The Astronomer Royal's Reply to Professor Challis. 145 

 city in the direction of y, --j*- = w = velocity in the direction 



of z. I have no objection to this (although it was inconve- 

 nient at first for the examination of Professor Chaliis's ex- 

 pression, because Professor Challis had not given the form of 

 <p adopted by him), and, taking the general equations for <p as 

 given by Professor Challis, I will show that Poisson's solu- 

 tion is tenable, and that Professor Chaliis's solution is unte- 

 nable. The general equations are these : — 



<p must be possible as a function of a?, y, z, t y 



dP ~ U \da* + dy* + dz* 



)• 



Velocity in the direction of radius = —, — . 



J dr 



I proceed now with the substitution of the two expressions. 



I. Poisson's Solution. 



tt {fir -at) fir-at)-\ a 



Here <J> = -l *^ A — ^ — - — J ~ - > cos 8 



- ■ T f(r-*ti f'{r-at) \ 

 , \ r 3 r 2 •/' 



d v x d v d y 

 Differentiating, and remarking that -s — = — , -j— = -f- > 

 " s dx r dy dr 



d r z 



dx ~ \_ 7 b r 4 r 3 J 



^*_J Sf(r-at) ^ 3/T {r-a t) f"^r-at)\ 



dx Z ~ Z \ $ + ,4 9 3 J 



' \ r 7 r 6 ' r 5 



r 4 J * 

 Similarly, 

 J!A_.X 3/(r- C Q 3/(r-a<) /"(r-a/) l 

 r/3/ 3 I r 5 T r 4 **~J 



*,V J -4- 15/(«--gQ 15^ (r-g/) 6f»(r-at) 



/ w (r-g<) 1 

 r 4 J' 



Phil. Mag. S. 3. Vol. 19, No. 122. Aug. 1841. L 



