OF ARTS AND SCIENCES : FEBRUARY 8, 1870. 211 



adding the results thus obtained, after having corrected them for the 

 deviation of the value of e from that adopted in the tables. After 

 which the value of 8 could be obtained from a table of natural sines. 

 For the case of the moon, the first function would need tabulation only 

 between the limits A = 0° and A = 5° 17' ; it might be tabulated for 

 every 10". The second would have to be tabulated from 0° to 90°; 

 it might be given for every minute of arc. The number^ in these ta- 

 bles might be rendered always positive by adding a constant to them ; 

 as, for instance, 0.1 to the first function, and 0.2 to the second; and 

 thus the addition of the three terms of sin 8 be made easier. 



We should then have to subtract 0.5 from the sum, in order to get 

 sin 8 ; or we might prepare a special table, which, with the argument 

 0.5 -f- sin 8, should give 8. But by the addition of these constants, the 

 extent of the tables would be doubled, as it would be necessary to tab- 

 ulate the numbers which correspond to negative values of the argu- 

 ments. 



The factor by which tan — ^ must be multiplied to obtain tan 



a -4- fi h 



— ^ — is always positive, and, e being regarded as constant, is a func- 



id 



tion of b -f- 8, and, for negative values of b -f- 8, its value is the recip- 

 rocal of that which corresponds to positive values of b -4- 8. Moreover, 

 when b -j- 8 is a tolerably small angle, it does not differ much from 

 unity, and varies very uniformly. In the case of the moon b -{- 8 

 rarely exceeds the limits ± 34°, and the common logarithm of this 

 quantity lies .between 9.9447979 and 0.0552021 ; and its rate of change 

 per minute of arc in b -\- 8 varies only from 262 to 289 units of the 

 seventh decimal place. We may, with the better advantage, tabulate 



the function 



. e — A e -f- A 

 log cos — log cos , 



for every minute of arc of the argument A from 0° to 34°, with the 



I _t_ 90° 

 precept that it is to be subtracted from log tan ■— ^- — when b -|- 8 is 



a positive angle, but added when b -4- 8 is negative. It will be neces- 

 sary to append to the table the variation of the function for a change 



in e. The functions log tan (45° -j- — ) arid log tan (45° -f- — ) can be 



found from the logarithmic tables, but some labor would be spared had 



we tables which gave log tan (45° -\- — ) with the argument A both 



