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 y, £, 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 
df _ -y drj _ -y (It, _ y 
dt ’ dt 5 dt 
£ = L + Zy-Y*,f = M + x,-z*,£ 
= N + Yx - Xy 
( 6 ). 
) 
Let X, g, % and H, $ be the components and moments 
of the impulse relatively to the axes PA, PB, PC moving with 
the body. We have 
= X (A, X) + g (B, X) + Z( C, X) 
X = 1L (A, X) + ;P % (B, X) + $ (C, X) + %y — XEz 
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 
as = 0, y = 0, 2 = 0, 
dx dy dz 
= U , = v , —— 
dt dt dt 
10 , 
(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)_ 
~di <r ’ dt dt p 
d(A, Z) 
dt 
= ” P 
d{ B, Z) 
dt 
= 
cKG, Y) _ _ 
dt 
= ~ 7Z 
Using (7), (8), and (9) in (6) we find (1). 
* See “ Vortex Motion,” £ 6, Trans. Roy. Soc. Edin. (1868). 
