( 359 ) 
be avoided by a grouping into classes of different proper motion. 
This can be done in still another way. 
By substituting the value (3) for v in (I) it becomes 
= u eos p sin À, maximum. 
As this holds also for stars whose proper motion is included 
between determined limits, it also holds for stars with absolutely 
the same proper motion «== tj. For such a group the condition 
becomes 
(II) cos p sin À, maximum 
and as each value of the proper motion leads to such a condition, 
it must also be satisfied by all the stars together. 
The equations for the coordinates of the Apex, obtained in this 
way contain only the directions and are entirely independent of the 
amount of the proper motion. 
It seems to me however that the condition (I), at least if it is 
applied to stars whose proper motions are included between pretty 
narrow limits, is preferable to (II) especially for this reason, that 
the former is a more direct consequence of hypothesis H on which 
the investigation is based. 
8. Derivation of the Apex from the condition (1). 
To determine the coordinates of the Apex in such a way that 
condition (1) is satisfied, the differential quotients in regard to A and 
D of Zvsinhy must disappear. Consequently we have with the 
aid of (5) 
Oe Cok Sees 
(13) EBL A hapa tN 
which for stars at one point of the sky is reduced to the single equation 
= t= 0, as of course was to be expected. 
Let A, and Do be approximate values of A and D and dd, dD 
the required corrections of these. All the quantities computed with 
the aid of these approximate values will be distinguished by means 
of an appended ,. 
So vo and zt, will represent the projections of the proper motion 
w on the great circle through the star and the approximate position 
of the Apex and at right angles to it. 
We thus have in the equation (13) 
