April 14, 192 1] 



NATURE 



207 



star, the fainter may its absolute luminosity 

 be. One of Russell's diagrams, in which 

 the absolute magnitudes are referred to a 

 distance of 10 parsecs, is reproduced in Fig. 6. 

 In this diagram the small dots represent individual 

 stars, the large circles mean values for bright 

 stars of small proper-motion and parallax. 

 It vi^ill be seen that the general distribution of the 

 dots is along two lines inclined at an acute angle 

 and intersecting at type B ; that this distribution 

 is not the result of the selection of stars for paral- 

 lax determination on the ground of brightness or 

 size of proper-motion was conclusively shown by 

 Russell. It will also be seen that for the red stars 

 there is a complete separation between the two 

 classes, so that a very red star is intrinsically 

 either very bright or very faint. These facts have 

 given rise to the "giant" and "dwarf" hypo- 

 thesis, and have led to a recasting within the last 

 few years of the ideas as to stellar evolution which 

 were formerly generally accepted. 



The following results emerge from Russell's 

 investigation : (i) Stars of all types occur brighter 

 than- zero absolute magnitudes, ^ and mostly be- 

 tween o and — 2M — say, about 150 times the 

 luminosity of the sun. These are called "giant" 

 stars. (2) There are no B-type stars, and very 

 few A-type stars, fainter than zero absolute mag- 

 nitude, or, in other words, all the white stars 

 are intrinsically very bright. (3) All the faint 

 stars, less than, say, 1/50 the luminosity of the 

 sun, are red and of types K and M. These are 

 called "dwarfs," and coraprise all the near stars 

 of large proper-motion. (4) In the intermediate 

 classes, F and G, there is no separation between 

 the giants and the dwarfs. Our sun (50M) is a 



- The unit of absolute magnitude u«ed here is that which corresponds lo a 

 parallax of one-tenth of a second of arc. 



typical G-type star. In view of these remarks, 

 it is obvious that no precise meaning attaches to 

 a statement such as "The average absolute magni- 

 tude of all stars is -J-2-7M." 



Shapley's work on the magnitudes of stars in 

 clusters, combined with his determination of the 

 distances of clusters, has shown that the giant 

 stars in clusters, which are the only ones suffi- 

 ciently bright to appear on the photographs, are 

 of about the same magnitude as the giant stars 

 in our more immediate neighbourhood. Two 

 further points of interest emerge from the investi- 

 gation : one is that in all the clusters examined 

 in detail the intrinsically brightest giant stars are 

 red stars ; this may also be true for the stars near 

 the sun, although the determinations of their abso- 

 lute magnitude are probably not sufficiently accurate 

 to show it ; the other point is the apparent im- 

 portance of an absolute magnitude of about 

 — 0-2M. Shapley finds that all Cepheid variables 

 and cluster variables exceed this brightness ; 

 moreover, in the luminosity curve which connects 

 the number of stars of any given absolute magni- 

 tude with the magnitude, there is a maximum in 

 the curve corresponding to the same magnitude. 

 In Shapley's opinion, this magnitude — correspond- 

 ing to a luminosity of about 100 times that of the 

 sun — indicates a critical stage in stellar evolution, 

 and, in all probability, is of significance in the 

 theory of a gaseous star. It seems, in fact, prob- 

 able that by the new methods recently discovered 

 for estimating great distances, combined with the 

 advantages afforded by the large reflecting tele- 

 scopes at Mount Wilson, we may learn more about 

 absolute magnitudes from a study of clusters at 

 distances corresponding to parallaxes of the order 

 of o"oooo5 than from the study of the stars 'which 

 immediately surround us. 



Dynamics of 



'T'HE physical principles underlying the flight 

 ^ of a golf ball were clearly laid down by the 

 late Prof. Tait between the years 1890 and 1896.^ 

 In view of the present agitation over the 

 standardising of the golf ball, it may be of advan- 

 tage to reconsider some of the problems attacked 

 by Tait and largely solved by him. The investiga- 

 tion led him into a series of researches on impact 

 so as to obtain data for measuring the resilience 

 of the material of which golf balls were then 

 made. Also, by means of a specially constructed 

 ballistic pendulum, measurements were made of 

 the speed of a golf ball impinging on the pendulum 

 placed at a distance of about 6 ft. from the tee. 

 By attaching a tape to the ball, Tait was able to 

 obtain direct measurements of the amount of 

 underspin communicated to the ball at the instant 

 of striking it. Outside observations were also 

 made of the heights of the trajectories of well- 

 driven balls, and of the ranges and times of flight. 

 All these data were skilfully introduced into the 

 mathematical discussion of the form of the tra- 



1 " On the Path of a Rotating <5pherical Projectile." Trans. R.S.E., '893 

 and i8q6; " Some Points in the Physirs of Oolf," Naturk, vols, xlii., xliv., 

 and xlvlii. ; " Long Drivinc;," Badminton Magazine, 1896. 



NO. 2685, VOL. 107] 



Golf Balls. 



jectory, a problem so difficult as to be capable of 

 solution only by approximate methods. This was 

 done before the days of the rubber-cored ball, and 

 the steady improvement in the manufacture of the 

 golf ball has enabled even very ordinary players 

 to exult in lengths of drive which in Tait's days 

 were beyond the powers of the mightiest exponents. 

 What Tait established beyond all controversy 

 was that the range of the trajectory of a properly 

 driven ball depended as much upon the underspin 

 as upon the speed of projection. The combined 

 effect of the linear speed and the rotation about a 

 horizontal axis brought into play a force perpen- 

 dicular to the direction of motion of the ball. Tait 

 gave sound reasons for regarding this force as 

 being proportional to the product of the velocity 

 and the spin. Thus, although the possibility of 

 a long trajectory depends primarily upon the velo- 

 city of projection, the range actually attained in 

 any particular case will be governed by the 

 amount of underspin communicated to the ball. If 

 this is too great, the ball will rise too high, and 

 the range will be correspondingly diminished. If 

 the underspin is too small, gravity will pre- 



