115, 116.] KINETIC ENERGY IN SPECIAL CASES. 131 



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

 assumes the form 



2T = Au 9 + Bv* + Cw* 



+ PI? + Qr + K&amp;gt;* 



+ 2X wq + 2M&quot;vr ........................... (36). 



The directions of permanent translation are parallel to the three axes 

 of co-ordinates. The axis of x is the axis of one of the permanent 

 screws (now pure rotations) of Art. 114; and those of the other 

 two intersect it at right angles (being parallel to y and z re 

 spectively), though not necessarily in the same point. 



(c) If, further, the body be one of revolution, about x, say, 

 the value of 2T given by (35) must be unaltered when we write 

 v, q, w, r for w, r, v, q, respectively ; for this is merely equi 

 valent 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 of Art. 115 we have M =*X . These 

 conditions can be satisfied only by M&quot; = 0, N = 0, so that 



(37). 



(d) If in (b) the body have a third plane of symmetry at 

 right angles to the two former ones, then taking this plane as that 

 of yz we have, evidently, 



(38). 



The axes of co-ordinates are in the directions of the three permanent 

 translations ; they are also the axes of the three permanent screw- 

 motions (now pure rotations) of Art. 114. 



(e) Next let us consider another class of cases. Let us sup 

 pose that the body has a sort of skew symmetry about a certain 

 axis (say that of x), viz. that it is identical with itself turned 

 through two right angles about this axis, but has no plane of sym 

 metry*. The expression for 2T must be unaltered when we 



* A two-bladed screw-propeller of a ship i* an example of a body of this kind. 



9-2 



