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to each equation — ceteris paribus — the same weight. In the first 
supposition the weights ought to be proportional to g?. As we sup- 
pose with Kaprryn (Proceedings p. 357) that the distances from the 
sun to the stars, whose motions are considered, as equal, the two sets 
of normal equations are identical on both the suppositions. 
The three normal equations for X, Y and Z, proposed by Airy, 
are derived from the equations for the two components 7 and wv. 
If however for some stars one of the components is unknown, we 
can deduce three normal equations from the other component, or if 
both components are known for all stars, we can, starting from each 
of the components separately, construct two sets each of three normal 
equations. KAPTEYN follows the last method, and so shall we. 
It is of course immaterial what are the directions 7 and U of the 
components of the proper motion and we may choose those that are 
the most appropriate. AIRY uses the direction towards the north 
pole of the equator and the direction of the parallel; KAPTEYN 
chooses the direction towards a point near the Antapex and the 
direction at right angles to it; we shall also follow the latter method. 
_ 2. Meaning of the symbols according to KAPTEYN: 
A, and D, right ascension and declination of the assumed Antapex ; 
Ao the distance from the star to this point; 
xo the angle made by the declination circle with the direction 
towards this point; 
vo the component of the total proper motion « according to the 
latter direction ; 
zo the component perpendicular to the preceding ; 
Po the angle, made by the total proper motion with the parallactic 
proper motion. 
_ The symbols without index (,) will be used when the real, instead 
of the assumed, Antapex is meant. 
Let « be the angle made by the directions of the star towards 
the assumed and the real Antapex, then we can put the equations 
(A) in this form: 
h h 
To =H 8in py = — sindsine; Uo = 4 c08spyp = —sinheose . . (A) 
5 Q 
The two last terms of the equations for 7o and vo are both 
