on the principles of Hydrodynamics. 363 



now required to treat equation (/3.) by the same process as 

 that applied to equation (y.), for the purpose of deducing the 

 particular form of the integral which defines the motion trans- 

 verse to the axis of z independently of any arbitrary disturb- 

 ance. It may be presumed that such a form exists, because 

 the motion along the axis is already so defined. I have ex- 

 hibited the mathematical reasoning by which an integral of 

 this nature is obtained, in ihe Postscript to my communication 

 to the Philosophical Magazine for last February (p. 96). The 

 following is another process which conducts to the same re- 

 sult. \i (^=x-\-y V — I, SLVidiy=x—y V — 1, the general inte- 

 gral of (|8.) is 



/=F(f.) + G(v)-.{vFi(^) + ^G,(v)} 



-&c., 

 where 



Fi(,.)=/F(^y/., F,(,.)=/Fj(,.)rf^,&c., G,(v)=/G(vyv, 



G2(v)=/Gi(v)^v,&c. 



Now a specific form may be given toy* by supposing the arbi- 

 trary functions to be arbitrary constants. Let, therefore, 

 F(jx) = c, and G(v) = d. Then 



F,(^)=c^, F,(^)=^, F3(^)=-j^^,&c. 



G,(v)=c'v, G,(v)=^, G3(v) = -^^,&c. 

 Hence 



/=s(c+c') {\^euv+ ^^ - ^jT^r^^ +&c.j, 



by putting r^ for x^-\-y^. Buty has already been required to 

 satisfy the condition, y= 1 when r=0. Consequently c + c'=l, 

 and the arbitrary constants disappear of themselves. Thus 

 we obtain 



/=l-^r^+ jl;^, ~jr^+&c., . . (8.) 



a result independent of all that is arbitrary. This form of/* 

 indicates that the motion is the same in all directions trans- 

 verse to the axis of z. 



By thus obtaining, prior to any supposed case of disturb- 



