364 Sir W. Thomson on the Motion of 



latively to the body, and (A, X), (A, Y), . . . the cosines of the 

 nine inclinations of these axes to the fixed axes OX, OY, OZ. 



Let the components of the "impulse"* or generalized mo- 

 mentum parallel to the fixed axes be f, 77, f, and its moments 

 round the same axes X, jm, v ; so that if X, Y, Z be components 

 of force acting on the solid, in line through P, and L, M, N 

 components of couple, we have 



&-*' #-*- &-*> 



^=L + Zy-Yz, J=M + X*-Z*, J=N + Y*-Xy. 



Let ^, 9, % and %, $&, jl be the components and moments of 

 the impulse relatively to the axes PA, PB, PC moving with the 

 body. We have 



f=f(A,X) + »(B,X) + Z(C,X), 



X=H(A,X)+Pl(B,X)+^(C,X) + %2/-^, ' • ( 7 ) 



Now let the fixed axes OX, OY, OZ be chosen coincident 

 with the position at time t of the moving axes PA, PB, PC : we 

 shall consequently have 



a?=0, ?/=0, z=0, -i 

 dx dy dz y ( 8 ) 



dt= u > tr v > it= w > j 



(A,X) = (B,Y) = (C,Z) = 1, 



(A, Y) = (A, Z) = (B, X) = (B, Z) = (C, X) = (C, Y) = 0, 



d(A,Y) d(B, X ) *(C,X) 



— uir =<J > —dr = - a > s — * 



d(A, Z) _ _ d(B, Z) _ d(Q, Y) _ 

 5 ~ pi dt "**' dt "" 



rw 



Using (7), (8), and (9) in (6), we find (1). 



One chief object of this investigation was to illustrate dyna- 

 mical effects of belicoidal property (that is, right or left-handed 

 asymmetry). The case of complete isotropy, with helicoidal 

 quality, is that in which the coefficients in the quadratic expres- 

 sion for T fulfil the following conditions. 



* See " Vortex Motion," § 6, Trans. Roy. Soc. Edin. (1868). 



