V62 PROCEEDINGS OP THE AMERICAN ACADEMY 



each particle we liave the component veh)cities u, v, w, parallel to the 

 fixed axes OX, OF, OZ. These have the following expressions: — 



^ ^ ^ dt' dt' dt' 



X, y, z, t being independent variables, and «, /3, /, 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 sub- 

 stance which retain their parallelism, — we must have, — 



Eq. (2) : — = ^ 'b z='i°- ^ =z^Ii - 



dz dy dx dz dy dx 



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

 specified." But these equations express that a d x -\- ^ dy -\- y d z, i& 

 the differential of a function of three independent variables ; and there- 

 fore, in order that there may be no rotation, a strain potential must 

 exist. The forces which solicit the particles of tlie 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 ai'ising from gravitation. It is evident, from a 

 consideration of these forces, that, after the drop has suffered a strain 

 at the surface, every poilion 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 api)roaching an 

 oblate spheroid to that of a sphere. Equations (2) do not liold, and a 

 strain potential does not exist, and tlierefore this drop must rotate. 

 This rotation is not in general of the ring form. If, on tiie 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 im- 

 pact 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 form in its passage through 

 the liquid. On tiie 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 rota- 

 tion, is to assume that each particle is compelled to move in restrained 

 limits, which do not exist. For the components X\ Y^, Z^, of the 

 attraction, which tend to make the non-diffusible drop reassume its 

 spherical form, we have in the case of the diffusible the components 



