( 360 ) 
t =4,,-F (55). dA + (Go). dD = Tod vo (54) as + vo (54) ap 
(dt GA) (saan)? 
Evidently these equations hold only as long as we do not approach 
the Apex or the Antapex within distances of the order of dA and 
dD where the terms of a higher order may not be neglected. It 
will be best therefore to exclude entirely the stars close to the 
approximate position of the Apex. This cannot cause any considerable 
loss of weight. I find e. g. that of the stars of BRADLEY only a 
fourteenth part have sind < 0.40 and less than one eighth part have 
sinA < 0.50. 
The first of the equations (13) now becomes 
et 
+ dD = jv (54) (Bt Eo (545), fu == = ra( 2 sin ho 
The quantities 7 must be in all parts of the sky as often positive 
as negative. According to what we have discussed this is an 
immediate consequence of the hypothesis A (compare form. (10) ). 
dA 
Dor a will already disappear for limited parts of the sky. 
0 
The same holds @ fortiori for the sum extended over the whole sky. 
dy ; : De 
ET (55) differs from the preceding sum only in the quanti- 
0 
ties t being computed with an approximate Apex, the coordinates 
of which need still the corrections dA and dD, This quantity will 
thus be of the order of dA and dD and may be neglected in the 
coefficient of dA. The same holds for all quantities containing 7, in 
the coefficients of dA and dD. So_the above equation is reduced to the 
former of the two following ones, in which the sums are indicated 
with the notation used in the theory of least squares: 
osinn ($4) ] dA-- ina, SE (4) aD=— sind ($2) | 
| ont) a dA + | vosinds ay Jap AGN 
(14) 
