SEPTEMBER 28, 1899] 
NATURE 
519 
LETTERS TO THE EDITOR. 
The Editor does not hold himself responsible for opinions ex- 
pressed by his correspondents. Netther can he undertake 
to return, or to correspond with the writers of, rejected 
manuscripts intended for thts or any other part of NATURE. 
No notice ts taken of anonymous communications. | 
The Life of a Star. 
THE letter of Prof. Perry on ‘‘ The Life of a Star,” published 
in your issue of July 13, is of interest to astronomers; and as the 
author of it evidently aims to be fair, I think it worth while to 
set right a misconception into which he has fallen. His reference 
to my paper in the Astronomical Journal (No. 455) shows that 
he has misconstrued the meaning of the symbol K in the formula 
a: =e . That paper was unfortunately very much abbreviated, 
and as I was not concerned with the analytical investigation of 
K, this constant was not sufficiently explained. Yet in my first 
note on this law in the 4. 7. (453), which he probably overlooked, 
it was anncunced that ‘‘ K isa constant different for each body.” 
Thus the constant K is not, as Prof. Perry supposes, the same 
for the whole universe, but varies from star to star, being a 
function of the mass, specific heat, emissive power, &c., into 
which we need not go at present. 
At the time of writing the paper in the 4. 7. (455) I had not 
seen the early paper by Ritter in Wredemann’s Annalen, 1878, 
S. 543, where he has reached by a different process a similar 
formula 
: Tyo za Tr, 
and anticipated a number of the conclusions to which I came 
independently. Ritter even applied this law to the temperature 
of the solar nebula when its periphery extended to the orbit of 
Neptune, and sagaciously observes that his conclusions do not 
agree with current views, but are yet uncontradicted by known 
facts. 
As I have prepared for the St. Louis Academy of Sciences a 
paper in which this whole matter is discussed with some detail, 
I will here merely summarise a few of the chief results. Suffice 
it to say that the formula T = = is shown to express a law of 
the utmost generality, for masses composed of one kind of gas ; 
and that even when the body is of heterogeneous constitution, 
made up of interpenetrating globes of different gases, the law 
suffers no essential modification except at very long intervals, 
when it would take the form 
T = — Bt) 
where 8 is a certain small secular coefficient, and ¢ the time. 
For a great epoch the term depending on 8 might be wholly 
neglected. 
The only hypothesis underlying the investigation is that of 
convective equilibrium, the validity of which is generally 
recognised by physical investigators. In order that the reader 
may see how far from metaphysical my argument really is, I add 
an elementary derivation of the law of temperature. 
Suppose a gaseous globe of radius Ry to be held in convective 
equilibrium by the attraction, pressure and temperature of its 
particles (the density and temperature decreasing from the centre 
co the surface), and let the temperature beneath the surface layer 
be T,. Let P, be the gravitational attraction exerted upon the 
thin layer of matter covering a unit surface of the globe, which 
may be regarded as the base of an elemental cone extending to 
the centre. Then suppose the globe to shrink by loss of heat 
toa radius R. If the original element of mass still covered a 
unit area, the pressure exerted upon its lower surface would 
thereby become P = Py ) But since the area of the initial 
\ix 
sphere surface has shrunk to S =8,(—) the area of the ele- 
a) 
mental conical base has diminished in the same ratio. As the 
force of gravity is increased, while the area upon which the 
element presses is decreased correspondingly, it follows that in 
the condensed condition of the globe, the gravitational pressure 
/ 4 
exerted upon a unit area is P = Py (R): The forces counter- 
1p Prof. Nipher, in preparing the excellent papers which he has contri- 
buted to the St. Louis Acadeiny of Sciences, first drew attention to this 
reference. 
NO. 1561, VOL. 60] 
acting this increased pressure are obviously the resistance due to 
the increase in the mean density, and a possible change in 
temperature which might affect the elasticity of the gas. But the 
density of the original mass was a), and hence ¢ = on (=). 
By hypothesis the equilibrium of the globe is maintained by the 
elastic force of the gas under the heat developed by the gravit- 
ational shrinkage of the mass. If therefore the globe was in 
equilibrium when the gas just beneath the surface layer had a 
mean temperature Tp, to remain in equilibrium in the condensed 
condition, T, must be multiplied by a As T, Ry = constant, 
we may write the law of temperature. 
SI 
This law of course applies to each layer of the globe, and 
thus to its mean temperature, and is’ obviously general for all 
gaseous celestial bodies condensing under ‘the law of gravitation. 
Some persons who do not fully understahd*the problem under 
consideration have ‘asserted thatthe functions. which express 
the distribution-of density and temperature with respect to the 
radius, are altered by shrinkage, ’so ‘that the! law then breaks 
down, or «rather is not proved to hold true. It is perhaps 
worth while to show the error-of this'view, ‘ 
Lord Kelvin: has shown (Ph2/. Mag., 1887, p. 287) that the 
temperature distribution throughout the globe must satisfy the 
differential equation a Tails 
ae : a0, 6° 
where @ is the temperature, x a function of the radius, and « 
a constant. | If @=(v) be a particular solution of this equation, 
the second differential coefficients 
ox) = — oe)er 4, 
and 
P'(mx)= — {p(mx)\ea tad 
and the general solution is shown to be of the form 
6=Co{xC- i(k), 
where C and « are constants. 
Under convective equilibrium the mass will contract in such 
a way that the particles in any concentric sphere surface do not 
penetrate those surfaces adjacent, that is, the new ordinate & of 
any particle is defined by the equation f=ax, where a is a 
numerical coefficient smaller than unity ; and hence 
8=CpleC-(«)} . 
will be a solution of exactly the same form as the first... A 
curve defined by the equation 
V=¥(7) 
will give the absolute temperature from the centre to the 
surface. In like manner another curve 
n=(7")\* 
will give the distribution of density with respect to the radius. 
Shrinkage by which the variables become p=ar will not 
change the character of these two functions; and hence the 
distribution of density and temperature is rigorofisly the same 
after contraction as before. This result continues to hold so 
long as the body is wholly gaseous and obeys the laws of con- 
vective equilibrium, 
Prof. Perry has examined at some length the question of radi- 
ation, and he deserves our thanks for the interesting suggestions 
he has advanced. Yet I have considered our knowledge based 
on terrestrial experiment too limited to furnish any conclusion 
which can be confidently applied to the conditions existing in 
the heavens, except that the masses are always in convective 
equilibrium, and that all shrinkage is determined by this condition. 
Accordingly the foregoing conclusions would seem to be valid 
generally. It seems fair to conclude that there are few branches 
of physical science which offer such an unexplored field as the 
one which relates to the life-history of stars. And though it 
may be assumed that forces are at work in space, of which we 
have little or no experimental analogy up to this time, yet it is 
always Safe to apply known laws to the phenomena of the 
heavens with a view of explaining observations, and of suggesting 
unknown causes which may become the subject of future 
research. PROS Jomskoies 
Washington, D.C., August 5. 
