1897-98.] Dr J. Burgess on Log. Tangents of Small Arcs. 269 
But for larger arcs, the above differences do not yield a simple 
empirical formula,* and as the log. cosine is easily obtainable by 
the usual process, it is better, in cases where extreme accuracy is 
required, to employ the formula : log. tan. a = log. sin. a - log. cos. a. 
Or, we may find the difference to be added to log. tan. a for an 
increment of x" by the formula — 
A log. tan. a = log. ( 1 
sin. x 
sin. a cos.(a + £) 
■> 
Thus to find the log. tan. 80° 2' 37" or log. cot. 9° 57' 23", 
The log. sin. 9° 57' 23" = log. cos. 80° 2' 37" is found as before.! 
log. sin. 37" 
log. sin. 80° 2' 
log. cos. 80° 2' 37' 
•00105339 . 
. 6*2537766 
. ar. co. 0-0066041 
. ar. co. 0*7622087 
. log. 7-0225894 
1-00105339 
log. tan. 80° 2' 
log. tan. 80° 2' 37" 
log. 0-0004572 
. 10-7551611 
. 10-755G183 
But, it is easier to use the sine and cosine, thus — 
log. sin. 80° 2' 37" . 9’9934096 
log. cos. . . . ar. co. 0*7622087 
log. tan. 
10-7556183 
* Approximately the correction may be represented by +^(14,685$ 
+ ,016015a 2 + ,00008447a 3 + ,0000797a 4 - ,0000011566a 5 + ,000000010737a 6 ). 
The comma is used to indicate the separation of the 7th and 8th places of 
decimals. 
f 9° 57' is nearly 10° ; and - ,123 x 10 x 23"= - 28,3 = corr. 
log. sin. 9° 57' (35820") . 9*2375153 
log. 35843" .... 4-5544044 
log. 35820 . . . ar. co. 5’4458744-10 
corr. ..... - 28 
log. cos. 80° 2' 37' 
. 9-2377913 
