ME. J. H. JEANS ON THE DISTEIBUTION OF MOLECULAE ENEEGY. 
409 
whereas Boltzmann’s theorem would lead to 
.y 1 ^ . 
* 9)1 ^ 71 -f* 3 
The equation (xxiv.) will only hold in the limit when the temperature 0 = 0. At other 
temperatures y will have a slightly different value, since the F energies cannot be 
entirely neglected. Our results as before only hold up to the temperature at which 
the gas begins to emit an appreciable amount of radiant energy, and this temperature 
may be supposed to be somewhat above the point of incandescence. 
Up to this point, Fj, Fo, . . . will always be in the same proportion to one 
another, so that the brightness of the various lines in the spectrum will be in a 
constant ratio, each being proportional to 6^ 
§ 14. We have been working on the assumption that there is a complete absence 
of frictional forces acting on K, and on E^, Eo, .... These assumptions, however, 
are not necessary. In the steady state we have from equation (xxlil.), 
F, = pIU q 
where is a quantity which depends only on the construction of the molecule.^ Tlie 
temperatures which have been considered have been those for which —y^K, —y^K, 
&c., are all very small. But if for any single degree of freedom, say that for which 
the energy co-ordinate is F^, either is exceptionally small or exceptionally great, 
the range of temperature will he greatly restricted on this account. At temperatures 
at which —.y/K is large while the remaining similar quantities are small, it is clear 
^1 
that Fi must be treated as an E co-ordinate. 
At zero temperature all the energy co-ordinates to which friction corresponds 
must be regarded as F co-ordinates. As the tem])erature increases we must supjDOse 
these co-ordinates one by one to change from being F co-ordinates, and after 
occupying a position intermediate between that of an F and that of an E co-ordinate 
to finally become E co-ordinates. If there is a co-ordinate for which e is extremely 
small, or yS very great, that is to say, a co-ordinate corresponding to a degree of 
freedom which is only very slightly retarded by friction, or to one from which energy 
passes freely, then such a co-ordinate will become an E co-ordinate at such a low 
temperature that it may be regarded as always being an E co-ordinate. 
* It may be noticed that the value of fSi supplies a measure of the facility with which energy is 
exchanged between the Fi mode and the other modes. If 0i — 0, it is impossible for such an exchange 
to take place, and the Fi mode does not satisfy the condition of continuity of path. Thus if friction 
dissipates the energy of the Fi mode, the value of Fi will finally be zero. If, how'ever, we have fSi = 0, 
together with €i = 0, the value of Fi is indeterminate. The rotation of the loaded sphere about the axis 
of symmetry supplied a good illustration of a mode of energy for which /? = 0. 
VOL, CXCVI.—A. 3 G 
