420 PHILOSOPHICAL TRANSACTIONS. [ANNO 1798. 



taking out two numbers, are by this means exchanged for the addition of the con- 

 stant logarithm of - : the quotients - and q -, once found, being constant numbers. 



8 — 2cc 



Thus, the numerical value of even - __ . , where r and s are single figures, is 



more easily obtained by — X ng-*-r ;> tnan by tn e former expression. 



11. But it will appear on trial, that the arithmetical value of any 3 terms, 

 p' + q'cc + r'c 4 , in -which p', q\ and r, are constant quantities, and cc consists of 

 5 or 6 places of figures, may, in general, be more easily obtained by logarithms, than 



the arithmetical value of - X r— -jl — • And since the difference of the values of 



s q -i- s ±cc 



these 2 expressions is inconsiderable in the present case, I shall make no further 

 use of the fractional expression ; but observe, that the logarithm of q'cc, in the 

 other expression, being found, the logarithm of r'c 4 will be had, by adding to it 



the logarithm of -r cc; for q'cc X ~gr cc = r'c*. And, since the logarithms of the 



numbers which stand in the places of q' and - may be taken out and reserved for 

 use, and the logarithms of cc and a, once found, will serve for all the terms in 

 which these quantities occur, it will appear by an example, that neither many 

 logarithms, nor many numbers corresponding to logarithms, need be taken out 

 of other tables, in computing the value of a or b. 



12. It will now be proper, since the literal expressions of the values of a and b 

 have been exchanged for others which are more convenient, to bring the new equa- 

 tions together in one view, and then give an example of the numerical calculations 

 by them. It appears, by art. 9, 10, 13, 16, and 1 7 of the preceding paper, and 

 6, 7, and 8 of this, that 



1. A = 



4r + e +f cc + g c * 



3.5 



* (a + 6)1 ) + a + :L ace + ~ a c 4 



8 '8.8 



, 1 \ — -| r- h 4- ice -f he 4, 



1+ *'= r- V V- CC T 3 , 3 5 3.5.21 



sr(a + b)v 



4.12.32 



2a 



*i 



p -\- lec -{- mc 4 



b *b{a + b)i 1 _L. 2a + ±<*CC -f ~-q uc\ 



4. B '=f(A'a-A). 



In which equations, the values of the coefficients are as follow: 



e — 0-1931472, h = 00823604, p = 1*3862944, 



/= 0-1036802, i = 0*0551502, / = 03465736, 



g = 0-0687064, k = 0*0408309, m = 0* 1793226. 



The constant numbers which will be wanted, in computing the arithmetical 



values of a and b, are those denoted by e, h, and p, which are given in the 



preceding article. Mr. H. then sets down the work of an example, in a very neat 



manner, viz. for the 2 planets Venus and the earth. 



