Prof. Helmholtz on Integrals caressing Vortex -motion. 489 

 nates are X> V) fr we ma y P ut at that point 

 u = A, 



du _ dw __ dv 

 dx ~ ' dy ~~" dz 



t, dv , du dw n 

 » =B ' Ty= h < Tz = T X =^ 



r . dw du du 



dz dx dy ' 



whence we have for a point x } y, z indefinitely near p 3 \), ft, 



u = A + a{x-r)+y(y-i))+j3(z-l), 



w = C+@{x-v) + *{y-t>) +c{z-%): 

 But if we make 



$=A(*-tf+B(y + |>) + C(*-g) 



+"(y-v) (z-j) +£(*- x) (*-j) +ir(*—lO(y-W 



then we have 



<7d> </6 cfci 



u=-r~) v= -f-i w= -~* 

 dx dy dz 



It is known that we can in one definite way, by change of 

 axes to those of x x , y v z l (j 4 , \), J being the origin), reduce the 

 expression for <f> to the form 



= A 1 a: 1 4-B 1 y 1 + C 1 2' 1 + l V] 2 + ^ 1 7/ 1 2 + lc^ 2 ; 



and the components of the velocity parallel to these new axes 

 are 



u l ^=K l + a l x v v 1 = B 1 -hb i y v w l = Q l -\-cz v 



The component u x is thus the same for all points for which x 1 

 has the same value; hence particles which at the beginning of 

 the time dt are in a plane parallel to one of the coordinate planes 

 will at the end of dt also lie in such a plane. Hence an inde- 

 finitely small parallelepiped with its edges parallel to the axes 

 of x v y v z x will move parallel to itself in space, and suffer 

 only dilatations or compressions (cases (1) and (2) above). 



Let us return to the first system of axes, and suppose that 

 in addition to the above motions there are, for the element of 

 the fluid just considered, angular velocities f, tj, f about axes 

 through X) Vt 3 parallel to the coordinate axes. These give 



