954 On Non-Newton'an Mechanical Systems, fyc. 



energy, then the number which have energy e is Me~ e/ * T , 

 the number having energy 2e is Me~ 26 RT , and so on. Thus 



M - M(l + e- € ! liT +e-*f* 1! +.. . .) = M/(l-e-*/ RT ), 



and if E is the total energy of the X vibrations, 



E = Me(,'-' :1IT + iV-- eliT + .. .) 



: -r 



(18) 



which gives Planck's law on taking e = hv. 



It will, I think, be Found that this calculation of Planck's 

 Formula is based on exactly the same physical ideas as those 

 of the original theory of Planck. One essential and necessary 

 feature or the theory is that it supposes the unit of energy 

 for vibrations of moderate wave-length to be so great that 

 the chance of a vibration having even one unit of energy is 

 very slight. We notice that only a Traction (N — M)/N, or 

 ( "' Kl , or the total number of vibrations possess any energy 

 at all. At wave-length A maI ., e/liT = 4-905, so that only one 

 wave-length in IK) possesses any energy. At wave-length 

 one-half of this the proportion is about one in 20,000. 



1-4. An interesting question is whether, if this theory is to 

 be accepted at all. it oughl not also to he expected to account 

 for the failure of certain other decrees of freedom to receive 

 the share of energy allotted to them by the theorem of equi- 

 partition. Many types of motion, such as the internal 

 vibrati ns of the atom, and the rotations of atoms or mole- 

 cules, must have direct interchange of energy with the aether 

 vibrations, so that if the latter are in temperature equilibrium, 

 the former might be expected to be so. A rough estimate 

 of the energy possessed by such degrees of freedom is 

 furnished by the values of the specific heats. For a degree 

 of freedom which has one thousandth of its equipartition 

 energy, e/RT must be about 9*1, and only one degree of 

 freedom in 9100 will have any energy at all. This result, 

 when applied, for instance, to the rotation of the atoms of 

 mercury vapour, is somewhat startling. 



Cambridge, Aug. 17. 



