On some Poinds relating to the Theory of Fluid Motion. 513 



If w, Vi w be the resolved parts of the velocity of a given par- 

 ticle of fluid in motion, and we suppose that 



la!=uU ly-i^t Iz—'w'^t, 

 the above equation becomes 



d"!/ d^ d^ d^ ^ H 



dt dx dy dz 



The signification of this equation depends entirely on the na- 

 ture of the curve surface defined by the equation •^{x^y,z^t)^=^Qi. 

 If, for instance, this be the surface of a fixed or moveable 

 boundary with which the fluid is in contact, the equation 

 affirms that the same particle remains in contact with the 

 boundary in successive instants. If the function vj/ be the 

 general expression for the pressure/?, then since /> = is the 

 equation of the free boundary, the above equation would ex- 

 press in this instance the condition that a given particle is 

 situated on the free boundary in successive instants. Let now 



(«4')= r dj;+ —dy+ —dz. 



AAA 



Then, as is known, 4/=:0 is the equation of a surface cutting at 

 right angles the directions of the motions of the particles 



through which it passes. The factor — is applied for the 



A 



sake of generality, because it may be assumed that such a sur- 

 face always exists, and consequently that the right-hand side 

 of the above equality is an exact differential, although it can- 

 not be affirmed that uda,- + vdy + wdz is always an exact differ- 

 ential. Assuming, therefore, the integrability of 



—■dx+ — dy+ —dz, 



AAA 



it follows that 



«-i"('-) -^S"(^-) »-i"(«-)- 



Hence, substituting in the foregoing equation, 



d^> ^^(d^'^ dV .d:\,^\ 



This is the new equation which it was proposed to obtain. 

 The course of the investigation shows that this equation ex- 

 presses the condition that the directions of the motion in a 

 given element are in successive instants normals to surfaces of 

 continued curvature. The fulfilment of this condition ensures 

 the continuity of the motion ; and the above equation may 

 Phil. Mag. S. 3. No. 232. Suppl. Vol. 34. 2 L 



