( 356 ) 
We have to express that these stars satisfy the hypothesis H. 
The very first condition, the one which I shall here use exclusively, 
resulting from this hypothesis is this, that the sum of the projections 
of the peculiar proper motions on any direction is equal to zero. 
This can also be expressed as follows: the resultant of the pecu- 
liar proper motions must be equal to zero. 
If we project the proper motion v (fig. 2) on 
C A 
} 
S h sind B G 
o 
Fig. 2. 
the direction SG to the Antapex and at right angles to it, this 
condition is expressed by the equations: 
(7) Zveosa =0 Zvene =v. 
Now along with the peculiar proper motion each star has a paral- 
: ie ee qe 
lactic motion — sin A, in the direction of SG to the Antapex. Con- 
¢? 
sequently the entire proper motion « of each star has for components 
L 
’ * ! 
v= vcos a’ + — sind T=V Sina 
0? 
so that if we take the sum for all the stars of the group we obtain, 
according to the conditions (7). 
U 
(8) Svs = = t=O, 
The resultant of all the total proper motions « is directed along 
SG, from which we conclude at once that the sum of the projections 
of the motions w on this direction is greater than on any other. If 
therefore the direction in which the Antapex is situated as seen 
from the group of stars under consideration is unknown, it can be 
derived from the observed proper motions by the fact that for this direction 
