MOTION OF THE SOLAR SYSTEM. 
103 
If we now substitute for dpJ the difference between the angles and * 4 /, or the 
value of as found by equations (1.) and (3.), we shall have an equation in 
which dA and c?D are the only unknown quantities. Every star furnishes a similar 
equation ; and the values of dA and dD deduced from the whole of the equations by 
the method of least squares give the corrections to be applied to A and D, the assumed 
right ascension and declination of Q. Before this method can be applied, however, 
it is necessary to consider how the observations are affected by the situation and 
other circumstances of the individual stars, in order that all the equations may be 
reduced to the same degree of precision. 
In the present inquiry it is assumed that the positions given in the catalogues, and 
the reductions from the first epoch to the second, are equally precise for all the stars ; 
and in respect of the true proper motion, it has already been stated that we are not 
possessed of data to enable us to make any distinction between one star and another, 
and must therefore assume that, in this respect, all the equations have the same 
weight. Confining our consideration, therefore, to that part of the apparent motion 
which is caused by the displacement of the sun, the relative weights of the equations 
are determined as follows : — 
Let the parallactic motions in right ascension and declination, in the unit of time 
(here assumed to be one year), be denoted by A a and AS respectively, and the cor- 
responding motion in the arc of a great circle by As, and we have the equations 
As sin - 4 /= cosSAa; As cos 4;'= AS, 
by differentiating which we get 
dAs sin -4/'+ As cos 'p'd-p'—d(coslAci) > 
dAs cos 4 /— As sin -p'd-p' =dAS, 
whence Asdp'= cos pJd{ cos SAa) — sin pJd AS. 
Denoting in general the probable error of any quantity x by e(x), and its square by 
i 2 (x), and observing that if x=y+z the theory of probable errors gives 
*(*) = */{£%) +s 2 (s)}, 
we have in respect of the above equation 
Ass(\p')= s/{cos 2 4 A 2 (cosSA a )-j- sin 2 ^'^(AS)}. 
Now the probable errors of Aa and AS depend manifestly on the precision and number 
of the observations from which the places of the stars have been determined ; and as 
minute accuracy is not attainable in the present case, it may be assumed that the 
places of all the stars in each catalogue have been determined with equal precision. 
It may also be assumed that in respect of an equatorial star the probable error in 
right ascension is equal to the probable error in declination, and, generally, that 
e(cos SAa) =g(AS). Denoting therefore the constant error by e , these two assumptions 
give e(cosciAa)=£(A($)=e, and the above equation becomes 
As s(%p') = e ; 
whence it appears that the probable error of -p' is inversely proportional to As. 
