444 Prof. Challis on the Principles of Hydrodynamics. 



Putting, for convenience, e for j-^, fi for z + at, and vfor ^r— «^, 



the integral of the above equation in a scries proceeding accord- 

 ing to powers of e is 



</,=?(/.) + G(v) 



+ &C., 



where 



Y,{,j,)=/Yi^)dp,, F,{f^)=J%{,ji)d^, G,(0=yGWr/v,&c. 



As the fujictious F and G satisfy the equation (?;) independently 

 of each other, it is permitted to consider them separately. Let, 

 therefore, 



c^=G(v) + ./.G,(v)+ ^G^M + j^ . G3(v) + &c. 



This value of (p, containing arbitrary quantities, is not generally 

 apphcable to the present inquiiy, which is antecedent to any 

 case of arbitrary disturbance. It is, however, to be remarked 

 that <j) has particular forms, expressible in finite terms, if forms 

 of the function G can be found, which will satisfy the equality 



dv 

 for every integral value of n. Now, 



Hence, by the above equality, 



dv »/ 



or 



^^+A-G„(v)=0. 



The upper sign gives a logarithmic form to the function G, 

 which is incompatible \\dth any general law of fluid motion, as 

 also with the value of (j) already obtained. Taking the lower 

 sign and integrating, we have 



G„(j')=Acos(Av + c), 



which determines the form of the function G. In conformity 



