Captain Hall’s comet seen at Valparaiso. 57 
By equation (10) we easily deduce an account of the 
smallness of H 
\T— 'l H 
77 77 sin 1 "sin {it— it) 
and with sufficient exactness 
v+ dV=* - *'+ dv—d*'+ + 
• • 1 sin («• — w) 
therefore 
dV = — ( dir — dir ) + ^ — ' 7Z> — V + 
z sin 1 " cos it ' cos it {(3+d@ — 0' — -d|3) 1 
sin {it — i/) 
and the same expression serves for dV f changing 7 /, ft 1 and V 
into tt", ft" and V'. 
Hence, substituting the values of ft, ft', ft", % , vr", tt” &c. & c. 
we obtain 
dV = — 3 44* +,3446.2:' — 439o.r+N (2680— ,4672.2;+, 21 88^)* 
dV '= — 703 +.2954Z"— 4390x+N , (4845— ,4672X+,io65<r'7 
where log N = 5.63801 and log ^=5.47328 
and finally, equations (8) and (9) give 
N (2680— 2534Gfr+63i5o^>) 2 — 364^ — 150649^=2428 (12) 
NX4845 — 3i66^+83572c//)) 2 — 452^— 227619^=3212 ( 13) 
From these equations the values of dp and dt may be derived. 
The indirect solution seems to be the shortest, as we know 
the value of dp within narrow limits. 
The first error of p as deduced from the approximation of 
Laplace, rarely indeed will amount to + ,01. 
1. Let us suppose dp— — ,005. Then equation (12) gives 
^=+4,294 and —1,194. The positive value is too great to 
be admitted, taking t=— 1,194, res ult from equation 
(13) is 0= — 468. 
* This number 344'— 6° 16' 44" — 6° 11* should be strictly 367"— 6° 16' 44" 
—6° 1 1' 37" ; but the accidental omission of 37" is not of the smallest consequence 
in the result. 
MDCCCXXII. 
I 
