424 
ME. C. W. MEEEIFIELD ON A NEW METHOD OE APPEOXIMATION 
then we shall also have, and with great approximation, 
log s=log {t . cos^ L)+(^— ^ . cos^ L) 
=:log [c . cof L)+(c+c. cot^ L), 
log c=log (s . tan^ L)i(s+s • tan^ L) 
=log {t . sin^ . sin^ L), 
log ^=log (s . sec*^ L)+(s — s . sec^ L) 
=log (c . cosec^ L )+(<? — c cosec^ L). 
I have given the whole set of six, but my Table was computed with the pair for log c. 
By way of example, I add a specimen copy of one of my working sheets. The use of so 
many as ten figures is not altogether unnecessary, because otherwise, when A is nearly 
1— A ? 
equal to unity, the value of log (1 — A) or of log cannot be had with exactness. 
21. The following formulse will also be found in many cases preferable, both for 
exactness and facility, to the ordinary use of logarithmic tables by means of difierences. 
These formulae, as well as those of the previous paragraph, are but applications of 
Taylor’s theorem, reduced to a shape fit for the computer. Even where only seven 
figures are required their application is frequently much easier, and gives more exact 
results, than interpolation by difierences. In what follows, x is supposed to be thf 
nearest tabular entry. 
22. To find \o^y from log tan y . — Let us assume simultaneously 
log 3 /=loga^+^, logtan «/=log tanar+^. 
Putting M=loga;, 0 =logtan^, we have 
du sin 2x 
dz 2x 
B/UcI 
sin 2a? 
2x 
^cos 2x— 
sin 2a?\ 
2x ) 
5 
M being the modulus of the logarithms. 
Hence, by Taylor's theorem, 
, , sin 2a? T.,,/sin2a? , 
Taking the logarithm, this becomes 
losl=logfi^)+t(;^-cos2.) 
. /t . sin a? . cos a?\ t . sin a? . cos a? , , „ , . „ 
^ ^ +?^ + 2#sin^a;, 
The latter is the better shape for a working formula, because log sin x and log cos x 
are found in the same page and line as log tan x, while log sin 2x must be looked for 
elsewhere. The first term alone is sufficient when x is small ; but when x much exceeds 
45°, cos 2,x changes its sign, and even the entire formula is insufficient. The maximum 
value of the coefficient of t in the second term is 1‘0631, corresponding to a’=78° 33' 26"’5. 
In many cases, where the first term alone is insufficient, a rough interpolation, made at 
