276 PROCEEDINGS OF THE AMERICAN ACADEMY 



infer that the distance of the Sun from the plane of these stars is small 

 compared with its distance from them. If the stars lay exactly in 

 one plane we might infer their distances from the Sun from these 

 residuals. As the residuals of the brighter stars show no systematic 

 arrangement, it seems probable that the variables of the fourth class 

 lie nearly, but not exactly, in a plane. This plane approaches that of 

 the Milky Way, but does not coincide with it. The pole of the latter 

 is nearly in R. A 12* 40"' and Dec. -f- 28°. Evidently the residuals 

 in column three would be greatly increased if we moved the pole from 

 its assumed position of R. A. 13* and Dec. -\- 20°, by more than 10° 

 to the pole of the Milky Way. The position of the Milky Way, as 

 given in the '' Atlas Coelestis Novus " of Ileis, agrees, however, more 

 closely with the plane of the variable stars. 



It is not certain whether the stars of longer period given in the 

 third section of Table XII. should be included with those of the 

 fourth class of variables. With two exceptions, IF Virglnis and 

 u Herculis, they lie near the plane of the others. 



The total number of stars in the Durchmusterung north of the 

 equator is 315,048. Since the area of the hemisphere is 20,626 square 

 degrees, this corresponds to 15.3 stars per degree, or an area of 236 

 square minutes to each star. A circle having a radius of 8'.7 would 

 have an area equal to this. If, then, a circle having this radius is de- 

 scribed around any star as a centre, it will be an even chance that 

 another star will be contained within it, provided that the presence of 

 the second star is no way affected by that of the first. For circles of 

 other radii the chances will vary as the squares of the radii, or as the 

 areas. We know from the existence of clusters and multiple stars that 

 one star is not without influence on the presence of another, and that 

 this effect may extend to some distance, as is shown in the Pleiades 

 and in Praesepe. This principle may still be used in comparing diff'er- 

 ent classes of stars, although the distance 8'.7 should be diminished. 

 It is, therefore, surprising that the average distance of the companions 

 of stars of the fifth class is as great as 10'.6, especially as two of them, 

 S Cancri and X Tauri^ lie near the large clusters Praesepe and the 

 Hyades, where the average intervals between the stars is much less. 

 A circle of radius 10'. 6 has only two-thirds the area of that of 8'.7, 

 hence these companions are only two-thirds as thickly placed as the 

 stars in other parts of the heavens. This effect extends to the square 

 degree, as is shown in the sixth column. It appears to be probable 

 that there is no physical coimection of these stars with the variables, 

 and that their sparseness is due to their distance from the Milky Way. 



