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



C', R', N, L" also must vanish, so that the expression for 2T 

 assumes the form 



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 z 

 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 



Rr* ........................... (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 t 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" = N'. If we further transfer the origin to the point 

 defined by Art. 122 (viii) we have M" = N'. Hence we must have 



and 2T = Au* + B(v* + w 2 ) 



(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, "On Hydrokinetic Symmetry," Quart. Jo/urn. Math., t. xx. 

 (1885). 



