Mr. Hellins's Second Appendix, &c. 87 



of finding some easier and shorter method of performing the 

 whole business, without the use of any trigonometrical tables, 

 in which time is required, not only in searching for logarithms, 

 but also in making proportions for the fractional parts of a 

 second ; and, after some consideration, I discovered that which 

 I am now to explain. 



This method, then, together with some further observations 

 which I have made for facilitating and abridging the work of 

 computing the values of A and B, will make up the contents 

 of this Paper. 



2. The H.L. - — ~, which was denoted- by a, both in 



the solution of the problem and in the Appendix, is = H. L. 



— — . — — ■ — fy 3 * &c. and if, for the sake of distinc- 



C 3 2.2 2.4 4 24 6.6' ' 



tion, the Roman letter a be put for H. L. — , we shall have a, = a 



cc 3 c* 



— , &c. fof which series, the first three terms are suffi- 



4 32 ' * 



cient for our present purpose) ; and this value of a being written 

 for it in the expression a, 1 1 -f- \ cc + "Jx c *)> which occurs in 

 the first theorem in Art. 12. of the first Appendix, we shall 

 have (1 + \cc + -fj- c*) x (a — j - - 3 F c*) ; that is, by ac- 

 tual multiplication, 



* Since H. L. % is = — — -f 3 C * + 3 ' 5 ' , &c, (See Art. 2. of 



1 + v /, ( 1 ~ cc ) 2 - 2 2 44 2.4.6.6 



, r , j- s i_ tt t ' + \/(l— CC) .... CC 3C 4 35C 6 



the first Appendix.) the H. L. ■ v will be — — —2 <■ ; > 



rr 2 2.2 2 4.4 2.4.6.6 



&C. and consequently H. L. L±k^Zf£J , I - ± x i±^Zffl ) will be - H. L. 



2 cc 3 c 4 . . 



c 4 4.8 



