268 Proceedings of Boyal Society of Edinburgh. [sess. 
half a unit, in the seventh decimal place, by 7,39 x a°, and for x 
the correction will be - 0,1232 ax". 
Let, as before a + x = 2° 42' 56"*44, to find the sine. 
Here a — 2-7 and x = 56*44 
-0,1232 x2*7= -0,33 and - 
Then, log. sin. 2° 42' . 
a = 2° 42' = 9720" . 
a + x" = 9776*44 
log. sin. 2° 42' 5 6"* 4 4 
»,33 x 56*44= - 18,4 = correction. 
8*6730804 
ar. co. log. 6*0123337- 10 
. log. 3*9901807 
correction - 18 
8*6755930 
This value is correct to the last figure of the decimal. 
For all angles to about 50°, but especially for those above 20°, 
the correction is expressed very nearly by — 
- x"{0, 1228a + (, 00000313a - ,00003)a 2 } . 
Thus, at 40°, it will be — 
- #"(4,913 + ,0000962 x 1600) = - x x 5,067. 
For the tangent, the differences or errors for 1' are as follow : — 
at 5° 
+ 73,883 
at 30° 
+ 504,922 
10° 
+ 149,237 
35° 
+ 620,914 
15° 
+ 227,885 
40° 
+ 756,284 
20° 
+ 311,752 
45° 
+ 918,424 
25° 
+ 403,138 
To about 8° the correction would be +14,8 per degree for 60", or 
+ 1 x x\ or + 0,246a x x". 
60 
Thus, for log. tan. 2° 42' 5 6" *44, we have — 
0,246 x 2°*7x56''*44= +37 + 
And, log. tan. 2° 42' .... 8*6735628 
a = 9720" 
a + z" = 9776"*44 
ar. co. log. 6*0123337 - 10 
log. 3*9901807 
correction +37 
log. tan. 2° 42' 56"*44 
8*6760809 
