204 Prof. Challis on a Mathematical Theory of Heat. 



all values of r that are very small compared to X. If therefore 

 such values be aloue considered^ the first term, and consequently 

 the condensation a, may be neglected ; and since the velocity of 

 propagation in the setherial medium is extremely great, and may 

 be regarded in this problem as infinite, we shall have very ap- 

 proximately the same value of V as if the fluid were incompres- 

 sible, viz. •V=--^. 



In order to include terms involving the second power of the 

 velocity, the equation (1) is to be integrated, retaining the last 

 term. We thus obtain 



o CdY V^ 



«VNap.log^+j^*.+ ^=0; 



and if p = /3,(l + cr), to the second power of a 



This equation gives the condensation a at any distance r from 

 the centre of the atom from which the waves are supposed to be 

 propagated, whence the pressure of the fluid on another atom situ- 

 ated at a given distance from the first might be inferred. But 

 so far as the condensation is periodic, it will only give rise to 

 oscillatory motions of the second atom, and may thei'efore in the 

 present inquiry be left out of consideration. Hence since V, as 

 we have seen, is necessarily periodic, and the above term aff'ected 

 with the sign of integration is consequently periodic, this term 

 may be omitted. Also it will sufiice to substitute for V in the 

 last term the value given by the first approximation. Hence 

 since 



V=/cacr 2'^ msin — (»' — ««/ + c) , 



by substitution in the foregoing equation and neglecting periodic 

 terms, we shall finally obtain 



This result shows that the part of the condensation which has 

 the effect of giving a permanent motion to the atom, varies in- 

 versely as the fourth power of the distance, and increases with 

 the distance, since o- is negative ; so that the second atom is by 

 this pressure urged towards the first atom. But as the diflferen- 

 tial pressure urging it varies directly as its radius, and inversely 

 as the Jiftk power of the distance, the accelerative force from this 

 cause must be extremely small, and not comparable with other 

 forces that have to betaken into consideration. 



