132 ON THE MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. V. 



change the signs of i\ w, q, r, so that the coefficients 7? , C , Q } R , 

 M t N, L t L&quot; must all vanish. We have then 



2T = Au* + IW + CW + 2A vw 



+ 2r(M&quot;v + N&quot;w) ........................ (39). 



The axis of x is one of the directions of permanent translation ; 

 and also the axis of one of the three screws of Art. 114, the pitch 



being j. The axes of the two remaining screws intersect it 

 ^L 



at right angles, but not in general in the same point. 



(f) If, further, the body be identical with itself turned 

 through one right angle about the above axis*, the expression (39) 

 must be unaltered when v, q, w, r are written for w, r, v, q, 

 respectively. This requires that B = C, A = 0, Q = R, P = 0, 

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

 chosen in Art. 115 we must have N =M&quot; } and therefore N .= 0, 

 M&quot; = 0. Hence (39) becomes 



+ 2M (vq + wr) ........................... (40). 



( g) If the body possess the same properties of skew symmetry 

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

 evidently must have 



+ 2L(pu + qv + rw) ..................... (41). 



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. 114, of pitch --r. The form of (41) is unaltered 



^ci. 



by any change in the directions of the axes of co-ordinates. 



* Some four-blacled screw-propellers are examples of bodies of such forms. 



