324. Mr. Airy m reply to Prof. Challis 



d . log p _ k z dv _ rj 



Tz V'Tt ~ 



d . lo2 



p \ ( dx) du d u \ 2u ,„ 



r \ d.r du dz ) r 



dt r\ dx dy 



In order that the supposed motion may be possible, these 

 expressions must satisfy the following equations : 



^ '' dy ~ dx • ^ '' dz ~ dy '' ^ '' dt " dz ' 



Equation (I.), on performing the differentiation, becomes 



kxy du /: ci- d° v _ kxy r/u ky d^ v 



r* 'ITt V ' dydt ~ ' r^ ' 'dt VTxdt* 



d^u d^v 



dy dt "^ dx dt 



r du 11- ^-1 1 dco 1 d CO 



Let -r- = w; then this equation becomes — . -r— = — . ^— . 



dt * y dy X dx 



If we solve this equation, in the usual way of solving partial 

 differential equations of the first order, we find that w must 

 be a function of .t' + t/^. 



Treating equation (2.) in the same manner, we find that 

 CO must be a function of t/'^ + z^. 



It is evident that these results can be united only by the 

 supposition that m is- a function of x- + y' + ^% or of /-, or 

 of r. 



That is, —r- is a function of r and t only ; and the quan- 

 dt 



titles .r, 3/, ;:, do not enter into it except as combined in the 



formula x^ +y^ + z^. 



Integrating this function, = R+ S, where R is a function 

 of r and t, and S is a function of a',j/, z. 



kz rf R 



To form equation (3.), we remark that Z = f •-tt> 



and therefore —r- = ~ , , . Also that part of T which 



dt r dt' 



I / dR X dR 7/ , dR z\ 



depends on R is - -(^^- ^ . - + j/^ ' t -^ "dV- ' 1^ ) 



_2R^_clR__2R p,,s'for-i-f.f + v^ 

 r dr r r \ dx - dy 



rfS\ 2S ™ rr ^R 2R , ^j, , rfT 



+ z~r-) . Then r = -, + S': and -r- 



dz/ r dr r dz 



— y ' flj.^ yi ' fly J.3 fl^ • 



