UT' 



PAPER BY PROF. HELMHOLTZ. 49 



is obtained from the equation (1) by multiplying tbe first by n, the sec- 

 ond by r, the third by tc, and adding ; whence results : 



/ du dv dw\ f dp, dp. ?p\ 



When we multiply both sides by dx dy dz, then integrate over the 

 whole volume of the liquid mass, and recall that because of (I4) 



( ti y^ -f r ^-~- -\- ic-~ ) dx dy dz=zz —J^ tp q cos ddcj, 



where ^' denotes a function that is continuo us and univalued throughout 

 the interior of the liquid mass, we obtain, 



^=/ dcj {p -hU + h1i(f)q cos (i (0Z>) 



dt 

 When the liquid mass is entirely inclosed within rigid walls then 

 at all points of the surface q cos 6 must be zero, therefore then will 



/7 AT 



— = 0, or TT become constant. 



dt 



If we imagine this rigid wall to be at an infinite distance from the 



origin* of coordinates and all vortex filaments that may be present to be 

 at an infinite distance from this origin, then will the potential functions 

 ly, J/, N [of imaginary magnetic matter], whose masses ^, //, C, 



or densities ~^, ^^, ^^ , each and all are equal to zero, diminish 



1_ 2n: 27C 27t J 



at the infinite distance 91 as — and the velocities [which are the par- 



W 



tial differential coefficients ofL, M, N], will varv as - -, but the element- 



ary surface dao, if it is always to correspond to the same solid angle at 

 the origin of the coordinates, will increase as W. The first integral in 

 the expression for K, equation (6a), which is extended over the surface 



of the liquid mass, will diminish as — and therefore will be zero for 9t 



equal to infinity. 



The value of K then reduces to the expression, 



K= -h fff {La + Mi] 4- NZ) dxdydz (6c) 



and this quantity is unchanged during the movement. 



V. RECTILINEAR PARALLEL VORTEX FILAMENTS. 



We will first investigate the case where only rectilinear vortex threads 



exist parallel to the axis of 0, either within a liquid mass of infinite 



extent or which comes to the same thing, in one that is bounded by 



two infinite planes perpendicular to the vortex filaments. In this case 



80 A 4 



