184 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI 



The form of this expression is unaltered when the axes of y, z 

 are turned in their own plane through any angle. The body is 

 therefore said to possess helicoidal symmetry about the axis of x. 



8. If the body possess the same properties of skew symmetry 

 about an axis intersecting the former one at right angles, we 

 must evidently have 



+ 2L (pu + qv + rw) ..................... (7). 



Any direction is now one of permanent translation, and any line 

 drawn through the origin is the axis of a screw of the kind con- 

 sidered in Art. 122, of pitch - L/A. The form of (7) is unaltered 

 by any change in the directions of the axes of coordinates. The 

 solid is therefore in this case said to be ' helicoidally isotropic.' 



124. For the case of a solid of revolution, or of any other form 

 to which the formula 



.(1) 



applies, the complete integration of the equations of motion was 

 effected by Kirchhoff * in terms of elliptic functions. 



The particular case where the solid moves without rotation 

 about its axis, and with this axis always in one plane, admits of 

 very simple treatment "f, and the results are very interesting. 



If the fixed plane in question be that of xy we have p, q, w = 0, 

 so that the equations of motion, Art. 121 (1), reduce to 



dU - - dV =-T^U,l 



(2). 



Let x, y be the coordinates of the moving origin relative to 

 fixed axes in the plane (xy) in which the axis of the solid moves, 



* I.e. ante p. 167. 



t See Thomson and Tait, Natural Philosophy, Art. 322; and Greenhill, "On 

 the Motion of a Cylinder through a Frictionless Liquid under no Forces," Mess, of 

 Math., t. ix. (1880). 



