A. — MATHEMATICS AND PHYSICS. 48 



The molecular weight can scarcely go beyond this range,* and 

 for the conclusions I am about to draw it does not much matter which 

 limit we take. Probably 90 per cent, of the giant stars have masses be- 

 tween ^ and 5 times the Sun's, and we see that this is just the range in 

 which radiation-pressure rises from unimportance to importance. It 

 seems clear that a globe of gas of larger mass, in which radiation-pres- 

 sure and gravitation are nearly balancing, would be likely to be unstable. 

 The condition may not be strictly unstable in itself, but a small rotation 

 or perturbation would make it so. It may therefore be conjectured 

 that, if nebulous material began to concentrate into a mass mucli greater 

 than 5 times the Sun's, it would probably break up, and continue to 

 redivide until more stable masses resulted. Above the upper limit the 

 chances of survival are small ; when the lower limit is approached the 

 danger has practically disappeared, and there is little likelihood of any 

 further breaking-up. Thus the final masses are left distributed almost 

 entirely between the limits given. To put the matter slightly differently, 

 we are able to predict from general principles that the material of the 

 stellar universe Avill aggregate primarily into masses chiefly lying 

 between 10^' and 10="* grams; and this is just the magnitude of the 

 masses of the stars according to astronomical observation.^ 



This study of the radiation and internal conditions of a star brings 

 foi-ward very pressingly a problem often debated in this Section : 

 What is the source of the heat which the Sun and stars are continually 

 squandering? The answer given is almost unanimous — that it is 

 obtained from the gravitational energy converted as the star steadily 

 contracts. But almost as unanimously this answer is ignored in its 

 practical consequences. Lord Kelvin showed that this hypothesis, due 

 to Helmholtz, necessarily dates the birth of the Sun about 20,000,000 

 years ago; and he made strenuous efforts to induce geologists and 

 biologists to accommodate their demands to this time-scale. I do not 

 think they proved altogether tractable. But it is among his own col- 

 leagues, physicists and astronomers, that the most outrageous violations 

 of this limit have prevailed. I need only refer to Sir George Darwin's 

 theory of the earth-moon system, to the present Lord Eayleigh's deter- 

 mination of the age of terrestrial rocks from occluded helium, and to all 

 modern discussions of the statistical equilibrium of the stellar system. 

 No one seems to have any hesitation, if it suits him, in caiTying back 

 the history ci the earth long before the supposed date of formation 

 of the solar system ; and in some cases at least this appears to be justified 



^ As an illustration of these limits, iron has 26 outer electrons ; if 10 break 

 away the average molecular weight is 5 ; if 18 break awav the molecular weight 

 is 3. Eggert {Phys. Zeits. 1919, p. 570) has suggested by thermodynamical 

 reasoning that in most cases the two outer rings (16 electrons) would break away 

 in the stars. The comparison of theory and observation for the dwarf stars 

 also points to_ a_ molecular weight a little greater than 3. 



6 By admitting plausible assumptions closer limits could be drawn. Taking 

 the molecular weight as 3.5. and assuming that the most critical, condition is 

 T^^ 3 of gravitation is counterbalanced (by analogy with the case of rotating 

 spheroids, in which centrifugal force opposes gravit^ation and creates instability), 

 we find that the critical mass is just twice that of the Sun, and stellar masses 

 may be expected to cluster closely round this value. 



