MOTION OF THE SOLAR SYSTEM. 
95 
that the agreement would hold good in so great a majority of instances if it were 
purely the effect of chance. But a much more certain conclusion will be arrived at 
from the combination of the whole of the observations by the method of least squares. 
The method of forming the equations of condition has already been explained 
generally, and the formulae for computation will be given in subsequent paragraphs ; 
but as some of the stars are more favourably circumstanced than others for deter- 
mining the question at issue, it becomes necessary, before proceeding with the solu- 
tion, to assign weights to the equations, in order to reduce them all to the same pre- 
cision, and obtain the most probable values of the corrections to be applied to the 
assumed position of the solar apex. For this purpose some special considerations 
are required. 
Admitting the hypothesis of the sun’s motion, it can hardly be supposed that any 
star is absolutely at rest. The apparent motion of a star, therefore, as it is made 
known to us by a comparison of observations, is the effect of the combined motion of 
the sun and the star. Now, with respect to the true proper motion, we are in igno- 
rance of all the circumstances by which its apparent or visible effect is modified. We 
know nothing whatever respecting the magnitude or nature of the orbit described by 
the star, or the absolute velocity with which it moves. Hence it is necessary to assume 
that all the stars move with the same absolute velocity, in which case (putting the sun’s 
motion out of consideration) the apparent velocity will be inversely as the distance. 
But we are equally ignorant of the relative distances, and are therefore reduced to the 
necessity either of disregarding the distance altogether, or of making some precarious 
assumption respecting it, — for instance, that the distances of the different stars are 
inversely proportional to their magnitudes (as in the method of Otto Struve), or in- 
versely as the quantities of the apparent proper motions. In the present inquiry no 
greater probable accuracy could be obtained by the adoption of either of these as- 
sumptions, and consequently errors to which differences of distance as well as of abso- 
lute velocity give rise, are regarded as constant. The only remaining circumstance 
by which the apparent effect of the true proper motion is modified is its direction ; and 
as there is no a priori reason for assuming that a star is more likely to move in one 
direction than another, all directions must be regarded as equally probable. The con- 
clusion, therefore, is, that in respect of the true proper motions inter se, we have no 
sufficient grounds for making any distinction as to the relative precision of the results 
given by different stars, so that the errors arising from this cause must be treated as 
accidental errors of observation, and all the equations be allowed to have the same 
weight. 
With respect to the part of the apparent motion depending on the displacement of 
the sun, the case is different, inasmuch as the parallactic effect depends not only on the 
distance of the star, but also on its situation with respect to the apex of the sun’s 
motion. The effect of the sun’s motion on the observed position of a star (as will be 
shown more particularly further on) is directly as the sine of the star’s distance from 
