COS' 



206 Prof. Challia on a Mathematical Theory of Heat. 

 flexion, we have 



a:V Nap. log ^ - ^ cos ^ - ^2!^ cos^ 6>=0. 



Let p=/Dj(l + a). Then to small quantities of the second order 



r \ 2K^av-^ 2n 



where the term involving a^ may, if we please, be retained ; but 

 as it is probablj' wholly insignificant on account of the small 

 magnitude of a, for the sake of brevity it will be omitted. The 

 whole effective pressure on the atom in the direction AA' is 



Stt/ji fflV cos ^ X a sin x udd, 



taken from ^=0 to 6=-^, because by hypothesis the incident 

 velocity is unaccompanied by condensation, so that for values of 

 6 greater than ^ the density is equal to p,. Hence it will be 

 found that the whole pressure is equal to 



The first term of this result is periodic, on account of the factor 

 /'(/), and therefore indicates no motion of translation of the 

 atom. The other term is partly periodic and partly non-pe- 

 riodic, and shows that the atom is subject to a pressure, which 

 causes a motion of translation from A, and varies inversely as 

 the fourth power of the distance. It is evident that this repulsion 

 from A, is a force of a higher order than the attraction towards 

 A, found in the former part of the investigation, because the 

 latter was multiplied by the ratio of the radius of the atom to 

 r. Hence the resultant of the forces acting between two atoms 

 is a repulsion. 



It may be also remarked that, since 



f{t) — m sin {)>t + c) + ??/ sin {hH + c') + &c., 

 C/(0/ = ^ + ~n +&C. + periodic terms. 



Hence the effect of several sets of vibrations acting simulta- 

 neously, is the sum of the effects they would produce if they 

 acted separately. 



Assuming the dynamic repulsion between two atoms obtained 

 by the preceding reasoning to be identical with the repulsion of 

 heat, since it was found that the motion of translation was due 



