VOL. LXXXVIII.] PHILOSOPHICAL TRANSACTIONS. 4\Q 



3.5.7.9-5.3 , 3 . 5 .7-9. H-7.5 3 . 5 . 7 . 9 • 11 . 13 . 9 • 7 « 



= 4 . 6 . 8 . 10 . '6 . 4 . 2 ~*~ 4 . 6 . 8 . 10 . 12 . 8 . 6 . 4 ' 4 . 6 . 8 . 10 . 12 . 14 . 10 . 8 . 6 y 



which is the value of another series of the foregoing paper. 



II. Observations, tending to facilitate and abridge the numerical computations of 

 a. and b in the preceding paper. — 6. The radical factor a/(1 — cc), in the literal 

 expressions of the values of a and b, may be taken away, by multiplying the other 

 factors by its equivalent 1 — — — - - — -g, &c. in consequence of which, other 

 expressions will be obtained, better adapted to the purpose of numerical calcula- 

 tion. This will appear by the following operations. The product of >/(l — cc) 



X the other factor in the expression of the value of a, in art. 9 of the preceding 



22 

 paper, viz. — + a + ^cc + vc 4 , will give - + e + fee + gc 4 , &c. where, e,f, and 



g, are = a — 1, ^ — 4-A — ^-, and t — 4-//, — -fA — i, respectively; in numbers 

 = 0-1931472, 0-1036802, and 0-0687064, respectively. And this expression, 

 which is evidently more simple than the former, is somewhat nearer than that to 

 the value of the whole series. 



7. In like manner, the product of the 2 factors in the value of a', in art. 13, 

 viz.— + — + a' + pec -f /c 4 , will be = — + - + h -f- ice + kc\ &c. which 

 expression also is more simple than that from which it is derived, while its accuracy 

 is not less, as is pretty evident on inspection. And that the numerical values of h t 

 i, and k, are very easily attainable from the values of a', (/, and /, given above 

 in art. 3, 4, and 5, of this paper, is very obvious. 



8. And the product of the 2 factors in the value of b, in art. 16, may also be 

 exchanged for a more convenient expression, by a like process. 



viz. p + ccc + tc 4 X 1 — - — —, &c. = p -f lec + mc 4 , &c. which expression 



also is more accurate than that from which it is derived, as well as more simple. 



9. The numerical calculation of the other member also, in which % enters, may 

 be facilitated and abridged, by the following considerations. If c be put for the 

 sine of an angle, radius being 1, then will J + \/(l — cc) be the versed sine of 



the supplement of that angle, and jr~z — \ w ^ ^ e = ^ e tan g ent of half that 



angle; from which it follows, that the reciprocal of this quantity, viz. 1 + ^ *~ cc l 9 

 is = the co-tangent of half the angle of which the sine is c. The common loga- 

 rithm of may therefore be taken out from a table of common loga- 

 rithms, and then converted into an hyperbolic logarithm, by table 37 of Dodson's 

 Calculator, or by table 7 of Dr. Hutton's Logarithms. 



10. An expression of this kind, - ~ rcc , when c is the only variable quantity, 

 consisting of several figures, and r and s are likewise long numbers, will be much 

 better adapted to the use of logarithms, when put in this form, - X u. *$ be- 

 cause the multiplications of r and s into cc, or additions of their logarithms and 



3 H 2 



