36 Prof. Challis on Hydrodynamics. 



this equation into the two following, 



<fe* + dy- 



+ J.,2 + „S — U > 



which are both linear with constant coefficients. It is evident 



that, both in this and in the former resolution, Zl is the value 



dH d 2 f 

 of -v4- + -t4t, where a? = Oand ?/ = 0. The particular solution 

 dx l dy~ 



of the first of these equations is the series for /already given, 



in which the value of r may now be taken without limitation. 



From the particular solution of the other equation, combined 



with the equations 



-ft ^/f=°> 



we obtain 



aa j. . 27r 

 (o= — = /7ttsm— [z—fcat-\-c). 

 k A, 



Also the transverse velocity 



, df m\ df 2tt . . 



= 0^ = — 5T— • -f • COS-r-(£ — tffl£ + c). 



^«r 27r dr \ K 



The definite expressions thus arrived at apply to the principal 

 parts of the condensation, and of the velocities parallel and 

 transverse to the axis — that is, to the parts which are of the same 

 order as the velocity and condensation indicated by the first 

 terms of the series for w' and a'. It has already been argued 

 that the parts of the velocity and condensation along the axis 

 indicated by the other terms necessarily coexist with those that 

 are principal ; but with respect to points distant from the axis, 

 expressions for the velocity and condensation to terms of a higher 

 order than the first are not deducible by the preceding investi- 

 gation, because the assumptions made respecting the properties 

 of the functions/" and cf> are not satisfied beyond terms of the 

 first order. It may, however, be assumed that the motion and 

 condensation are symmetrically disposed about the axis, this 

 having been shown to be exactly the case in its immediate neigh- 

 bourhood, and very nearly so at all other positions. In fact, by 

 supposing that (dyfr) = (d .f<j> + ^%) = udoc + vdy + wdz, % being 

 a function of z, r, and t, and by substituting in the general 

 differential equation to terms of the second order, of which yjr is 

 the principal variable, viz. 



