212 PROCEEDINGS OF THE AMERICAN ACADEMY 



in arc and time; which tables would be useful in many other cases, since 

 this function is frequently met with in trigonometric formulas. 



The modifications necessary in applying this method to the inverse 

 problem of determining the longitude and latitude from the right as- 

 cension and declination are obvious. The variations due to the change 

 of the obliquity might perhaps be neglected in using the tables, espe- 

 cially in the* case of the declination, and computed at the end by means 

 of the very simple formula? 



da ^ 



-3— = — tan 8 cos a, 



a e 



d8 



-r- = sin a. 



de 



Take this example for illustration : — 



On January 14.0, 1871, G. M. T. we have in the case of the moon, 



I = 206° 40' 35."9 * = 23° 27' 19."81 



b = + 5 3 16.0 From Tab. I., Arg. b, + 0.0808224 



— 1.7 X (A f =- 019 ) 

 / 4- b = 211 43 51.9 From Tab. II., Arg. 1 + b, — 0.1046706 



— 11.7 X Ai +2 

 I — b = 201 37 19.9 From Tab. II., Arg. I — b, — 0.0733354 



— 8.2 X Ae +2 



b— — 5 34 37.16 sin 8 — 0.0971832 



b + 8 = — 31 21.16 log tan 148° 20' 17."95 9.7900662 



a — 13 h 46 m 19M2 From Tab. III., to be added, 0.0008223 



4- 0.09 XAf 



n 



log tan 9 h 53 m 9 8 .56 9.7908885 n 



The objection to this method is, that so many arguments I -\-b,l — b, 



b -\- 8, 45° 4 > an d a from 45° -|- -5 are to be formed ; but this is 



confessedly less fatiguing than the taking of tabular quantities from a 

 table. 



It may be allowed to notice here a series, which determines a in 

 terms of I, viz. : — 



2 t _ e 4 __ b 4- 8 

 2 



a = I 4- ~ tan z tan - ~T " cos I 



— - tan 2 - tan 2 "T" sin 2 I 



— 7 tan 3 - tan 3 — £ — cos 3 I 

 3 2 2 



