( 364 ) 
We see at once that the direction which causes the + 7!) of the 
proper motions of the stars in S to disappear is the bisectrix SO 
of the angle ASB. Likewise the line which in S' makes = r = 0 
will be the bisectrix SO of A’'S'B’, 
From the given proper motion we conclude therefore, according 
to the method proposed by me, to a position O of the Apex. 
On the other hand the direction which makes minimum the = r? 
of the proper motions of the stars in S is evidently the line S0' at 
right angles to the bisectrix; likwise in S' the line satisfying this 
condition is the line S'O' at right angles to S'O. 
According to the condition 2 7? min. of Airy we conclude there- 
fore to a position for the Apex (or Antapex) in 0'. 
If we leave the angle ASB unchanged, but reduce B'S'4' in 
such a way, that the bisectrix does not change place, then in the 
moment that that angle passes through 90° the Apex (Antapex) 
according to Airy's determination, will suddenly leap over from 0' 
to 0", where it remains when the angle B'S'A' is still more reduced. 
If the angle B's'A’ had retained its original value (> 90°) and 
angle ASB had been reduced in the above stated way, the Apex 
would have leaped from O' to 0”, If then again we had reduced 
4'S'B', at the moment of this angle passing through 90°, the Apex 
(Antapex) would have leaped from O" to 0, 
In the nature of the problem there seems to be no reason whatever 
for such leaping ®) and in our determination the Apex remains 
where it is notwithstanding the changes introduced. 
Moreover it seems very little plausible indeed to assume for the 
Apex one of the positions O', O", 0". 
The place determined according to both methods coincides evidently 
only when both angles are acute. 
2nd example (see fig. 4). For stars in the region S let the line 
Antapexr 
) According to the statement in $3 the condition for stars at one point of the 
sky is equivalent to © r= 0. 
2) There are still many other cases in which the condition (18) leads to discontinuous 
changes in the Apex for continuous changes of the data of the problem. 
