328 J. Trowbridge — Vortex Rings in Liquids. 



in which it cannot diffuse, the conditions of its motion just after 

 the instant of its striking the surface of the liquid of less density 

 are indicated by the general equations of heterogeneous strains* 

 " For each particle we have the component velocities «, v, iv, 

 parallel to the fixed axes OX, OY, OZ. These have the fol- 

 lowing expressions : 



Eq. (1): 



_ da d/3 



x, y, z, t being independent variables, and a, /3, y, functions of 

 them. If the disturbed condition is so related to the initial 

 condition that every portion of the body can pass from its initial 

 to its disturbed position and strain, by a translation and a strain 

 without rotation, — i. e., if the three principal axes of the strain 

 at any point are lines of the substance which retain their paral- 

 lelism, — we must have — 



■K»«I-36 



and, if these equations are fulfilled, the strain is now rotational, 

 as specified." But these equations express that adx+pdy+ydz, 

 is the differential of a function of three independent variables ; 

 and therefore, in order that there may be no rotation, a strain 

 potential must exist. The forces which solicit the particles of 

 the drop when it rests upon the liquid of less density in which 

 it cannot diffuse are evidently their mutual attraction, a force 

 arising from the superficial tension of the liquid, and one aris- 

 ing from gravitation. It is evident, from a consideration of 

 these forces, that, after the drop has suffered a strain at the 

 surface, every portion of the drop cannot pass from its initial 

 position to the next following by a translation and a strain 

 without rotation. For the drop tends to return from a shape 

 approaching an oblate spheroid to that of a sphere. Equations 

 (2) do not hold, and a strain potential does not exist, and there- 

 fore this drop must rotate. This rotation is not in general of 

 the ring form. If, on the other hand, the drop of liquid can 

 diffuse itself in the liquid through which it falls, each particle 

 with the velocity u, v, w, is solicited at the moment of impact 

 by a superficial tension, by the force of gravitation, and by a 

 force arising from the rate of diffusion. In this case, there is 

 no tendency of the body to reassume the spheroidal or spheroid 

 form in its passage through the liquid. On the other hand, to 

 assume that each particle in the next state of the drop very near 

 that which it assumes on striking the free surface of the liquid 

 of less density, is translated without rotation, is to assume that 

 each particle is compelled to move in restrained limits which 



* Thomson and Tait's Natural Philosophy. 



