428 
ME. C. W. MEEEIEIELD ON A NEW MIETHOD OF APPEOXIMATION 
As these inverse tangents range generally from 45° to 60°, I computed them hy the 
shortened formula of paragraph 22, namely log ^ j + ^ ; this being sufficient to 
give eight figures of decimals accurately. I found 
log tan"* I cos 45° tan 60°} =9-94747 15296, 
log tan" cos 30° tan 60°} =9-99246 23739, 
log tan-*-|A(30°, 45°)tan60°} =0-00766 92607, 
log tan-*}A(30°, 67^). tan 60°} =9-99727 33807, 
log tan-* }A(30°, 221). tan 60°} =0-01665 09657. 
I hence obtained the following values : — 
For the positive terms. 
7-10091 3039 
4-37212 6152 
2- 70421 6251 
14-17725 5442 
3- 69590 9978 
8)10-48134 5464 
"l 
For the negative terms. 
1-13483 2441 
0- 72539 4027 
1- 66839 9478 
0-16728 4032 
3-69590 9978 
1-31016 8183 value required 
A more exact value of the integral, otherwise obtained, is 
1F(30°, 60°)+tan-* =1-31016 8161, 
which differs from the previous value by 2 units in the eighth decimal place. 
28. In order to find how many places ought to have been accurately obtained, I 
13 
observe that the method followed gives N=Yg, r=l, whence 
log x/Nj = 9-95491 =log sin 64° 20' 30". 
The corresponding meridional parts are 5086-5, which must be multiplied by ^ = 8, 
giving 40692-0. Referring to the Table in paragraph 14, I find that this nearly corre- 
sponds to tm places correct, and therefore that the integral ought to be correct to at 
least that extent. That it is not so, is due to my having curtailed the formula for 
finding the logarithms of the inverse tangents. But my object was only to give seven 
decimals correct, and my going beyond that was simply because, with a ten-figure 
Table, putting down the additional figures gave me less trouble (once I had to use more 
than seven) than abbreviation would have done. This remark may at first sight seem 
strange to any one who has not had some practice in using large Tables. But the loga- 
rithmic corrections are given in the shape of arithmetical complements : with reference 
