Prof. Challis on a Mathematical Theorij of Heat. 205 



Let us now investigate the effect produced by tbe impinging 



of the part of the velocity expressed above by — 2 on any atonij 



supposing thisvelocityto be unaccompanied by condensation. The 

 function /(/) must be periodic, and, taking account of its origin, 

 may be put under the form S [m sin [ht + cj] . It may be sup- 

 posed that the atom on which this velocity impinges -is at rest, 

 because when an equilibrium of caloric action is established, the 

 motions of the atom will be oscillatory, and may be left out of 

 account in calculating the mean effect of the incident velocity. 

 Conceive the centre (A) of the atom from which the velocity is 

 propagated, and the centre (A') of that on which it is incident, 

 to be joined by a straight line, and let 6 be the angle which anv 

 radius of the latter makes with this straight line. At this point 

 of the reasoning I must take for granted what I have elsewhere 

 much insisted upon, viz. that the reflected velocity at any point 



fit) 

 of the surface corresponding to the angle 6 is -~ cos d. It may 



suffice to state here, that this value is required by the general 



law of axes of rectilinear transmission, which I have shown to be 



an inference from the general hydrodynamical equation which 



expresses the law of continuity of the motion. (See the article 



in the Philosophical Magazine for December 1852, Prop. X.) 



The resolved part of the velocity along the surface is consequently 



fit) 



-^ sin 6. We can therefore calculate the amount of pressure 



on the hemispherical surface on which the velocity is incident, 

 by the hydrodynamical equation applicable to impressed or con- 

 strained curvilinear motion, viz. 



K^a^dp dV W_Q 

 pds dt ds ' 



in which ds is the differential of the line of motion. But the 

 lines of motion in this instance are the intersections of planes 

 passing through the line AA' with the surface of the atom. 

 Hence, if « be the radius of the atom, ds = udd. Consequently 

 integrating, and considering r to be constant, which is allowable 

 on account of the small size of the atom and because the change 

 of p due solely to change of r has already been considered, we 

 obtain 



«2«2 Nap. log p - ^^ cos d + 2lV_ sin^ e=^}r{t). 



Determining the arbitrary quantity so that where ^ = — p is 

 equal to a constant p„ because at these positions there is uo re- 



