A SWAEM OF METEORITES, AND ON THEORIES OF COSMOGONY. 
10 
total internal kinetic energy of agitation of the sphere of gas at minimum temperatui'e 
limited by the radius a is ■§ jjiM/a) M = ^ very nearly. Now, the energy 
lost in the concentration of a homogeneous sphere M from a condition of infinite 
dispersion is exactly f yiM^ja, It might, therefore, be suspected that •39723 is only an 
approximation to -f, which may be the rigorous value. But my numerical calculations 
were carried out with so much care that I find it almost impossible to believe that 
there is an error as large as 3 in the third place of decimals, or, indeed, any error at 
all in the third figure. Moreover, it would be expected that, if this very simple 
relationship is rigorously correct, it would be possible to prove it rigorously, just as 
it is rigorously shown above that ivjp = ; but I am unable to find any analytical 
relationships by which the minimum value of yS® can be deduced. If my arithmetical 
process be correctly carried out, then we ought to find that, when r = ’6204, dyjdx-^ 
should be equal to — x^ d'^yjdx-^ or e^^lx^. Now, I find that, when r = '6264, 
1‘57703 and e^'-jx^ = 1‘5770, so that the two agree to four places of 
decimals. I conclude, therefore, that the true minimum of ^ is •3972.* 
It will be observed that, as a/cq increases to infinity, y8^ terminates by being equal 
to 4. M. Ritter has found that it rises above and oscillates about that value an 
indefinite number of times with diminishing amplitude, gradually settling down to ^ 
as a/a^ becomes infinite. The values in the preceding table are not, however, carried 
far enough to exhibit these oscillations of A consequence of this result is that 
there are a number of modes of equilibrium of a gas at a given temperature, provided 
that the temperature lies within certain narrow limits. This very remarkable 
conclusion is rendered more intelligible by Mr. Hill’s treatment than by M. Ritter’s. 
This point has, however, no bearing on the present investigation. 
In any one of the solutions comprised in Table II. we may complete the table of 
densities by the formula (24), viz., 
W 
p dydclx-^ (q = ay a) ’ 
and I shall later proceed to do this in the one case which has interest for our present 
problem, namely, where the temperature is a minimum, so that a/a^ is •6264, The full 
numerical results may be more conveniently given hereafter, and it will only be now 
necessary to indicate how they are to be computed. 
When, for example, r= •! X a-^, rla= •l/^6264 = •1596; thus, our equidistant values 
of the density and other functions will proceed by multiples of ^1596 a up to ^9578 a, 
and the limit of the isothermal sphere is where r = a. 
When the temperature is a minimum ^ — ^39723, and we have = Yp ; there¬ 
fore, ivjiVQ = wl\ p, and, therefore, if y^^Q be the value of y^, when r = *6264 
* This is confirmed by Mr. Hill. His equation s = z is equivalent to x-y dry-^jdxy + x-^ dy^dx-^ = 0, and 
it appears from his tables that s ■= z — 2'517. Now, s = 3/yd-, and the reciprocal of 2’5]7 is '397. 
D 2 
