APPLICABLE TO ELLIPTIC AND ULTEA-ELLIPTIC EUNCTIONS. 
421 
14. A table of meridional parts, such as is given in the books on Navigation, if carried 
far enough, would solve this equation. I have calculated an auxiliary Table for the 
purpose, as follows : — 
Let cosec <p—l=z, log^tan then 
i'=^log.^=^log2--^log2+|log (l+^^) 
— 16“^48 128^" 
To bring this formula to the same unit as the common Table of meridional parts, 
we must multiply it by the number of minutes in the arc equal to unity, or by 
L=:3437‘74677 07849 4, whence we have \ L logj 2 = 1191*43224 08243 2, and 
^ Llog, 10 = 3958*85223 39129 100. These data give the following Table, the argument 
being the common logarithm of z with its sign changed ; that is, the number of places 
which are correct: 
-logz. 
y- 
1 
5234-14859 
2 
9117*70966 
3 
13068*84816 
4 
17026-92712 
5 
20985-70200 
-log 2 -. 
y- 
11 
44738-80681 
12 
48697-65905 
13 
52656-51128 
14 
56615-36352 
15 
60574-21575 
-log 
y- 
6 
24944-54650 
7 
28903-39796 
8 
32862-25012 
9 
36821-10235 
10 
40779*93458 
/N 
15. As a simple example, let N= 3, r= 2 ; .*. -y- = sin 60° : the meridional parts for 
60°= 4527 ; and in order that the error may not exceed unity in the tenth place of 
40780 
figures, we must have m or ? = -^^y = 9; so that we must make ^=9 at least, for the 
10th figure to be correct. 
16. These methods of course only exhibit the degree of approximation on the surd 
itself. The proportionate approximation is generally greater on the integral than on the 
simple surd, because the first approximant is usually so chosen as to be identical with 
the surd at one of the limits, and it is only near the other limit that the discrepancy 
tells. 
Section II . — Details of Reduction and Computation. 
17. The chief assistance, which can be provided a priori for the computer, consists in 
the exhibition and discussion, for the ordinary forms, of the integral 
and of the auxiliary functions which present themselves in its reduction. 
18. In applying these methods to elliptic integrals, the radical and the first approxi- 
mant r must both be of a simple form, and it is advisable thatr^ — N or N— should be 
