Prof. Challis on a Mathematical Theory of Heat, 203 



waves. It may be supposed that an unlimited number of such 

 waves are produced by series of plane-waves traversing the me- 

 dium in all directions, and that, when an equilibrium of heat is 

 established, these reflected waves are propagated equally in all 

 directions from an atom, or the resulting velocity and condensa- 

 tion are functions of the distance from the atom. To determine 

 the dynamical action of waves so propagated from a centre, is a 

 hydrodynamical problem, which I now proceed to consider. 



The article in the Number of the Philosophical Magazine for 

 February 1853, contains, under Prop. XIII., an investigation of 

 the differential equations to the first approximation applicable 

 to motion propagated equally in all directions from a centre. 

 The exact equations obtained on the same principles are, 



"-^-^T>-f =0. a) 



the first of which differs from the equation usually obtained for 

 this problem by having Ka in the place of a, the value of k being 

 1-18545. (See the Article " On the Central Motion of an Elastic 

 Fluid " in the Philosophical Magazine for last Januai-y.) 



The known integrals of these equations applicable to propa- 

 gation from a centre are, to the first approximation, 



^_f'{r—Kat + c) _f{r—Kat-\-c) 



r r^ ' 



fir—Kat + c) 



Ka<x= -^-^ —\ 



r 



Respecting the function /it is to be observed that its form is not 

 entirely arbitrary, because the secondary waves under considera- 

 tion have their origin in waves whose velocity and condensation 

 are expressed by periodic functions. It is evident, in fact, that 

 V must have as many plus as minus values, since there can be 

 no permanent motion of translation of the fluid to or from the 

 centre. Hence we may assume as follows : — 



fir—Kat + c) 1 vf • Stt , ^ ,"1 

 -2 '=-^.2,^msm — {r-mt + c)j, 



fir—Kat + c) 1 „r27rmr ^ir , ^ /I 



'-^—, ' = -,.l.l-^cos-{r-Kat + c)j. 



As the ratio of the maximum values of corresponding terms 



27rr 

 on the right-hand sides of these equations is -— •, it follows that 



A. 



the first term in the value of V is much less than the other for 



P3 



