Prof. J. Trowbridge on Liquid Vortex-Rings. 291 



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 parallelism, we must have 



d/3 _dy dy _ doc doc _d/3 _ ,. v 



dz dy' dx dz y dy~ dx ' 

 and if these equations are fulfilled the strain is non-rotational, 

 as specified ; but these three equations express neither more nor 

 less than that otdx + ($dy + ydz is the differential of a function of 

 three independent variables." In other words we have a 

 strain-potential ; and in the case of strains rotation is incon- 

 sistent with the existence of a strain-potential. The forces 

 which solicit the particles of a 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 upon which the drop rests, and a force 

 arising from gravitation. It is evident from a consideration 

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

 liquid surface, every particle of the drop cannot pass from its 

 initial position to the next following position 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. Then equations (1) do not hold, and a strain-poten- 

 tial does not exist, and the drop must rotate. This rotation 

 is not in general of the ring-form. If, on the contrary, the 

 drop of liquid can diffuse itself in the liquid through which it 

 falls, each particle, with the velocities u, v, w 9 is solicited at 

 the moment of impact by a superficial tension, by the force of 

 gravitation, and, on account of the tendency to diffuse, the 

 forces of attraction which tend to make the non-diffusing drop 

 reassume the spherical shape are very small. To assume that 

 each particle of the drop, in the next state to that which it 

 assumes on striking the free surface of the liquid, is translated 

 without rotation, is to assume that each particle is compelled 

 to move in restrained limits which do not exist. 



If we follow the notation of Poisson* and Helmholtzf, we 

 shall have for the general equations of motion of an internal 

 particle of a liquid, 



-^r 1 dp du du du du -> 

 X— - . -J- = — +U-r- +v-j- +w~r-> 

 h dx at ax ay dz 



v 1 dp _^dv dv dv dv 



li dy~~ dt dx dy dz \- ' ' (^) 



7 1 c?p _ dw die dw die 



h dz ~ dt dx dy dz , 

 * Traite de Mecanique. t Orelle's Journal, vol. lv. 1858. 



U2 



