166 MR WALLACE'S Formula for finding the Logarithms 



log s/cos x cos (a> + A x} (16) ; then A log tan \ x may be found 

 by formula (M) . 



When tan |- (x -f- A x} is to be found from tan (x + A x), 

 we have, in the first place, - - A log cos x sin 2 x . A log tan x , 

 by formula F, art. 12., then, 



log Vcos .r cos (X + AX) log cos # sin* .r A log tan x, 



so that we may express fomula (O) in logarithms thus, 



log (A log tan | x) log {cos * A log tan x} sin 2 A log tan x. (P) 



In like manner, if we make A x negative, we may find 



tan 4- (x A x) from tan (x A x) by this formula, 



log ( A log tan x 4) =: log {cos A T ( A log tan x)} + \ sin* . ( A log tan x) (P') 



the two, (P) (P') enable us to take for x the angle expressed 

 by the nearest even number of minutes, whether greater or less 

 than the angle corresponding to the given tangent. 



As an example of the application of both formula?, I shall 

 take an unfavourable case, and suppose the tangent of 

 89 1'= 1176587928 to be given, to find the tangent of 

 44 30' 30". 



Employing the first formula, we have 



Tan (a- + A a-) = 11-76537923 



tan o-(89) = 11-75807853 



A log tan a; = -00730070 



sin 2 ^ 19-9998676 COS.T 8-2418553 



A log tan x 3-8633645 A log tan x 3-8633645 



3-8632321 4^10521 98 



sin* x A log tan x -0072985 sin* x A log tan x = -0036492 



A log tan x -00012635 4-1015706 

 tan | x (44 30') = 9 99241975 



tan J (* + A *) = 9-99254610 



