STRUCTUEE OP THE UNIVEESE KAPTEYN. 307 



The answer must be, of any individual insect, no; but of the cloud 

 as a whole we can, provided that the cloud as a whole has not moved ; 

 or expressed more mathematically: Provided that the center of 

 gravity of the cloud has not moved, we can derive the average dis- 

 tance '^ of all the insects. We shall be sure of the immobility of the 

 center of gravity if we know that the direction of the motions of the 

 insects is quite at random; but this is by no means required. The 

 motion may be preferentially in a horizontal plane or along a de- 

 termined line, say along the longer axis of the pond, provided only 

 that the motions in any two opposite directions are equallj'^ frequent. 



Not only that, even if the cloud, as a whole, is not immovable, we 

 are not necessarily helpless. For, if the insect cloud and>the photog- 

 rapher were both on a sailing vessel, circumstances would be the same 

 as on the mainland, though now the cloud is in motion. Only, instead 

 of the absolute displacement of the photographic apparatus, we must 

 laiow the displacement relative to the ship, or rather relative to the 

 insect cloud. This, then, finally is the real thing wanted. We may 

 obtain the distance of the insect cloud, or what comes to the same, the 

 average distance of its members, as soon as we are able to find out the 

 displacement of our point of view with regard to the center of gravity 

 of the cloud. 



Our case is much the same in the world of the stars. 



We shall be able to determine the average distance of the members 

 of any arbitrary group of stars, provided that we can find the motion 

 of the solar system, both in amount and in direction, relative to the 

 center of gravity of the group. 



Now, astronomical observations such as those which led the elder 

 Herschel to his discovery of the solar motion through space enable us 

 to determine the direction of the sun's motion relative to such groujjs 

 as the stars of the third, fourth, etc., magnitude. Spectroscopy en- 

 ables us to determine the amount of that motion. 



We must be able, therefore, to find out the average distance of the 

 stars in these groups. For other groups, such as the stars having an 

 apparent centennial motion of 10", 20", etc., there is a difficult)^ 

 Still, however, we have succeeded in overcoming this difficulty by a 

 somewhat indirect process, and pressing into service the stars of which 

 the individual distances are known. This, then, is the upshot of 

 astronomical work on the distances. 



" The expression " average distance " ought, strictly spealiing, to be replaced 

 by the distance corresponding to the average parallax. For sake of clearness I 

 have ventured here and in what follows to substitute one expression for the 

 other. 



