the Theory of Fluid Motion, 517 



is a differential equation of the same surface. Consequently 



(#) = F'(^)(^^). 

 But 



(#)=iW; 



therefore 



A= 



Also by integration, 

 Hence 





Consequently by substituting in (4.), and having regard to the 

 value of A, we obtain 



df.dfdfdf:^ 



dt "^ dx' ■** ^y "*■ dz'' "^ F(f) ""• • • ^^'^ 



Thus we have an equation involving the same variables as 

 the general equation (w) already referred to, and yet not iden- 

 tical with it. The interpretation of this analytical circum- 

 stance is, that the function <^ has not an arbitrary, but apar- 

 ticidar form ; and it is a matter of importance to ascertain 

 what that form is. The following investigation may perhaps 

 suffice for this purpose, but is not the most general that might 

 be adopted. For the sake of avoiding Umg processes I shall 

 confine the reasoning to the first order of approximation. 



The equation (8.) may be put under the form, 



dt ^ ds^ + F(^) -"' .... {9,} 

 where ds is the increment of a line drawn in the direction of 

 the motion, so tha 

 ving V^, we have 



and by integration, 



F(?)+xW=e(5). 

 Hence 



the motion, so that -^ =V. Now neglecting the term invol- 



