123] SPECIAL FORMS OF SOLID. 183 



5. If, in the last case, the form of the solid be similarly 

 related to each of the coordinate planes (for example a sphere, or a 

 cube), the expression (3) takes the form 



2T = ^(^ 2 + ?; 2 + w 2 ) + P(^ 2 + 2 2 + r 2 ) ............... (4). 



This again may be extended, for a like reason, to other cases, 

 for example any regular polyhedron. Such a body is practically 

 for the present purpose ' isotropic,' and its motion will be exactly 

 that of a sphere under similar conditions. 



6. We may next consider another class of cases. Let us 

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

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

 through two right angles about this axis, but has not necessarily a 

 plane of symmetry*. The expression for 2T must be unaltered 

 when we change the signs of v, w, q, r, so that the coefficients 

 B', C', Q', R', M, N, L', L" must all vanish. We have then 



2T= An? + Bv 2 + Cw* + 2A 'vw 

 + Pp 2 + Qq 2 4 Rr* + 2P'qr 

 + 2Lpu 



+ 2q (M'v + N'w) 

 + 2r(M"v + N"w} ........................ (5). 



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

 and is also the axis of one of the three screws of Art. 122, the pitch 

 being L/A. The axes of the two remaining screws intersect it 

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



7. If, further, the body be identical with itself turned through 

 one right angle about the above axis, the expression (5) must be 

 unaltered when v, q, w, r are written for w, r, v, q, respectively. 

 This requires that B = G, A' = 0, Q = R, P'=0, M* = N" t 

 N' = M". If further we transfer the origin to the point chosen 

 in Art. 122 we must have N' = M", and therefore N' = 0, M" = 0. 

 Hence (5) reduces to 



q + wr) ........................... (6).f 



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

 f This result admits of the same kind of generalization as (3), e.g. it applies to 

 a body shaped like a screw-propeller with three symmetrically-disposed blades. 



