the Theory of Fluid Moiion, . , MW 



Also supposing that 'H '.^ iff . "^iviul -»\m tiM 



grzh flo? + /3j/ +gx^ -{- hxy + ^y + &c., 

 it follows that 



^-ojr ^-0 and ^-0 



The coefficient h may be made to disappear by changing the 

 direction of co-ordinates, and g and k must be supposed to be 

 independent of the time. Hence by substituting in (10.) we 

 obtain for determining the form ofy the following equation ; 





Putting m for the coefficient of/", and v for the quantity of 

 which /is a function, the integral of this equation becomes 



/=Ae'"''+B. 



The form of/ is thus ascertained. Since q is of arbitrary 

 value, we may multiply it by v'—l, and the value of /will 

 then become 



/= Ae~(^*^'''"y^^e'"(*+^-'''^^~^+ B. 



The equation (10.) being linear, may be satisfied by the sum 

 of this value and that which results by changing the sign of 

 \/ — \. So that putting x — Q and y — % and suppressing the 

 constant B, the final result is, 



<^—\i.QQ%m{z-\-d^ct), 

 By putting — for w, the resulting value of c is the following, 



,=,(,+ te±S.^)*. 



This value agrees with what I have previously obtained in a 

 different manner. The reasoning in the present method is 

 more direct, in consequence of the use that has been made of 

 the new hydrodynamical equation. I shall conclude this investi- 

 gation with the remark, that the results arrived at are wholly 

 incompatible with those deduced from the supposition of 

 plane-waves, although the reasoning proceeded on the general 

 hypothesis that udx + vdy-{-isodz was an exact differential, and 

 ought not, if that supposition were allowable, to have led to 



