BURNS— RELATIVE AND ABSOLUTE PARALLAXES. 497 



with respect to the sun. In that case the two last-mentioned 

 equations are identical; that is, the motion across the line of sight 

 with respect to the sun is equal to the motion in the line of sight 

 afifected by the factor 1.57, for the mean of a large number of 

 objects distributed and moving at random. The unit of V is kilo- 

 meters per second, and of [i is seconds of arc per year. Substitut- 

 ing for Vm we find 



Tt" = 77 = 3.02 irr- 



In case the data can be represented by curves somewhat simi- 

 lar to the probability curve, the average parallax is given by the 

 formulae of the last paragraph by comparing average radial ve- 

 locities with average proper motions. If the data fit the probability 

 curve well, we may use mean values of fx and V , as the ratio of 

 the mean is the same as that of the average. We may still use 

 mean values if, as is the case with the data under discussion, the 

 curves for ii and V are of the same form and depart but little from 

 the error curve. By " average " is meant such a value that there 

 are as many data larger as smaller. The mean is 18 per cent, 

 larger than the average, so we have for the mean parallax 



7r" = 3.56^- (a) 



Formulae similar to the foregoing are only strictly valid in 

 case the objects involved are distributed at random and moving 

 at random. Any tendency toward systematic motion places certain 

 restrictions on the use of these formulae. For instance, if we had 

 accurate proper motions of all the planetary nebulae, the parallax 

 as derived from the tau components would be too small, for the 

 radial velocities of these objects show that they are moving sensibly 

 parallel to the galactic plane. The apices of the sun's way lie near 

 to this plane, therefore the tau components are very small in com- 

 parison with the radial velocities and total proper motions. The 

 use of the upsilon components would be free from this latter dif- 

 ficulty. The use of total motions also avoids this difficulty, and 

 has the further advantage of saving a great deal of labor. 



