PAPER BY PROF. HELMHOLTZ. 121 



The quantities p and f are dependent on each other as soon as the 

 form of the space is given for whose boundary they hold good; so that 

 we can put 



p=f. 9t 



where )K indicates a value that depends only on the size and form of this 

 space. Hence there results 



2* 2 ' 2 W ' ' 



When therefore >H experiences a change S 8ft then if f remains un- 

 changed we have 



8L=* f 2 . <y$R 



on the other hand, when p remains unchanged we have 



2 W 2 ' 



tf p = 0. 



Both variations therefore have tlie same values with opposite signs. 

 We can therefore, instead of 



( j4> -6 L = 

 dp l =dfo=0. 



which is the form of variation for the stationary condition where the 

 variation of 3 L is deduced from the variation of the form of the 

 region, also write 



o- €> +6 L = 



6 f, =6 f 2 = 0. 



The quantities f according to their definition have the value: 



pX, y + A 



V = / (u. dx+v. dy) 



x,y- 



the integral being taken for any value that leads from the point (x, y) 

 to the point (x, y + A). When we choose the stream-line ?/'= constant 

 for this path between these points then the integral also iudicates a 

 path along which a series of material liquid particles would flow. The 

 value of the integral f 1? as computed for such a series of material flow- 

 ing particles as is well known remains unchanged, whatever motions 

 may otherwise be going on in the liquid, provided there are no differ- 

 ences iu the sum total of the pressures and potentials of the exterior 

 forces between the beginning and the end of the series, and provided 

 there is no friction. This is the same sum that also remains unchanged 

 in the vortex motion in every closed ring of material particles. We 

 can therefore in fluid motions consider s, u and s 2 f 2 as the moments of 



