91 
435] ON THE SIX COORDINATES OF A LINE. 
and hence the displacement in the direction of the line is 
= cos a 8x + cos /3 8y + cos 7 8z, 
which attending to the significations of (a, b, c, f g, h) is 
= ap + bq + cr +fl + gm + hi, 
and we have thus the theorem in question. 
62. It thus appears that for a system of rotations 
Aj about the line (a u b 1} c 1} f u g 1} h), 
^•2 » (®2> b 2> c 2 , f%, g^, h^), 
&c. „ &c. 
the displacements of the point (x, y, z) rigidly connected with the several lines are 
8x = . — y%h\. + zXg\ — 2aX, 
8y = x%h\ . — z'Z/X — 'i.b’i,, 
8z = — xtg\ + y'tfX . — 2cA, 
and when the rotations are in equilibrium then the displacements (Sic, 8y, 8z) of any 
point (x, y, z) whatever must each of them vanish; that is, we must have 
1,\a = 0, 2A b = 0, 2 Ac = 0, 2 A/ = 0, 2Ag = 0, 2 a h — 0, 
which are therefore the conditions for the equilibrium of the system of rotations 
Aj, A2, &c* 
63. And it further appears that for a system of forces acting on a rigid body, 
A,j along the line (a lt b lt c u f lt g lt h), 
A 2 „ (Q'2j 82, c 2 , fo, g-i, h), 
&c. 
the conditions of equilibrium as given by the Principle of Virtual Velocities is 
2A (ap + bq L cr+fl+gm + hn) = 0, 
or what is the same thing, that we have 
2Xa = 0, 2a6 = 0, 2Ac = 0, 2A/=0, 2x^ = 0, 2XA = 0, 
for the conditions of equilibrium of the system of forces X x , X 2 , &c. The conditions 
of equilibrium are thus precisely the same in the case of a system of rotations 
(infinitesimal rotations) and in that of a system of forces. 
64. It now appears that the greater portion of the investigations in the first 
part of the present paper are applicable, and may be considered as 1 elating, to. the 
equilibrium of forces (or of rotations; but as the two theories are identical, it. is 
sufficient to attend to one of them), and that we have in effect solved the following 
12—2