( 365 ) 
to the Apex be determined by a number of proper motions (not 
shown in the figure) which, to avoid complication, we shall suppose 
to be all direct. If now one more star be added, whose proper 
motion SA makes an acute angle with the line towards the Apex 
(which therefore is retrograde) we easily: see that according to the 
condition 27? min. (Airy) the line SR towards the Apex will have 
to be turned somewhat more into the direction RV, whilst the con- 
dition (10) demands a movement in the direction RW |). 
a h ae: 
12. The condition = (— sin A — v) minimum. 
The equations of condition are of the form 
h 
(23) —sindA =v 
Q 
They contain the distances which are as a rule unknown. This 
is certainly the chief objection to the use of these equations. They 
seem therefore much more suitable to give information about mean 
parallaxes of definite groups of stars when once the Apex is known, 
than to assist in determining the position of the Apex itself. 
For the calculations according to AIRy’s method different ways 
have been followed to escape the difficulty arising from the unknown 
distances. One of the commonest practices (STUMPE, PoRTER, etc.) 
is to divide the stars into groups included between narrower or 
wider limits of proper motion and then to assume the distance of 
the stars of each group to the sun to be the same. 
If this be true in the mean of great numbers of stars for diffe- 
rent parts of the sky, it might seem for a moment that we might 
really derive trustworty values of dd, dD and the mean value of 
i ; 
kn from a treatment of the equations (23). Meanwhile we must 
Q 
bear in mind that at all events a new hypothesis has been introduced, 
viz., that the mean parallax of stars with equal proper motion in 
different parts of the sky, is the same. If this is not the case the 
position obtained for the Apex too will be in general erroneous. 
However there is another and decisive objection to the use of 
equations (23) if we have grouped the stars according to their proper 
1) A practical advantage of our method over Atry’s may still be mentioned here: 
In Atry’s method the large proper motions have a much more predominant influence 
on the results even than in ours: This is easy to see from the normal equations to 
be given in art. 15, 
