Thus in the case of fig. 36 it may be required to connect the infinitesimal rotations [xi], [eta], [zeta] about OA, OB, OC with the variations of the angular co-ordinates [theta], [psi], [phi]. The displacement of the point C of the body is made up of [delta][theta] tangential to the meridian ZC and sin [theta] [delta][psi] perpendicular to the plane of this meridian. Hence, resolving along the tangents to the arcs BC, CA, respectively, we have [xi] = [delta][theta] sin [phi] - sin [theta] [delta][psi] cos [phi], [eta] = [delta][theta] cos [phi] + sin [theta] [delta][psi] sin [phi]. (3) Again, consider the point of the solid which was initially at A´ in the figure. This is displaced relatively to A´ through a space [delta][psi] perpendicular to the plane of the meridian, whilst A´ itself is displaced through a space cos [theta] [delta][psi] in the same direction. Hence [zeta] = [delta][phi] + cos [theta] [delta][psi]. (4) [Illustration: FIG. 40.] To find the component displacements of a point P of the body, whose co-ordinates are x, y, z, we draw PL normal to the plane yOz, and LH, LK perpendicular to Oy, Oz, respectively. The displacement of P parallel to Ox is the same as that of L, which is made up of [eta]z and -[zeta]y. In this way we obtain the formulae [delta]x = [eta]z - [zeta]y, [delta]y = [zeta]x - [xi]z, [delta]z = [xi]y - [eta]x. (5) The most general case is derived from this by adding the component displacements [lambda], [mu], [nu] (say) of the point which was at O; thus [delta]x = [lambda] + [eta]z - [zeta]y, \ [delta]y = [mu] + [zeta]x - [xi]z, > (6) [delta]z = [nu] + [xi]y - [eta]x. / The displacement is thus expressed in terms of the six independent quantities [xi], [eta], [zeta], [lambda], [mu], [nu]. The points whose displacements are in the direction of the resultant axis of rotation are determined by [delta]x:[delta]y:[delta]z = [xi]:[eta]:[zeta], or ([lambda] + [eta]z - [zeta]y)/([xi] = [mu] + [zeta]x - [xi]z)/[eta] = ([nu] + [xi]y - [eta]x)/[zeta]. (7) These are the equations of a straight line, and the displacement is in fact equivalent to a twist about a screw having this line as axis. The translation parallel to this axis is l[delta]x + m[delta]y + n[delta]z = ([lambda][xi] + [mu][eta] + [nu][zeta])/[epsilon]. (8) The linear magnitude which measures the ratio of translation to rotation in a screw is called the _pitch_. In the present case the pitch is ([lambda][xi] + [mu][eta] + [nu][zeta])/([xi]² + [eta]² + [zeta]²). (9) Since [xi]² + [eta]² + [zeta]², or [epsilon]², is necessarily an absolute invariant for all transformations of the (rectangular) co-ordinate axes, we infer that [lambda][xi] + [mu][eta] + [nu][zeta] is also an absolute invariant. When the latter invariant, but not the former, vanishes, the displacement is equivalent to a pure rotation.