284 G. H. BRYAN. 



and tlie law of probability of distribution of coordinates niust be such 

 as to satisfy tliis condition. This law of distribution tlius dépends on 

 tlie energy E, affordiug an analogj witli the tliermodjnaniical property 

 that the state of a body dépends on one variable (eiiergy or température) 

 besides the controllable coordinates of the body. 



If we assume a kuown relation to exist betweeu the distributions of 

 coordinates and velocities^ the frequency function instead of being 

 /'(.^■) <P ('') fJx do must be taken to be i''(.r, r) dx do and in the équation 

 of mean accélération of kinetic energy (1)^ [;?-] [^/^ /''/''''•''"] must be 

 replaced by \_o^ drj'jdx^']. The conclusions will ])robably be of some- 

 what the same gênerai character as in the case considered, but will not 

 admit of such simple mathematical discussion. 



For siiiipli! ItarMonic moiioii, the e(|uation of energy-equilibrium may 

 be easily verified^ for we hâve .r= a s'm nt, r = a n cas ni, V^ ^ '"■" ''''*''" 

 dVjdx =^ 7Î^ mx , d^Fjdx"^ = /rm, also / {x) the probability function 

 co Ijv a: («^ 



[l '""'] = 



(the tirae intégrais being taken over a half period or quarter period) a 

 resuit agreeing with that found by calculating the time average of 



1 9 19 9 9 / 



^ '}iiv'- or 4 (^ I'" t'i COU" ni. 



Example 2. Tioo partides in a sfra'ujht Une atlractvnfj or repeUiny 

 eacJi ofher. 



Let two particles m^ and ni.^ at points .<•, and x.^ be moving with 

 velocities r, and v.^ in a straight line under a iield of external force, 

 and a force betweeu them which is a function of their distance a])art 

 x^ .^'i • 



The ])otential energy of the system is ï"^ ^ V^ -\- V^ -\- A^j , 

 where Vi = potential energy of m^ due to field, a function of x^ 

 K.^ = potential energy of m.^ due to field, a function of X2 

 / J.2 ^ mutual potential energy, a function of x.2 — Xi. 



