( 358 ) 
This is easily done. For it is seen at once that, whereas the former 
ef the conditions (7) ceases in general to be satisfied by such groups, 
the condition 
FE ysina' == 0 
will still hold. We immediately conclude to this from reasons of 
symmetry. Now as vsina@'=T, we shall also still have 27 =0 and 
the condition (9) will be satisfied which is equivalent to it. 
Consequently there can be no objection to the grouping in classes 
of determined proper motions. With this the only advantage which 
condition (11) might have over (9) disappears in a great measure. 
In what follows we shall therefore entirely neglect the conditions (11). 
6. Stars scattered about the whole sky or any considerable 
part of it. 
So every region of the sky gives a condition of the form (9). 
These might all be combined to one condition 
(12) 2v maximum 
where the sum is to be extended to the stars available in all parts 
of-the sky. In that way however not the most accurate determination 
of the position of the Apex will be obtained: 
To arrive at a more advantageous combination the following 
problem is to be solved: 
Given that for the various parts of the sky the accidental devia- 
tions from Hypothesis 4 are equal, to combine the conditions (12), 
which hold for the separate zones of constant 4, in such a way that 
the effect of those deviations on the coordinates of the Apex which 
have to be determined, be a minimum. 
The solution of this problem which gives rise to no particular 
difficulties, shows that the = v of each region must be multiplied 
by its corresponding value of sin 4, before they are combined into 
a single sum. 
Consequently for the whole sky we shall not have to satisfy the 
condition (12), but 
(I) ZwsinÀ, maximum. 
1, Second form of the method. 
As has already been remarked the objection, that by using (1) 
the large proper motions exercise a very predominant influence, may 
