PRESIDENTIAL ADDRESS. 363 



may be assumed that the peculiar motions neutralise one another in the mean. 

 For example, the average distance of stars of type A, or stars of the titth 

 magnitude, or any other group desired may be found. In this way Kapteyn 

 has found from the Bradley stars that the mean parallax of stars ot 

 mao-nitude m is given by the formula log 7r„j= -1-108-0'125 m. 



La conjunction with another observational law which expresses the number 

 of stars as a function of the magnitude, this leads to a determination oi 

 the density of stars in space at ditterent disUnces from us, and also of the 

 ' luminosity law,' i.e., the percentage of stars of different absolute brightness. 

 Professors Seeliger and Kapteyn have shown in this way that there is a con- 

 siderable falling off of star-density as we go further from the Solar system 

 It seems to me very necessary that this should be investigated in greater detail 

 for different parts of the sky separately. A general mathematical solution ot 

 general questions which arise in the treatment of astronomical statistics has 

 been given by Professor Schwarzschild. His investigations are of the greatest 

 value in showing the exact dependence of the density, luminosity, and velocity 

 laws on the statistical facts which can be collected from observation, ihe many 

 interesting statistical studies which have been made are liable to be rather 

 bewilaering without the guidance furnished by a general mathematical survey 

 of the whole position. tit 



When the proper motions are considered in relation to the spectral types 

 of the stars, the small average velocities of the hydrogen stars and still smaller 

 ones of the heliimi stars found from line-of-sight observations are confarmed. 

 If stars up to a definite limit of apparent magnitude, say, to 6-0 m., or between 

 certain limits, say, 8-0 m. and 90 m., are considered, then the Solar stars are 

 found to be much nearer than either the red or the blue stars. Thus both 

 red and blue stars must be of greater intrinsic luminosity than the Solar stars. 

 As regards blue stars, this agrees with results given by parallax observations. 

 But the red stars appear to consist of two classes, one of great and one ot 

 feeble luminosity, and it does not seem that a sufficient explanation is given 

 by the fact that a selection of stars brighter than any given apparent magnitude 

 will include the very luminous stars which are at a great distance, but only 

 such stars of feeble luminosity as are very near. 



The significance of these facts was pointed out by Prof. Hertzsprung and 

 Prof. Russell. They have a very important bearing on the question of stellar 

 evolution, a subject for discussion at a later meeting this week. From the 

 geometrical standpoint of my address these facts are of importance in that 

 they help to classify the extraordinarily large range found in the luminosities 

 of stars. Putting the matter somewhat broadly, the A stars, or hydrogen 

 stars, are on the average intrinsically 5 magnitudes brighter than the Sun, 

 whilst the range in their magnitudes is such that half of them are within 

 I magnitude of the mean value. The stars of type M, very red stars, are 

 of two classes. Some of them are as luminous as the A stars, and have a similar 

 range about a mean value 5 magnitudes brighter than the Sun. Others, on 

 the contrary, have a mean intrinsic brightness 5 magnitudes fainter than the 

 Sun and with the same probable deviation of | magnitude. Between the 

 types M and A there are two classes whose distance apart diminishes as the 

 stars become bluer. The facts in support of this contention are very forcibly 

 presented by Prof. Russell in Nature in May 1914. If this hypothesis is true, 

 and it seems to me there is much to be said in its favour, then the apparent 

 magnitude combined with the type of spectrum will give a very fair approxi- 

 mation to the distances of stars which are too far away for their proper 

 motions to be determinable with accuracy. 



In dealing with the proper motions of the brighter stars, the sky has* 

 been considered as a whole. Now that the direction and amount of the solar 

 motion are known, we may hope that, as more proper motions become available, 

 the different parts of the sky will be studied separately. In this way we 

 shall obtain more detailed knowledge of the streaming, and also of the mean 

 distances of stars of different magnitudes in all parts of the sky, leading to a 

 determination of how the density of stars in space changes in different direc- 

 tions. A second line of research which may be expected to give important 

 results is in the relationship of proper motions to spectral type. There is in 



