A RIGID BODY IN ELLIPTIC SPACE. 
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and the couple about it is 
P (af H- bg + ch' +fa + gb' + ch'). 
42. All the theorems about compounding linear and rotational velocities apply to 
systems of forces and couples. Thus any system of forces may be reduced to two, 
acting along lines which are conjugate to each other with respect to a certain linear 
complex. Any system of forces may be reduced to a wrench about a certain screw. 
We can now give an interpretation to the simultaneous invariant of two screws. 
For suppose LI is a twist about a screw a, b, c, . . . , and Q a wrench about another 
screw a', b', c, . . . Then Ha, fib, . . . are angular velocities about the six edges of 
the tetrahedron of reference, and Qa', Qb', . . . are couples about them. Hence if the 
body receive a small displacement about the first screw, while the wrench Q about the 
other screw is acting on the body, the rate of doing work will be 
flQ (af'+bg'+ch'-f-fa'+gb' +he') 
and therefore the simultaneous invariant of two screws is the rate of doing work, 
when the body has a unit twist about one screw, while a unit wrench on the other 
screw is acting on the body. 
43. The condition of equilibrium of any number of wrenches on given screws is the 
same as the condition that a number of twists about the screws should neutralise each 
other. If the wrenches reduce to couples or forces we have only to make (0,0) = 0, 
(1,1) = 0, . . . The conditions, therefore, for the equilibrium of any number of forces 
P 0 , P l5 . . . acting along given lines are, first, that the determinant 
(0,1) (0,2) ... 
( 1 , 0 ) • ( 1 , 2 ) ... 
(2,0) (2,1) . ... 
should vanish, and further, that the forces should be proportional to the square roots 
of the minors of the terms of the leading diagonal of this determinant. 
44. Since P —vnf, if we resolve the force P into the components X x , X 2 ,. . . X 6 along 
the edges of the fixed quadrantal tetrahedron, and if v x , v 2 , . .. v G be the components of 
f along these edges, we have 
mVi=Xi (i= 1,2,3,4,5,6). 
Applying these equations to all the particles of the body, and using D’Alembert’s 
principle, the equations of motion of the body become 
2mVi=2Xi (^= 1 , 2 , 3 , 4 , 5 , 6 ). 
