448 Prof. Challis on the Principles of Hydrodynamics. 



We have thus arrived at an equation for determining / which is 

 consistent with the original supposition that this quantity is a 

 function of x and y only. 



The next step is to ascertain the particular form of/ which 

 applies to the motion resulting from tlie mutual action of the 

 parts of the fluid. As the equation (v) is of the same form as 

 the equation {jjl), the same process that conducted to a particular 

 solution of the latter must conduct to a particular solution of the 

 former. In fact, by this process we obtain 



f=ciC(i?,{(jx-\-hij), 



which evidently satisfies [v), g and h being subject to the con- 

 dition 



Let ^ = 2 v'e cos 9. Then h = 2 Ve sin 6, and the above integral 

 may be put under the form 



f— ucos{'^ Ve{x CO?, 6 + ym\ 6)]. . . (tt) 



Now as it was argued that an exact and unique integral of [jj), 



the form of which was indicated by the analysis, I'eferred to the 



motion resulting from the mutual action of the parts of the fluid, 



by parity of reasoning, the integral (tt) of the equation (v) should 



receive the same interpretation. But it is to be observed, that 



since 



j.df 1 ,df 



r. = </,-^and. = </,^, 



the value of /given by the eqiiation (tt) indicates that the part 

 of the motion parallel to the plane of xy is parallel to an arbitrary 

 direction in that plane depending on the value of 0. There is, 

 however, an integral of (v) which gets rid of this arbitrariness by 

 embracing all directions depending on the arbitrary values of 6. 

 For since that equation is linear with constant coefiicients, it is 

 clearly satisfied by supposing that 



/=S.«8^cos {2 V e[x cos 6 + y %m 6)} J 



S9 being an indefinitely small constant angle, and the summa- 

 tion being taken from ^=0 to ^ = 27r in order to embrace every 

 possible direction of the motion. By performing the sximmatiou, 

 substituting ?-^ for x'^ + y'^, and determining « so as to satisfy the 

 condition that/=l where r=0, the result is 



This value of/, containing no arbitrary quantity whatever, indi- 



