Prof. Challis on Hydrodynamics. 211 



sumptive evidence that the third general equation cannot be 

 dispensed with. Accordingly I now proceed to inquire how 

 the problem of rectilinear motion perpendicular to a plane, as 

 well as other problems, may be solved by means of the equation (a). 

 The steps of the reasoning by which that equation was shown 

 to be applicable to motion along a rectilinear axis accompanied 

 by motion transverse to the axis, are given in arts. 14-21. Also 

 in arts. 22-24 a particular solution of it is derived from the in- 

 dications of the analysis prior to the consideration of any case of 

 motion, which solution is on that account to be regarded as ex- 

 pressing an independent law of vibratory motion due to the 

 mutual action of the parts of the fluid. The integration is 

 effected by successive approximations, and the result to the third 

 approximation is 



= »* ' cos q/i j2p±Bm2qfL-- -^[jjr - g 1 cos Hf*> 



V 3b 2 + 12/' 



2tt 



= a 2 -\ — s + m'-ty 



/jb being put for z— a L t + c, and q for — — If m be substituted 



for — qm! and da for a x> we shall have, to the same approxi- 

 mation, 



dd>. . 2roV ZrrP{7iP + l) . 



2 _ § , & m» 5/^ + 3 

 «i ~ a + ^ + 12 k' 2 -1 " 



These equations should be substituted for those in art. 31, the 

 value of k employed in obtaining the latter having been shown, 

 by an argument in the communication to the Philosophical 

 Magazine for last May, to be erroneous. In that communica- 

 tion the numerical value of «/, obtained on the supposition that 



b 2 

 a^= « 2 + -g, is found to be 1*2106. If this constant be repre- 

 sented by k, we shall have 



, 2 _ % ,_nP_ 5* a + 3 

 * "" + 12« 2 " * 2 -l " 



The expression for the condensation may now be derived from 

 the equation 



from which the arbitrary quantity F(/) has disappeared by sup- 



P2 



