MOTION OF THE SOLAR SYSTEM. 
105 
equations of conditions, are given to four decimal places, all the calculations were 
made from the tables of five-figure logarithms. 
The weights and probable errors of the values of dA and dD deduced from the 
equations of condition by the method of least squares, and the probable error of T, or 
probable value of sin %, are computed according to the following formulae : 
Putting the above equation (7.) under the form 
0 = ad A -f - bdD — n, 
and assuming, according to the usual notation, (aa) to denote the sum of the squares 
of the coefficient a, ( bb ) the sum of the squares of b, ( nn ) the sum of the squares of n, 
(ab) the sum of the products ab, and so on, and denoting also the weights of dA and 
dD respectively by w(dA),w(dD), the formulae of the method of least squares give 
(bb)(an) — (ab)(bn) (aa)(bn) — (ab)(an) 
(aa) {bb) — [ab) (ab) ’ C (aa)(bb) — (ab)(ab) 
w(dA) — 
(aa) (bb) — (ab) (ab) 
~(W~ 
w(dD)= M. 
Let h denote the number of equations, s(T) the probable error of T, s(dA) and s(dD) 
the probable errors of dA and dD, and assume 
(nn.2) = (nn) 
(an) 9 - (bn. I) 2 
where 
(aa) (bb. 1)' 
(bn.\) = (bn)-^ (an) ; (bbA) = (bb)~ (ab), 
then the theory gives 
effl =*67449 ; e(dA)= ^ 
V w(dA) ’ 
-.(dD)- 
\/ w(dD) 
It may be remarked that in these formulae (nn.2) denotes the sum of the squares of 
the remaining errors when the values of dA and dD, found as above, are substituted 
in the equations of condition. 
In the following table (which has been frequently referred to) the names of the stars 
are given, with their mean places for 1790, and the principal details of the calcu- 
lation. The columns headed “ Lacaille — Johnson,” i: Lacaille — Henderson,” 
“ Bradley — Henderson,” contain the proper motions as given by Mr. Johnson and 
Prof. Henderson in their respective catalogues for the whole interval between the 
compared catalogues, those in ZR being in seconds of time. In the column of differ- 
ence in 7R. the positive sign shows that the right ascension is greater in the ancient 
catalogue than in the modern, and has consequently been diminished, and the nega- 
tive that it has been increased, through the effect of the proper motion. In the 
column of difference in declination, the positive sign indicates a proper motion 
towards the south, and the negative a proper motion towards the north ; the stars 
being all in the southern hemisphere, and their declinations consequently supposed 
to be affected with the negative sign. 
MDCCCXLVII. P 
