Ath^adive and Repulsive Forces. 193 



means of the factor -. As there is an analytical distinction be- 

 tween these two cases, according to the before-cited rule there? 

 must exist, consistently w4th the principle of geometrical conti- 

 nuity, a corresponding difference in the actual circumstances of 

 the motion. It is possible to point out such difference by em- 

 ploying the following argument. In case \iidx] be an exact 

 differential, the equation [iidx'] =0 is the differential equation of 

 a surface the curvature of which is generally finite, and conti- 

 nuous through at least an indefinitely small extent. Hence it 

 foilovv^s, since the normals converge to two focal lines, that the 

 form of the element to which the coordinates x, y, z apply at the 

 time t is undergoing change. The same is the case with respect 

 to any other element, the continual change of form of identical 

 portions of matter being a general characteristic of a fluid mass 

 in motion. But it is also possible that a mass of fluid may 

 under certain circumstances move in such manner that each ele- 

 ment continues to be of the same form throughout the motion — 

 for instance, if the fluid rotate about a fixed axis, and the velocity 

 be a function of the distance from the axis. For by the prin- 

 ciple of easy divisibility the fluid may in that case be conceived 

 to be divided into indefinitely thin cylindrical shells, having as 

 their common axis the axis of the motion, the velocity of rota- 

 tion of any shell being at the same time a function of the dis- 

 tance from the axis. Also on the same principle each shell 

 might have, in addition, a motion of translation parallel to the 

 axis, the form of every element of the shell still remaining 

 constant. 



8. In articles 18 to 25 of the communication in the June 

 Number I have shown that for the above-mentioned cases of con- 

 stancy of form of the elements in motion, \udoc\ is integrable by 

 a factor, and that this analytical circumstance not only proves 

 that such motions are compatible with the principle of geome- 

 trical continuity, but also distinguishes them from all other mo- 

 tions. I have also in the same articles determined the condi- 

 tions under which these motions satisfy the general equations 

 (b) and (c), whence it appears that they must be steady motions. 

 I consider it therefore unnecessary to introduce these investiga- 

 tions here, and shall only remark further on this part of the 

 subject, that the cases of motion for which \udx\ is integrable 

 by a factor are applied in the hydrodynamical theory o^ galvanic 

 currents along slender wires. (See arts. 26-40 in the Theory of 

 Galvanism and JMagnetism, contained in the June Number.) I 

 proceed now to the discussion of the classes of motion for which 

 \udx] is integrable per se, 



Phil, Mag. S. 4. Vol. 44. No. 29.2. Sept. 1872. 



