738 



SCIENCE. 



LN. g. Vol. IX. No. 230. 



account for the observed radiation. His 

 principal conclusions may be summarized 

 as follows : 



1. That a shrinkage in the radius of 35 

 meters per year will generate sufficient 

 heat to sustain the annual output of radi- 

 ant energy. 



2. That on this basis the radius of the 

 sun would not shrink more than xo « o o P^''* 

 in 2,000 years, and this shrinkage could 

 not be detected by any measurements 

 which have been made within historical 

 time. For the mean value of the sun's 

 radius is about 961 seconds of arc, and is 

 still uncertain by about one-half second ; 

 TTTTT'o *^f t^i^ radius is thus but one-fifth of 

 the outstanding uncertainty in the sun's 

 semi- diameter, in spite of all the labor 

 which has been spent in finding its exact 

 value by refined measurement. As the 

 diameters noted by the ancients are much 

 less accurate than those which can be in- 

 ferred from the recorded duration of an- 

 cient eclipses in conjunction with the the- 

 ory of the moon, we can only say that 

 there is no evidence that the radius has 

 diminished since the earliest ages. Even 

 with the finest measurements now avail- 

 able, it would take ten thousand years for 

 the shrinkage to become clearly sensible. 

 There is, therefore, little hope that the 

 shrinkage of the sun can ever be observed, 

 yet from known mechanical laws we may 

 confidently compute its amount, with even 

 greater accuracy than we could hope to ob- 

 tain from direct measurement. 



3. That all the energy generated in the 

 mass of the sun by the falling together of 

 its particles would suffice to raise an aque- 

 ous globe of the same mass to a temperature 

 of over 27 million degrees Centigrade. 

 Pouillet estimated from experiments on 

 solar radiation that the heat annually lost 

 by the sun would raise the temperature of 

 such a globe 1.25° C. On this basis the 

 observed radiation of the sua could not 



have gone on uniformly in the past for 

 more than about 22 millions of years. As 

 more modern estimates increase the ob- 

 served radiation appreciably, when full ac- 

 count is taken of atmospheric absorption, 

 we shall adopt 18 million years as the past 

 duration of the sun, on the theorj' of uniform 

 radiation and homogeneous density as- 

 sumed by Helmholtz. 



4. Helmholtz further shows that all the 

 energy given up by the condensation of the 

 several planets amounts to but little more 

 than y^u^J^^ part of that developed by the 

 condensation of the sun, and that the energy 

 of the motion of the planets amounts to 

 only yJ-y of that resulting from the potential 

 of the homogeneous sun upon itself Thus 

 nearly all the energj' of the solar system 

 has resulted from the condensation of the 

 solar mass. 



I propose this evening to present the re- 

 sults of a determination of the potential of 

 the sun upon itself, when the mass is 

 heterogeneous, or made up of successive 

 layers of a uniform density, and the density 

 follows the laws found by our countryman. 

 Lane, just 30 years ago, for a gaseous body 

 in convective equilibrium. The density of 

 each layer can be found from Lane's theory. 

 Beginning at the center and proceeding out- 

 ward, we can thence determine the average 

 density of the included spheres when suc- 

 cessive layers of known density are added. 

 (The speaker here explained the theory of 

 the integration which he had developed, and 

 said that the mathematical discussion of 

 the process would appear in the Astronom- 

 ische Nachrichtcn.) From an astronomical 

 point of view the problem to be solved is best 

 treated by some process of mechanical 

 quadrature; and accordingly I have divided 

 the radius into 40 parts, and by successive 

 steps obtained an integral for the potential of 

 the heterogeneous sphere upon itself, which 

 is almost rigorouslj' exact. It turns out that 

 the condensation of the heterogeneous sua 



