258 REPORT—1905. 
antapex. (See Q.) One thing, however, must be clear, and we want no 
more for what follows ; it is that there will remain a bilateral symmetry, 
the line of symmetry being evidently the line aSa through the apex, the 
star, and the antapex. _ 
Near to this line, on the antapex side, the proper motions will be 
most numerous, and they will be greater in amount. 
This evident condition of bilateral symmetry would furnish probably 
the best means of determining the position of the apex. For if from all 
our data about proper motion we determine these lines of symmetry for 
several points of the sky and prolong them, they must all intersect in 
two points, which are no other than the apex and the antapex. 
In trying to realise this plan we must meet with the difficulty that on 
account of errors of observation and the restricted number of stars 
included in the investigation we must be prepared to find in reality no 
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such perfect symmetry as theory demands. For the lines of symmetry we 
shall thus have to substitute lines giving the nearest approach to symmetry. 
Their position will depend, at least to a certain extent, on what we choose 
to consider as ‘the nearest approach to symmetry.’ 
If we call the required line of symmetry the axis of the a, the line at 
right angles thereto the axis of the y, then we may, for instance, define 
that position of the a-axis as the line of greatest symmetry, which makes 
zero the sum of the y’s 
The lines of symmetry furnished by this definition, prolonged, will not 
pass through a single point ; they will all cross a certain more or less 
extended area, the centre of gravity of which might be taken as the most 
probable position of the apex. 
Drawing great circles through this apex, we must necessarily find them 
diverging somewhat from the lines of best symmetry in different parts of 
the sky. 
