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equal to zero; therefore if the Apex is determined according to 
Arry’s method, the conditions: 
[st sin | =0 and [3E sin 1] =0 Tete NN 
must be satisfied. The same conditions have been deduced by KAPTEYN 
from his fundamental hypothesis (p. 359). 
h : : 
If however — = 0, the coefficients of dA and dV in the equations 
v 
(B) are zero, and further Apex-determination is out of the question. 
As a first objection against KAPTEYN’s normal equations may be men- 
tioned that it is not self-evident that a solution of his equations is 
impossible in this case; on the contrary, with a given combination of 
r and v, the position of the non-existing Apex may be arrived at. 
3. We shall now try to prove that the conditions which, according 
to Kapreyn (Proceedings p. 362), may be derived from Arrv's method 
are not correct. 
When the position of the Apex and the amount of the solar motion 
have been found, and the apparent proper motion is resolved into 
the peculiar proper motion and. the parallactic one, the sum of the 
squares of the components of the peculiar proper motion, according to 
Arry, must be a minimum. As the component of the parallactie solar 
motion perpendicular to the direction of the true Apex is zero, the 
place of the Apex and the amount of the solar motion must be 
determined so, that: 
ee h 
[7°] = minimum and (e = emcee 2) | = minimum 
¢ 
(see KAPTEYN |. c.) 
Let g be the angle made by the motus peculiaris m with the 
direction of the star towards the true Antapex, whose Right Asc. 
and Decl. are A and D then: 
m 
T= — sing. 
If, however, we resolve the proper motion into two components, 
one in a direction towards a point outside the Apex (Right Asc. 
