88 
MR. GALLOWAY ON THE PROPER 
as to warrant the assumption of a systematic origin, or make evident the operation 
of a common cause. This mode of considering the question was first adopted by 
Argelander, in the memoir alluded to at the commencement of this paper. 
Argelander’s investigation, as already stated, is founded on the proper motions 
of 390 stars, determined by a comparison of their mean places in 1755, according to 
Bbssel’s reduction of Bradley’s observations, with their mean places in 1830, as 
given in his own catalogue*, deduced from observations made by himself at Abo, 
the interval between the epochs being seventy-five years. In this investigation every 
star was included which appeared, on comparison of the two catalogues, to have 
undergone a change of position to the extent of 7 ,, ‘5, or to have an annual proper 
motion amounting to 0"T in space. By reason of the excellence of both catalogues, 
the long interval between their respective epochs, and the very considerable number 
of stars employed, the result must be considered as by far the most satisfactory that 
had yet been given. 
The method of calculation employed by Argelander may be described generally 
as follows : — 1. From the variation in right ascension and declination given by the 
comparison of the catalogues, the angle (4) is computed which the apparent path 
described by the star makes with the circle of declination. 2. A point (Q) is assumed 
as the apex of the sun’s motion, and the direction in which the star would appear to 
move (if it had no real motion of its own) in consequence of the motion of the sun, 
is computed from the position of the star and the assumed position of the point Q, 
and expressed in terms of the angle (40 which it makes with the declination circle. 
3. The trigonometrical value of 4 ' is differentiated on the supposition that the right 
ascension (A), and declination (D) of the point Q are variable quantities, and in the 
resulting expression the numerical value of the difference of the angles 4 and 4' is 
substituted for dip', by which means an equation is obtained in which there are only 
two undetermined quantities, viz. dA and dD. Each star furnishes a similar equa- 
tion ; and as the effect of the real motion of the sun on the apparent displacement of 
any star is proportional to the sine of the star’s distance from the apex of the sun’s 
motion, each equation is multiplied by the corresponding sine of this distance, 
whereby they are all reduced to the same degree of precision. 4. The equations are 
then solved by the method of least squares, and the resulting values of dA and dD 
applied as corrections to the assumed values of A and D, which determine the 
situation of the point Q. With the corrected values of A and D thus obtained, the 
angles 4* may be recomputed, and after one or two repetitions of the same process, 
values of A and D will be obtained giving the position of Q which most nearly 
represents the whole of the observations. 
The effect of the sun’s displacement on the apparent proper motion of a star is 
inversely proportional to the distance of the star from the sun ; but this distance 
* DLX Stellarum fixarum positiones mediae ineunte anno 1830. Ex observationibus Abose habitis deduxit 
Fr. Argelander. Helsingforsise, 1835. 4to. 
