148 Tiie Astronomer Royal's Reply to Professor Challis. 



f 8f(r-at ) 5f{r-at) /"(r-atU 

 + ~ L r« r> + r 4 J 



(fyjr (Pjr d*_f 

 + dx* + ~dlf + ^* 



d 1 ty d* ty 

 dy 1 dz 2 ' 



a 1*1 r *« *fP* , ^ 2 4> , <* 2 4>\ 

 And therefore -^ - « 2 |^ 2 + j^+ jff 



~^ aZ r 4 + rf/ 2 a \dx* + Ip + dz* f' 



It is necessary for the possibility of Professor Challis's so- 

 lution, that the last expression be = 0. The question there- 

 fore is, Can \J/ be so determined as to make it = 0? It clearly 

 cannot. For, in order to make it = 0, ty must contain func- 

 tions of? 1 — at; but this is evidently inconsistent with its 



m qj 9 



primary condition, that it should be a function of — , — , — , 

 r j r r r » 



and /, and should contain r in no other way. Professor 

 Challis's solution therefore is not possible. 



In two or three passages (as page 64, line 21, and page 68, 

 line 2), Professor Challis seems to intimate that there is a di- 

 stinction to be made between the cases of motion in. the direc- 

 tion of the radius from a fixed centre and motion in the direction 

 of the radius from a moving centre. This distinction (as I under- 

 stand it) is groundless ; and for this reason, that the alteration 

 which it makes in the place of a particle of air is of the second 

 order with regard to the disturbance ; and quantities of the 

 second order are neglected through the whole investigation. 

 Besides, it is evident that if the resolved part of the motion 

 of the air, as measured from a fixed centre, is represented 

 by cos 6 x function of r, there will always be provided the 

 proper vacuum of air to be occupied by the sphere (the oscil- 

 lations being small). I trust, however, that I have misunder- 

 stood the meaning of the expressions. 



Although I do not entirely agree with Professor Challis in 

 attaching slight importance to this special problem, yet I con- 

 sider the correctness or incorrectness of his result as almost 

 insignificant when compared with the legitimacy or spurious- 



