BODY BY THE AID OE THE THEOEY OP SCREWS. 
37 
through the corresponding screw, then the two screws of zero pitch on the cylindroid 
are a pair of axes that fulfil the required conditions. If one of these axes were given, 
the cylindroid would be defined and the other axis would he determinate. 
42. On a property of a body possessing four degrees of freedom. — When a body has 
four degrees of freedom it is able to twist about every screw in space reciprocal to a 
certain cylindroid. Thus through any point a cone can be drawn the generators of 
which are perpendicular to the generators of a cylindroid ; and by assigning the proper 
pitch to each of the cone generators we have a reciprocal screw [art. 89], If, however, 
the point had been selected upon the double line of the cylindroid, the cone vanishes 
into a line and the pitch becomes indeterminate, thus giving the following general 
theorem in kinematics : — 
When a body has four degrees of freedom, there is always one screw about which the 
body can be twisted whatever be the pitch assigned to that screw. 
43. A point P is always to be moved in a plane A, determine the “ locus ” of screws 
about which P may receive small twists. — If the point P formed a portion of a rigid body' 
M, the condition imposed on P would still leave M five degrees of freedom. There is 
therefore one screw, S, reciprocal to the freedom ; S is to be found by the condition that 
a wrench about S shall be unable to disturb M. The only wrench which would not 
disturb M must be a force through P normal to A. The reciprocal screw S must 
therefore lie along this normal and have zero pitch. Any screw reciprocal to S will 
fulfil the required condition. 
44. A point P is always to move along a line AB, determine the “locus” of screws 
about which P may receive small twists. — A rigid body attached to P would have four 
degrees of freedom. The reciprocal cylindroid in this case reduces to the plane through 
P normal to AB. All the screws in this plane pass through P and have zero pitch. 
The “ locus ” required is evidently that of screws reciprocal to the cylindroid which 
has assumed this simple type. 
45. Generalization of a theorem due to M. Chasles. — If a system of forces be resolved 
into two forces, the volume of the tetrahedron of which the two forces are opposite edges 
is constant. This may be generalized into the following : — If a system of forces be 
resolved into wrenches about two screws of equal pitch, the volume of the tetrahedron 
of which the" wrenching forces are opposite edges is constant. 
Let 0„ 0 2 , 6 3 be three cocylindroidal screws about which wrenches X, Y, Z equilibrate. 
The volume of the tetrahedron formed by X, Y is 
\m (sin 20 2 — sin 24J sin (6 x — 6 2 ) XY ; 
but 
X Y Z 
sin (0 2 — y sm (0 3 — 0j) sin (®i — 0 2 ) ’ 
thence the volume reduces to 
-fmZ 2 cos (0i -f0 2 ) sin (0 2 — 0 3 ) sin (0 3 — 0J. 
