6 PROFESSOR ANDREW ORAY, 



which Hows into the tube, and the latter the mass which flows out, per unit of time. 

 Denoting this constant quantity by m, we obtain 



kz+pV+p^s^fe + v.+ft-^-v,-^ . . (ii). 

 j \ p x p 2 / 



Now consider the volume-integral on the right of (8). By the equation of con- 

 tinuity given in § 9 we have 



{p®dvs=-Jp d .P . . . (12), 



J j p~ 



where the integral on the right is to be taken along the tube here considered, in the 

 direction of flow. Integrating by parts, we find 



J \ds <t 3s/ \jOjj p 1 J Pi p J 



This added to the former result gives 



fiW+P^+PhndS + fp^ + ^d^^^yi + Y.-hl-Y,- f"^) . (13). 



Thus, since the left-hand side of (13) vanishes in the case of steady motion, we 

 obtain finally for any cross-section of the tube the well-known equation 



k 2 + V+ f^ = const. . . . . (14), 



J P 



which is generally obtained by another process. It is to be understood that the 

 integral, fdp/p, is to be taken along the stream-line, from any chosen starting-point 

 up to the cross-section to which q and V belong. It is thus shown that ^ 2 + V +fdpjp 

 is constant along a stream-line ; but its constant value, it is to be observed, may change 

 from one stream-line to another. 



11. We shall now find an expression for 



/: 



&* + tf + T + |* 



in the general case. This will show that when the fluid is destitute of what may be 

 called elemental rotation the value of this expression, which is a function of the time, 

 is the same throughout the fluid at any one instant, and becomes an absolute constant 

 when the motion is steady. 



The theorem thus arrived at will be found to lead at once to Lord Kelvin's 

 theorem of circulation, from which the permanence of vorticity, or non-vorticity, in any 

 portion of a perfect fluid, can be inferred. It will be shown also how another 

 relation can be obtained, which affords another view of the proof of the theorem of 

 permanence. 



Let \j/- denote 



