14-6 The Astronomer Royal's Reply to Professor Challis. 



Then 

 <*4> _/ /(>— at) /Hq , jtl 3/(r-afl 

 dz"\ ? P J + I 5* 



3f'(r-at) f'{r-aty 



+ 



-*fll 

 J3 J 



r* r 



^ - a jrl S/(r- a 1 3/'( r-afl f"(r-aty 

 dz* 



15 / (r-a t) 15/' (r- a t) \ 6/" (r-a£) 



f"'(r- aty 



r 4 



\ r 5 r 4 r 3 f 



■ ,3 f | 15/(r-flQ 15/* (r-a/) 



rr.i «■ SP « 9 tt <p . 



The sum, or Vj + -~ + -=-| is 



5^ 



< ^9 <^9 <fi$ 

 3/(r-at) 3/' (r-a t) /"(r—at) 



C_ 3j(r-at) Sf (r-a t) J" (r—at) -} 

 \ r 5 * r 4 r 3 J 



f 15/(r-at) \5f'(r-at) 6f"{r-at) 

 f"(r-aty 



r—at) ^ 



_ j /"(r-at) f»>(r-at) \ 

 ~ z \ r* P J* 



But ** = #„ f f"(r-at) _ / ff '(r-*fl \ 

 tfut rf<9 -« *^ p r2 J, 



Therefore || = a* {*£ + -0 + -g) : and therefore 

 Poisson's solution is possible. 



II. Professor Challis's Solution. 



Professor Challis has not, in his first papers, given a value for 

 9, but he has assumed (Phil. Mag. and Journal, December 

 ] 840), in page 463, line 25, " the velocity and density of the 

 fluid which passes the area w 2 are v and p" and in page 465, 



line 7, &c. o = f '<?-"') - ■/>-"'> . From page 463, 



line 14, it is evident that this is the velocity in the direction 

 of the radius : and from page 465, line 23, as also from page 

 131 (February 1841), line 6, that it is to have the factor cos 0, 

 without which it cannot be adapted to the different points of 

 the sphere's surface. Hence we have 



