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



C&quot;, R , N, L&quot; also must vanish, so that the expression for 2T 

 assumes the form 



2T = Au? + BV* + GW* 



r ........................... (1). 



The directions of permanent translation are now parallel to the 

 three axes of coordinates. The axis of x is the axis of one of the 

 permanent screws (now pure rotations) of Art. 122, and those of 

 the other two intersect it at right angles (being parallel to y and s 

 respectively), though not necessarily in the same point. 



3. If the body have a third plane of symmetry, viz. that of 

 yz, at right angles to the two former ones, we have 



(2). 



The axes of coordinates are in the directions of the three perma 

 nent translations ; they are also the axes of the three permanent 

 screw-motions (now pure rotations) of Art. 122. 



4. If, further, the solid be one of revolution, about %, say, the 

 value (1) of 2T must be unaltered when we write v, q, w, r for 

 w, r, v, q, respectively; for this is merely equivalent to turning the 

 axes of y, z through a right angle. Hence we must have B = C, 

 Q = R, M&quot; N . If we further transfer the origin to the point 

 denned by Art. 122 (viii) we have M&quot; = N t Hence we must have 



and 2T = An? + B (v 2 + O 



(3). 



The same reduction obtains in some other cases, for example 

 when the solid is a right prism whose section is any regular 

 polygon*. This is seen at once from the consideration that, the 

 axis of x coinciding with the axis of the prism, it is impossible to 

 assign any uniquely symmetrical directions to the axes of y and z. 



* See Larmor, &quot;On Hydrokinetic Symmetry,&quot; Quart. Journ. Math., t. xx. 

 (1885). 



