If the small displacements of a rigid body be subject to one constraint, e.g. if a point of the body be restricted to lie on a given surface, the mathematical expression of this fact leads to a homogeneous linear equation between the infinitesimals [xi], [eta], [zeta], [lambda], [mu], [nu], say A[xi] + B[eta] + C[zeta] + F[lambda] + G[mu] + H[nu] = 0. (10) The quantities [xi], [eta], [zeta], [lambda], [mu], [nu] are no longer independent, and the body has now only five degrees of freedom. Every additional constraint introduces an additional equation of the type (10) and reduces the number of degrees of freedom by one. In Sir R. S. Ball's _Theory of Screws_ an analysis is made of the possible displacements of a body which has respectively two, three, four, five degrees of freedom. We will briefly notice the case of two degrees, which involves an interesting generalization of the method (already explained) of compounding rotations about intersecting axes. We assume that the body receives arbitrary twists about two given screws, and it is required to determine the character of the resultant displacement. We examine first the case where the axes of the two screws are at right angles and intersect. We take these as axes of x and y; then if [xi], [eta] be the component rotations about them, we have [lambda] = h[xi], [mu] = k[eta], [nu] = 0, (11) where h, k, are the pitches of the two given screws. The equations (7) of the axis of the resultant screw then reduce to x/[xi] = y/[eta], z([xi]² + [eta]²) = (k - h)[xi][eta]. (12) Hence, whatever the ratio [xi] : [eta], the axis of the resultant screw lies on the conoidal surface z(x² + y²) = cxy, (13) where c = ½(k - h). The co-ordinates of any point on (13) may be written x = r cos [theta], y = r sin [theta], z = c sin 2[theta]; (14) hence if we imagine a curve of sines to be traced on a circular cylinder so that the circumference just includes two complete undulations, a straight line cutting the axis of the cylinder at right angles and meeting this curve will generate the surface. This is called a _cylindroid_. Again, the pitch of the resultant screw is p = ([lambda][xi] + [mu][eta])/([xi]² + [eta]²) = h cos² [theta] + k sin² [theta]. (15) [Illustration: From Sir Robert S. Ball's _Theory of Screws_. FIG. 41.] The distribution of pitch among the various screws has therefore a simple relation to the _pitch-conic_ hx² + ky² = const; (16)