502 Prof. Helmholtz on Integrals expressing Vortex-motion. 



and integrate partially, denoting by a, j3, y, and 6 the angles 

 which the inwardly directed normal of the element dco of the 

 fluid mass makes with the coordinate axes and with the resultant 

 velocity q ; we thus obtain, attending to equations (2) and (1) 4 , 



K = — \h (V<y[P</cos#4-L(«;cosy — wcos/3) + Mucosa — wcosy)! y 



dK 

 The value of —- is found from (1 ) if we multiply the first by u, 



CLZ 



the second by v, and the third by w, and add, 



7 / du dv dw\ (dp dp dp\ 



h \ u Tt +v It +w Tt)= - T& + ■ ty +w Tz) 



+h ( u ^i +v d i +w d i\- / i( u m .^m +w m). 



\ dx dy dz J 2 \ dx dy dz / 



If both sides be multiplied by dx dy dz and integrated through 

 the whole extent of the fluid, noticing that by (1) 4 



if yjr denote in the interior of the fluid mass a continuous and 



single-valued function, we obtain 



^= £ doy(p - hV+±hcf)q cos 0. . . .(6b) 



If the fluid mass be entirely enclosed in a rigid envelope, q cos 6 



must be zero at every point of its surface. Hence -^-=0, or 



K= constant. 



If we consider this rigid envelope as being at an infinite dis- 

 tance from the origin of coordinates, but the vortex-filaments at a 

 finite distance, the potential functions L, M, N, whose masses 

 tj, r\, f are each in sum equal to nothing, are, at an infinite dis- 

 tance K, proportional to R -2 , and their differential coefficients as 

 R~ 3 ; but the surface-element da> } if it always correspond to the 

 same solid angle at the origin, is as R 2 . The first integral in 

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

 of the fluid mass, will vary as R~ 3 , and therefore vanish for an 

 infinite value of R. The value of K thus becomes 



K=-h§§§{L£-\-M v + ~NQdxdydz; . . {6)c 

 and this value does not alter during the motion. 



