414 ASTROMETRY 



would probably have led him to the same conclusion which I am 

 now about to submit to you. 



In order to show clearly the anomaly in the distribution of the 

 proper motions here alluded to, it will be necessary to call to mind 

 how this distribution must present itself if our fundamental hypo- 

 thesis is really satisfied. 



For this purpose consider a great number of stars, very near 

 each other on the sphere. For the sake of convenience we shall 

 assume them to be all situated in the same point S (see page 399, 

 Fig. 1, P) of the sphere, though not at the same distance from the solar 

 system. 



The peculiar proper motions of these stars shall be distributed 

 somewhat in the manner indicated in Fig. 1, P. Now, as explained 

 before, if we compose the peculiar motions SB, SC, with the paral- 

 lactic motions which are all directed along Sr, we get the really 

 observed motions Sb, Sc which have been represented in Fig. 1, Q. 



From this it must be evident that, whereas, according to our 

 fundamental hypothesis, the distribution of the peculiar proper 

 motions will be radially symmetrical, this symmetry will be de- 

 stroyed for the observed proper motions. There will be a strong 

 preference for motions directed towards the Antapex (see Fig. 1,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 a Sx 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 furnishes probably 

 the best means of determining the position of the Apex. 



For if, from all our data about proper motions, 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 Antapex. 



In trying to realize this plan we meet with the difficulty that we 

 do not find in reality any such perfect symmetry as our hypothesis 

 demands. For the lines of symmetry we 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 demanded line of symmetry the axis of the x, the 

 line at right angles thereto the axis of the y, then we may, for in- 

 stance, define the line of greatest symmetry to be that which makes 

 zero the sum of the y's. 



The line of symmetry furnished by this definition, if prolonged, will 

 not pass through a single point; they will all cross a certain more or 



