152 MR WALLACE'S Formula for finding the Logarithms 



if x and x + A x be any two adjoining angles, we have in 

 general 



cos x cos (x + A x) A log cos x + sin # sin (# + A .r) A log sin x = nearly. 



And consequently 



(A) 



sin x sin (x + A #) A log sin * = cos x cos (a? + A x) ( A log cos x) *. 



This formula may be expressed independently of logarithms, by 

 considering that 



sin (x + A x) , cos (x + A x) 



A log 3in x = log *-i , A log cos x = log 5 -; 



sin x cos a 1 



Hence we have, nearly, 



(A') 



sin x sin (x + A j) /- -| cos * cos (* + A ,r) 



r*r\a I I A /v>\ I v ' 



{1 sin x sin U + A j) /- -v _ 



sm (x + A .r) f _ I cos (a- + A a?) f 



sin x f cos f 



2. The object I have in view being the determination of the 

 logarithmic sine and cosine of an angle, the one from the other, 

 to adapt the formula to that purpose, I express it in these two 

 ways. 



(B) 



A log sin x = cot x cosec x cos (#-f Aa 1 ) -: ; - r- ( A log cos x) ; 



sin (x + A x) 



,M V- "* (C) ^ 



- , COS X 



A log cos x = tan x sec x sin (x 4- A *) - -, - r A log sin x . 



' cos (x + A x) 



The same formulae, expressed in logarithms, will stand ready 

 for use thus : 



(B') 



Log (A log sin x) = log {cot x cosec x cos (x + &x)( A log COS.T)} A log sin/r; 



Log ( A log cos x) = log { tan x sec x sin (x + A#) A log sin a: } + ( A log cos x) ; 



* The symbol A log cos"* here indicates, that the sign of log x log (x + A x)^ 

 (which is a negative quantity), is to be changed to +. 



