362 CARNEGIE INSTITUTION OF AVASHINGTON. 



MATHEMATICAL PHYSICS. 



Moulton, F. R., University of Chicago, Chicago, Illinois. Investigations in 

 cosmogony and celestial mechanics. (For previous reports see Year Books 

 Nos. 4, 5, 8-14.) 



The investigations of the year which are as yet unpublished are: 



(1) Computations of periodic orbits. — Contrary to the report of a year 

 ago, there turned out to be one outstanding critical case in periodic 

 orbits which was needed to complete the w^ork which has been under 

 way a number of years. It proved to be one of great difficulty and 

 cost eight months' work. It is now believed to be complete. 



(2) Functions of infinitely many variables. — The work on infinite 

 systems of equations, reported on last year, has been greatly extended 

 in a number of dhections. An application of particular interest is 

 the dynamics of an infinite universe in which there are galaxies, each 

 composed of a finite number of stars; super-galaxies of the first order, 

 each composed of a finite number of galaxies separated by distances 

 which are great compared to their dhnensions; super-galaxies of the 

 second order, each composed of a finite number of galaxies of the first 

 order; and third-order and still higher-order super-galaxies, without 

 limit, similarly composed of finite numbers of super-galaxies of the next 

 orders lovv'er. Such an organization is strictly in hannony v/ith obser- 

 vational experience ; it is analogous to the organization from electrons 

 through atoms and molecules up to our galaxy ; it contains an infinite 

 amount of matter and an infinite amount of energy; it may have existed 

 roughly in its present condition for an infinite time in the past, and 

 may continue to exist without becoming cold and lifeless for an infinite 

 time in the future. ■Moreover, it has been shown that the dynamical 

 properties of such an infinite universe are the same as those of the visible 

 universe. This result is by no means trivial, for it has not been true in 

 some very important suggestions for infinite systems hitherto made. 



(3) Evolution of stars. — The consideration of the problem of the 

 evolution of the stars has led to establishing a number of laws of the 

 general type of Lane's law. Among these results are: 



(a) The absolute temperatures of stars of equal volumes of the 

 same monotomic gases are proportional to their masses. 



(6) The absolute temperatures of stars of the same density and 

 monotomic gases are proportional to the squares of their radii. 



(c) The absolute temperatures of monotomic gaseous stars are pro- 

 portional to the cube roots of the products of the squares of their masses 

 and their respective densities. All these laws hold irrespective of the 

 source of the heat of stars. 



(d) Under the assumptions that the heat of stars is entirely produced 

 by their contraction, and that their rates of radiation at different 

 temperatures satisfy Stefan's law, it is found that the rate of diminu- 

 tion of the radius of a star is proportional to the square of its mass. 



(e) Under the same hypotheses, the rate of increase of density of a 

 star is proportional to the cube root of the fifth power of its mass. 



