A. — MATHEMATICS AND PHYSICS. 41 



deviations of dense gas from the ideal laws and using our own Sun to 

 supply a determination of the unknown constant involved. We can 

 calculate the maximum temperature reached by different masses; for 

 example, a star must have at least \ the mass of the Sun in order to 

 reach the lowest spectral type, M; and in order to reach the hottest 

 type, B, it must be at least 2J times as massive as the Sun. Happily 

 for the theory no star has yet been found with a mass less than 

 |- of the Sun's; and it is a well-known fact, discovered from the study 

 of spectroscopic binaries, that the masses of the B stars are large com- 

 pared with those of other types. Again, it is possible to calculate the 

 difference of brightness of the giant and dwarf stars of type M, i.e. at 

 the beginning and end of their career ; the result agrees closely with the 

 observed difference. In the case of a class of vai-iable stars in which 

 the light changes seem to depend on a mechanical pulsation of the 

 star, the knowledge we have obtained of the internal conditions enables 

 us to predict the period of pulsation within narrow limits. For example, 

 for 8 Cephei, the best-known star of this kind, the theoretical period 

 is between 4 and 10 days, and the actual period is 5 J days. Correspond- 

 ing agi-eement is found in all the other cases tested. 



Our observational knowledge of the things here discussed is chiefly 

 of a rather vague kind, and we can scarcely claim more than a general 

 agreement of theory and observation. What we have been able to do 

 in the way of tests is to offer the theory a considerable number of 

 opportunities to ' make a fool of itself, ' and so far it has not fallen 

 into our traps. When the theory tells us that a star having the mass 

 of the Sun will at one stage in its career reach a maximum effective 

 temperature of 9,000° (the Sun's effective temperature being 6,000°) 

 we cannot do much in the way of checking it ; but an en'oneous theory 

 might well have said that the maximum temperature was 20,000° (hotter 

 than any known star), in which case we should have detected its error. 

 If we cannot feel confident that the answers of the theory are true, it 

 must be admitted that it has shown some discretion in lying without 

 being found out. 



It would not be surprising if individual stars occasionally depart 

 considerably from the calculated results, because at present no serious 

 attempt has been made to take into account rotation, which may modify 

 the conditions when sufficiently rapid. That appears to be the next 

 step needed for a more exact study of the question. 



Probably the greatest need of stellar astronomy at the present day, 

 in order to make sure that our theoretical deductions are starting on the 

 right lines, is some means of measuring the apparent angular diameters 

 of stars. At present we can calculate them approximately from theory, 

 but there is no observational check. We believe we know with fair 

 accuracy the apparent surface brightness corresponding to each spectral 

 type ; then all that is necessary is to divide the total apparent brightness 

 by this surface brightness, and the result is the angular area subtended 

 by the star. The unknown distance is not involved, because surface 

 brightness is independent of distance. Thus the estimation of the 

 angular diameter of any star seems to be a very simple matter. For 

 instance, the star with the greatest apparent diameter is almost certainly 



