Solution of a Problem in physical Astronomy. 91 



8. It is no less evident, that the three logarithmic terms a, 



-face, and-^ac 4 , mentioned in the preceding Art. are to the 



three logarithmic terms 2a, \&cc, and-^-rac 4 , which occur in 



the third theorem, in the ratio of 1 to 2, 1 to ±, and 1 to| 

 respectively ; or as 1 to 1 -j- 1, 1 to 1 + -j> and 1 to 1 -{-■§■; by 

 which mixed numbers, as was observed in the preceding Art. 

 the logarithmic terms in the third theorem may be more easily 

 derived from those in the first theorem, than by the fractions. 



9. The first of the logarithmic terms in the first theorem has 

 already been denoted by the Roman letter a ; now let the 



second and third, viz. -f a c c, and -|~ at 4 , be denoted by the 



Roman letters b and c respectively ; and let the sum of these 



three terms, viz. a -{- -f ace -|- -|'-|- ac 4 , now denoted by a -f- b-f- c, 



be put = S ; then, by Art. 7. the logarithmic terms in the 

 second theorem will be (1 — £) a, (1 — i) b, and (1 — ±) c; 



and the sum of these terms will be a-j-b-fc 1 — _ i 



e= S — — — -g- jj-, where S is given, it being = the three 



logarithmic terms in the first theorem, with which the compu- 

 tation ought to begin ; and the ^, i, and | of these terms respec- 

 tively, are very easily computed without the use of logarithms, 

 as will hereafter appear by an example. 



And the logarithmic terms in the third theorem will likewise 

 be denoted by 2a, (l-f-j)b, and (l-j-i)c respectively; the 



b c 



sum of which is = a-j-b + c + a + yb-f-ic = S-fa-[ \~— , 



where S, as well as a, b, and c, being given, the fractional parts 

 are very easily computed without the use of logarithms. 



10. Having now described these short and easy methods of 



N2 



