 x + A x) 

 2 cos 1 x sin 2 a; 



But in the applications to be made of the formulae, the angle A a? 

 is always small ; we may therefore reject the term sin A x, and 

 assume sin (2<a?+ A#) =. sin 2# ; the expression will then become 



cos 2 a? sin 2 x cos 2 sin x cos x 



5 =-5 = 3 =- = 2 cot 2 .r. 



2 cos' a? sin 2 x cos z x sin* x 



Thus we have 



cos * cos (x + A a;) A log cos x ) w (A a?)* 



\ = cot 2 x Q ' nearly. (7) 

 4- sin x sin (.r + A x) A log sin x ) 



The second member of this equation vanishes when x is half 

 a right angle ; and supposing the angle A a? to be one minute ; 

 when x=l, or when #=89, the function 



__ co t 2 x "(AJ?)* _ -0000000002, 



the sign being in the former case, and + in the latter; 

 therefore in any Trigonometrical Tables hitherto constructed, 



