412 Prof. Encke on the Calculations requisite 



moon have been calculated from the differences of u and 8, 

 which had already been calculated in examining the correct- 

 ness of the tabular values. In general the limit of accuracy 

 may be taken at 0'*1. But with regard to the 8 in the places 

 of the greatest changes, especially in the three first months, 

 errors of *3 may occur, arising from the circumstance of 

 these calculations having in the beginning only been made for 

 jthe upper culmination. Greater deviations, if any should be 

 found, must have been introduced by misprints. 



In deducing the places for the single values of t from the 

 culminations, it appears unnecessary to go further than to se- 

 cond differences. 



The tables for calculating the parallaxes for every value of 

 t will, however, considering the quick change of 8, not be of 

 the most convenient form if made with double entries ; and 

 I have likewise in this case found more convenience in the 

 method so frequently and successfully applied by Gauss, of 

 avoiding the double entry by a table of single entry and a 

 short logarithmic calculation. 



As the moon's place is only known to 0'*1 of a degree, the 

 parallaxes are only required to be known to 0'*01, in order to 

 be quite sure that by them no other error shall be introduced. 

 It will for this purpose be sufficient to have logarithms with 

 only four places of decimals, which everybody may easily com- 

 pile. Ten duodecimo leaves thus contain to the most conve- 

 nient extent the numbers, the trigonometrical functions, and 

 Gauss's auxiliary tables for addition and subtraction. 



The form which I have chosen is this: For every t from 

 15 h to 9 h I have calculated for all values of 7r from 10" to 10" 

 within the limits of tt = 53' and tt = 62' 10" : 



a = — g . sin w . cos <p . sin r . 3437*75 



COS * («' — a) 



tangy^tangp— ti^ 



sin y 



In calculating y, instead of \ (a' — a), has been adopted 

 \ a . sec .21°, the value of a being the one corresponding to 

 it = 57' 40"; sec 21° is about the mean value between the 

 greatest and smallest sec 8 which can take place for the moon. 

 The adoption of these mean values cannot sensibly change 

 the result of the calculation. A table was formed of the lo- 

 garithms of a and b to four places of decimals ; in these tables 

 the minute of a degree was considered as unity. Next the 

 quantities of the second order were arranged in tables of 



double 



