96 
MR. GALLOWAY ON THE PROPER 
the point Q towards which the sun is moving- ; and hence, in order that all the equa- 
tions may have the same weight, each must be multiplied by the sine of that distance. 
In other words, if e(¥) denote the probable error in the observed direction of the 
proper motion, or the probable value of for a star at the distance of 90° from 
the point Q, then s(¥) will also be the probable value of (\p — ^') sin % for a star 
whose distance from Q is measured by the angle %. Hence it follows that every value 
of 4 / must be multiplied by sin %. 
The eighty-one equations of condition are given in an appended table. They are 
of the following form, 
0= -j-adA-j-bdD — n, 
where a, b, and n are numbers deduced from the data, and dA, dD the quantities to 
be determined from the equations and applied as corrections to the assumed values of 
A and D. Forming the squares and products of these numbers, and adopting, 
according to the usual notation, ( aa ) to denote the sum of the squares of the coeffi- 
cients of dA, ( bb ) the sum of the squares of b, ( ab ) the sum of the products of a and b, 
and so on, the following values are found: — 
(nn) = 178660-4, (aa) = 38-5423, (bb) =26*7425, 
(ab) = — 5-6852, (an) = — 129*462, (bn)— - 105693, 
and consequently the two following equations for determining dA and d D, viz. 
0=+38-5423e?A— 5-6852rfD- 129-462, 
0 = — 5-6852c?A+267425^D- 105-693, 
the solution of which gives 
dA =-J-4°-070 with the weight 37*333, 
dD= +4 0, 817 with the weight 25'904. 
On computing s(T), or the probable value of (ip—\p') sin %, from the appropriate 
formula of the method of least squares, we find s(T) = 31°-98 ; whence, and from the 
above weights, the probable errors of dA and d D are respectively 5 0, 234 and 6°*285. 
The result, therefore, of the whole calculation from the assumed values of A and D, 
namely A=259° 46'-2, D=+32° 29'-6, gives the following values of A and D for the 
position of the point Q for the beginning of 1790, 
A =263° 50'-4±5° 14'-0; D = +37° 18''6+6° 17'*1. 
This result presents a very remarkable agreement with that obtained by Otto 
Struve from the Dorpat observations ; the values of dA and dD are, however, some- 
what greater than the probable errors of the hypothesis, according to the determina- 
tion of Argelander. 
Following out the principle of the method, the next step would be to recompute 
the angles • 4/5 and the equations of condition, with the values of A and D now found, 
so as to obtain a result having a smaller probable error ; but, in the present case, the 
labour attending a new calculation (by no means inconsiderable) is altogether unne- 
cessary, as will appear from the following considerations. 
