BODY BY THE AID OE THE THEOEY OE SCEEWS. 
35 
line in the plane passing through the point with zero pitch is a coordinate screw. This 
expresses the freedom enjoyed by S. 
Finally, if the body had only three degrees of freedom, the four equilibrating forces 
P, Q, E, S may be situated anywhere. 
The positions of the forces being given, their magnitudes are determined ; for draw 
X 15 X 2 , X 3 reciprocal to the freedom, and find the seven equilibrating wrenches about 
P, Q, E, S, X„ X 2 , X 3 . 
The last three are neutralized by the reactions of the constraints, and the four former 
must therefore equilibrate. 
40. Equilibrium of five forces applied to a rigid body. — The five forces must, if the 
body be free, form a coordinate system of four degrees of freedom. Draw the cylindroid 
reciprocal to the coordinate system of freedom. The five forces must therefore intersect 
both the screws of zero pitch on the cylindroid. We therefore have as a necessary 
condition that two straight lines can be drawn which intersect all the five forces. Four 
of the forces will determine the two lines, and therefore the fifth force may enjoy any 
liberty consistent with the requirement that it also intersects the two lines. This con- 
dition is also a sufficient one, so far as the positions of the forces are concerned. 
If P, Q, E, S, T be the five forces, the ratio of P : Q is thus determined. 
Let A, B be the two screws of zero pitch upon the cylindroid. 
Let X, Y be two screws reciprocal to P, Q. 
Let Z be a screw reciprocal to E, S, T. 
Construct the screw I reciprocal to the five screws 
X, Y, A, B, Z. 
Now the four screws X, Y, A, B are reciprocal to the cylindroid (P, Q) ; therefore I? 
which is reciprocal to X, Y, A, B, must lie upon the cylindroid P, Q. 
Since A, B, Z are all reciprocal to E, S, T, it follows that I, being reciprocal to 
A, B, Z, must be coordinate with E, S, T. 
Hence I is coordinate with P, Q and also with E, S, T. If, therefore, forces along 
P, Q, E, S, T equilibrate, the forces along P, Q must compound into a wrench about I. 
This condition determines the forces along P, Q. 
41. Equilibrium of six forces applied to a rigid body. — Professor Sylvestee has shown 
(‘Comptes Eendus,’ tome lii. p. 816) that when the six lines P, Q, E, S, T, U are so 
situated that forces acting along these lines equilibrate when applied to & free rigid body, 
a certain determinant must vanish. 
Using the ideas and language of the theory of screws, this determinant is simply the 
sexiant of the six lines, the pitches of course being zero. 
If x m , y m . , z m be a point on one of the lines, the direction cosines of the same line being 
a m , j3 m , 7m> the condition is therefore 
F 
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