( 369 ) 
Its position is no other than that of the group of stars itself. In 
the choice of the position of the Apex each region votes as it were 
for itself. Every line passing through this region thus passes through 
the Apex too, so that at the same time the condition (10) is satis- 
fied and it is not. 
This peculiarity of the method together with this second (which 
exists for stars of one part as well as for stars of all parts of the 
sky) that for the direction of the motion of an arbitrary number 
of stars we may substitute a diametrically opposite motion without 
the slightest effect on the coordinates of the Apex, appears to me 
sufficient to declare the method unsuitable for the determination of 
the direction of the motion of the sun. 
15. Abridged calculation. 
It is a very common practice in the derivation of the coordinates 
of the Apex, to abridge the work of computation by taking the 
mean of the proper motions of a greater or smaller number of stars 
situated close together. I wish to point out that in this way the 
result, derived by means of the various methods, will approach in 
general to those which will be found by the method proposed here. 
So, far from having been more or less impaired by this abridged 
calculation, the results must have gained considerably in accuracy. 
It must be borne in mind however that in this way, in all 
methods except in that proposed by me, the principle is sacrificed, 
at least in part. 
The proof of what has been advanced here will be best given by 
writing out in full and in a similar form for the various methods, 
the equations of condition and the normal equations ensuing from 
these. I begin by giving them. 
a. Method of Airy (as modified). 
I leave out of consideration condition (19), this being the only 
one dependent on the distances. 
As 
Or Or dz Ox 
i Seite ade in ean? Ven pias el oe ogy aay 
r= tt Ga dA +(5p), 4 Fy + vo (54), 4 1+ vo (54). D 
) 
the equations (16) become 
(30) wv, (54) aa ib 4 CARL rg 
which, treated with least squares, give the normal equations: 
27 
Proceedings Royal Acad. Amsterdam. Vol. IL. 
