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13. Method of ARGELANDER. 
In this method each star gives an equation of condition of the form 
(a4) p= 0 (weight sin? À,) 
They are treated with least squares. So in reality A and D are 
determined by the condition 
(25) Ep? sin? À, minimum, 
giving the minimum conditions 
0 = p sin® i, 8x __ 9, 
d 
SD 
(26) Ep sin? À 5D 
LG Sa 
For a single region of the sky the two are reduced to this one 
Qt) =p 6, 
so that here neither the condition furnished by hypothesis 4 is 
satisfied. 
The objection to the method of ARGELANDER consists chiefly in 
this that the retrograde proper motions have too great an influence. 
Let for instance the proper motions ge #2 #3 #4 (belonging to 
stars in the same region of the sky) make with an assumed direction 
towards the Antapex angles of +20°, +10°, —10°, —20°. As long 
as we know only these proper motions the assumed direction 
towards the Antapex, both according to my method and to that of 
ARGELANDER, will be the most probable. If however a proper 
‘motion «; is added, making with the assumed direction towards 
the Antapex an angle of 170°, this direction, according to ARGE- 
LANDER’s method will have to be corrected by 34°, whereas 
according to our method that correction will be only 2°.1 in the 
same direction. Since long it has been remarked moreover that in 
ARGELANDER’s method too, discontinuous changes in the place of 
the Apex may be caused by continuous changes of the proper motions. 
The following example will prove this clearly. 
In a definite region of the sky there are stars whose proper 
motion is in perfectly the same direction. This common direction 
is assumed as the approximate direction towards the Antapex. 
We now add one star, making with that direction the angle 
Py = 180 —@ 
