VOL. LXXXVII.] PHILOSOPHICAL TRANSACTIONS. 230 



Put n = % \a . s>, ± (h — h) — t, ±a . s*, i (h -f- h). 



We shall have 

 s, da -f- 2s 2 , ±da ,'t, a = n; 

 and s, da = n — 2.s 2 , -^da . t' , a. 



But s, da — Is, ^da . c, \da ; 



, s, da n — 2s % , Ada . '#- a 



' s -*-da — - = ~ * 



* ' ' ■ 2c, \da " 2c, Ida 



and s, da = n - <ls\ \da .t',a = n- It', a I * "*?**£''''' )*, 



1 3 * 7 v 2c, \da ' * 



, ( n — 2s 2 , Ida, fa > 2 n — 4» . s 1 , \da . t', a + 4s 4 , Ida . 't z , a 



because ( — -^ ) — 4x(i-i 5 TPo 



— j ± n . s , \da . t,a — n . s , \da . t, a 



-j- s 4 , ±da . 'f, a = n — 4-rc 2 . t' , a — -±-n 2 .'t,a . s 1 , ±da 



-f 2rc . '<?, a . s Q , i-da + 2n'* 9 , a . s 4 , 4-da — 2V, a . s 4 -f ^da, 



by substituting for s, \da its near value n, 



— n — \7?t', a — -^p- + -Ln 3 ?, a + in 5 '* 2 , a — |bV, or, 



where the last term but one containing the 5th power of n may be rejected, as it 

 has been omitted by M. de Lambre. 



As da is always very small, the arc da in parts of the radius, unity, = s, da in 

 parts of the same radius, therefore s, \" : 1" :: s, da (in parts of radius unity) : — 



X s, da = da in seconds, = — pr X (w — 2* 2 , £da . '/, a) = — -„ X (n — da . s, 



S, L S, 1 v 



, , N 1" x ra 1" x da .sAda /t,a , c \" , L 



^a . $ o) = — ^ ^-p ; v if we put n === — x :f }$ ±a.s,± (h- h) 



— t, \a . *°, 4- (h -f- h), and da = a number of seconds, we shall have da = n — da . 

 s, \da ,'t,a\ and, for the most part, without any sensible error, da = n — n . 

 s, 4-rc . % a. 



* Table 1 contains l ^~~ and Kfi&& i table 2 contains 10000 X Ai (h + h). 

 Table 3 contains the term — - n . s,^n . 't, a. The argument on the side is a, and 

 that on the top is n or the result found by the, help of the first 2 tables. If this 

 correction should be considerable, with the value of da, found after this correction 

 has been applied, enter table 3, again at the top, and with a on the side as before; 

 the number now found subtracted from n will give the correct value of da. 



By the investigation, da = \'t, \a .vs (h ^ih) — ^t, \a . vs, (h + h) — vs, da . 'ta, 

 where the upper or lower signs are to be used, according as the objects are on the 

 same, or on contrary sides of the great circle to which they are referred ; the 3d 

 term will be negative or positive, according as a is less or more than 90°.-f- If da 

 should come out negative, a will be less than a, or a greater than a. In the case 



* It does not appear what tables are here meant. 



+ Compute the two, which will give the approximate value of da, and make use of them in com- 

 puting the 3d term ; and join the 3 terms together according to their signs, which will give da still 

 nearer ; and, if this should prove considerable, compute the 3d term a 2d time with the new value of 

 ad. — Orig. 



