PAPER BY PROF. HELMHOLTZ. 39 



As is well known, we can combine rotations together after the method 

 of parallelograms of forces. If £, ?/, C are the velocities of the rotations 

 about the coordinate axes, then the velocity of rotation {q) about the 

 instantaneous axis of rotation is 



and the cosines of the angles that this axis makes with the coordinate 



axes are respectively — , ~. and — . 



If now we consider an infinitely small distance qe in the direction of 

 the instantaneous axis of rotation, then the projections of this distance 

 on the three coordinate axes are respectively f£, erj, and eC,. While 

 at the point x y z [at one end of qe] the components of the velocity are 

 u. v, w, they are at the other end of qe respectively 



dw dw dw 



^= w +^* +«*#+<*■ 



Therefore in the course of the elementary time dt the projection of 

 the distance of the two particles of liquid that at the beginning of dt 

 were distant by the quantity qe has attained a value that, considering 

 the equation (3), can be written as follows: 



£<§■+(«!— u)dt — e'y g+ ^r-dt J, 

 e v +{v l -v)dt=e( v + ~^dt\ 



et + (ic l -ic)dt=e( £+ %dt). . 



On the left are the projections of the new location of the connecting 

 line qe ; on the right are the projections multiplied by the constant 

 factor e of the new velocity of rotation. It follows from these equa- 

 tions that the line connecting the two liquid particles that at the be- 

 ginning of the time dt limited the portion qe of the instantaneous axis 

 of rotation will also after the lapse of the time dt still coincide with the 

 now changed axis of rotation. 



When we, as above agreed on, call a line whose direction through- 

 out its whole length agrees with the direction of the instantaneous axis 

 of rotation of the particle of liquid at each point, a vortex line, we can 

 express the proposition just found as follows : Every vortex line remains 

 permanently composed of the same particles of liquid while it progresses 

 with these particles through the liquid. 



