260 G. W. Colles, Jr. — Distance of the Stews. 



discussed it in two papers published in the Proceedings of the 

 Royal Irish Academy and in the Monthly Notices, respec- 

 tively.* It could, of course, be extended to multiple systems, 

 the principal drawback being that it necessitates the knowledge 

 of the inclination of the plane of revolution to the line of sight. 

 Obviously it can not be applied to any single body, unless its 

 exact direction, — that is, the ratio of the components of its 

 motion in and across the line of sight, — be known. In such a 

 case as that of the so-called " runaway star " (1830 Groom- 

 bridge) this might sometime be ascertained, if the star should 

 ever get far enough so that increase or decrease in its annual 

 angular motion could be measured. In general, however, the 

 limitations of our present knowledge of astronomy seem to 

 preclude this method of finding the distance of particular stars. 



The application of Doppler's principle which I had in mind, 

 however, is a very much wider one than any of these.f It is 

 to the problem of finding the mean distance of all the stars, 

 involving the theory of probability, and giving us a more Or 

 less reliable idea of the extent of our cosmus. 



For the demonstration, suppose a very large number of stars 

 — so large as to be practically infinite — distributed equably over 

 the celestial sphere ; that their motions are perfectly at ran- 

 dom, and represented by straight lines according to the usual 

 convention ; and that the velocity in the line of sight and the 

 proper angular motion of each star are correctly known. Let 

 r be the radius vector drawn from the earth to any star, $ and 

 <p the angles of reference of r to a fixed line and a fixed plane 

 passing through that line, respectively; v the line representing 

 the motion of any star and #' and <p' angles of reference similar 

 to # and cp, and of which r is the initial line. The mean value 

 of the projection of v upon ?\ which we will call M(a), is, 

 then, abstracting signs, 



M(a) 



' v f 2 P '"" cos B' sinS'^S' dep' 



2r 



2nv V 



Jo «/o 



47T 



sin 3' dS' d<p 



For it will be seen that, since all directions for r are equally 

 probable, and also all directions for v are equally probable, 



* See Proc. Roy. I. Acad., 2nd ser.. vol iv, jSTo. 6, and Month. Not. Roy. Ast. 

 Soc , vol. 1, p 302, " On the Parallax of Double Stars." 



f Suggested to me by Prof. H. A. Newton. I believe the problem has been 

 previously discussed, but with inaccurate results. The idea must have long been 

 more or less vaguely in the minds of astronomers. 



