( 692 ) 



If this succeeded we miglit then compare this mean p. m. with 

 the mean p. ni. of different classes of stars, the mean distance of 

 which is known with some approximation or, better perhaps, with 

 the mean radial velocity of the nebulae determined by the spectro- 

 scope. The comparison would lead at 0)ice to ideas about the real 

 distances. 



Unfortunately the mean of a great number of observed p. m. will 

 not be materially more correct than the individual values, if the total 

 proper motion is small. The cause of this lies in the fact that in such 

 a case the effect of a determined error of observation is not at all 

 cancelled by an ecpial but opposite error of obser\'ation. Suppose for 

 instance two nebulae both having in reality a p. m. of 0"01. For 

 the first let the error of observation be 0"10 in the direction of the 

 p. m. For the second assume an equal error in a direction opposed 

 to the p. m. The observed p. m. of the first nebula will be 0"11, 

 that of the second 0"09. Taking the mean of the two we are not 

 brought nearer to the real value. 



For this reason we shall not be led to any valuable result in 

 this way, even if our material consists of very numerous objects, as 

 long as the errors of observation exceed the real p. m. 



The difficulty here considered would vanish if, instead of the total 

 p. m., we could avail ourselves of some component of the p. m., 

 which in different direction would have different sign. In this case, 

 if systematic errors can be avoided or determined, the accuracy would 

 increase as the square root of the number of objects included. 



Such a component of the p. m. is that in the direction towards 

 the Antapex. From this component we may derive the mean paral- 

 lactic p. m. which is a measure of the mean parallax. 



I will not here stop to consider the hypothesis involved. It must 

 be sufficient to state that it assumes that the sum of the projections 

 on some determined direction of the peculiar p. m. vanishes in the 

 case of very numerous nebulae or, which comes much to the same 

 that the peculiar p. m. may be treated as errors of observations. 



Let 



h be the linear annual motion of the solar system ; 



Q the distance of a nebula from that system ; 



A the angular distance of this nebula from the Apex of the solar 

 motion ; 



V, T the components of the observed p. m. in the direction towards 

 the Antapex and at right angles to that direction ; 



p the component of the peculiar p. m. in the direction towards 

 the Antapex. 



