386 Proceedings of the Royal Society 
x, y , z. Let PA, PB, PC be three rectangular axes fixed relatively 
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 momen- 
tum, parallel to the fixed axes be £, rj, £, and its moments round 
the same axes A, y, 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 
dij _ d/Yj y df y 
dt ' ’ dt~ ’ dt~ 
( 6 ). 
dX _ _ _ dfi ,, v - dv AT v v 
j t = L + Zy - Yz , -£ = M + Xz - Zx , = N + Yx - Xy \ 
Let g, % and 3L, HU, be the components and moments 
of the impulse relatively to the axes PA, PB, PC moving with 
the body. We have 
| =$(A,X) +g(B,X) + Z(C,X) ^ 
A = n (A, X) + m (B, X) + $ (C, X) + %y - 
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 
£c = 0, y = 0 , z = 
dx _ dy _ dz 
It U 3 dt ’ dt 
( 8 ). 
(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) 
~~It * ’ dt 
d(A, Z) _ _ d(B,Z) 
dt P ’ dt 
“ o’ j 
d(C, Y) _ H 
dt 
Using (7), (8), and (9) in (6) we find (1). 
* See “ Vortex Motion,” \ 6, Trans. Roy. Soc. Edin. (1868). 
