104 
MR. GALLOWAY ON THE PROPER 
To determine this quantity, let C be the place of the sun at 
the beginning, and C' its place at the end of the time t, S the 
place of the star, and let straight lines joining CS and C'S 
meet the great circle whose plane contains the points C, C', S, 
in s and s'. This plane also contains the points Q and T, 
which, therefore, are in the circumference of the same great 
circle. Now the angle CSC', being the difference between QCS 
and QC'S, is the parallax, or angular variation of the apparent 
place of the star, which in consequence of the motion of the sun from C to C' will 
appear to have moved from s to s', and is therefore (making one year) the angle 
denoted by As. Hence the triangle CC'S gives 
sin A s= 
CC' x sin QC'S 
CS 
Let CC', which is constant, be denoted by R, and CS, the distance of the star, by r ; 
then, observing that QC'S the angular distance of the star’s place from Q is the angle 
which has been denoted by %, and that sin As= As sin l", the above equation becomes 
A 
r sin 1" 
Substituting this in the equation Ase(\p')=e, we have 
ffsiny , siny . ... 
or -^s(^') = a constant; 
whence it follows that in order to reduce all the equations of condition to the same 
degree of precision, it is necessary to multiply each by sin y, and to divide by a 
number proportional to r. The relative distances of the stars are however unknown, 
and in the present inquiry it has been assumed that they are all at the same mean 
distance, that is, r is assumed to be constant ; and accordingly sin y becomes the 
measure of the precision of the equation. 
Multiplying equation (5.) by sin y, the formula for the equations of condition be- 
comes 
sin yjd'V— |cos D sin S-f- sin D cos h cos (« — A)j^A-f- — — dD;. . (6.) 
or, since 
cos D sin 4/' 
siny sin (a — A)’ 
sin %d$= sm S ;;! X ) jcos D sinS-f- sin D cos l cos (a - A) je?A+ dD. 
( 7 .) 
The double equation affords some advantage in checking the calculations ; and it 
will be observed that the logarithms of all the sines and cosines required for com- 
puting the coefficients of dA and dD have already been used for computing y and \p'. 
It may be proper to state, that although in the appended table the logarithms of the 
sines and cosines of the different angles, and the coefficients of dA and dD in the 
