APPENDIX. 285 



the formulas of article 34 become 



, A* A* 1 



I - A * - A 0&quot; + li 0* + &amp;lt;H ^ + M ^ 9 + A 0&quot; v2 

 ! 24 



The second equation, in which v is expressed in terms of w, is that given by 

 BESSEL, Monatliche Correspondenz, Vol. XII., p. 197. He also gives the third coeffi 

 cient of the series, but has computed a table of only the first two. POSSELT, in 

 the Zeitschrlft fur Astronomic und verwandte Wissenschaflcn, Vol. V., p. 161, has given 

 the first equation ; he has also given three coefficients of the series, but a table of 

 the second only, since BESSEL S table will give the first coefficient simply by 

 changing the sign. POSSELT has changed the sign of the second coefficient also. 



Instead of the logarithms as given in the tables of BESSEL and POSSELT, the 

 corresponding numbers are given in table V a , and to avoid large numbers, O.OL 

 is taken as the unit of d. 



Patting 



tan i x = 

 the table contains 



T&amp;gt;_ - A A r 



10000(1+ ?) 4 



1 . fc I t 8 I -1 t6 I 41 W 1 t9 19 til 



TV _ ~T5&quot;S~TTB ^ &quot;t~ Tff * I 8Tnr &amp;gt; 3?^ ~fSU ? 



-&quot; 



So that when x = iv we have 



y = w 4- 

 And when x = v, 



w = v A (100 8} B (100 3f 



It seems unnecessary to recompute the table in order to be certain of the 

 accuracy of the last place, or to extend it further, as its use is limited. For 



