It is convenient to distinguish the two senses in which rotation may take place about an axis OA by opposite signs. We shall reckon a rotation as positive when it is related to the direction from O to A as the direction of rotation is related to that of translation in a right-handed screw. Thus a negative rotation about OA may be regarded as a positive rotation about OA´, the prolongation of AO. Now suppose that a body receives first a positive rotation [alpha] about OA, and secondly a positive rotation [beta] about OB; and let A, B be the intersections of these axes with a sphere described about O as centre. If we construct the spherical triangles ABC, ABC´ (fig. 38), having in each case the angles at A and B equal to ½[alpha] and ½[beta] respectively, it is evident that the first rotation will bring a point from C to C´ and that the second will bring it back to C; the result is therefore equivalent to a rotation about OC. We note also that if the given rotations had been effected in the inverse order, the axis of the resultant rotation would have been OC´, so that finite rotations do not obey the "commutative law." To find the angle of the equivalent rotation, in the actual case, suppose that the second rotation (about OB) brings a point from A to A´. The spherical triangles ABC, A´BC (fig. 39) are "symmetrically equal," and the angle of the resultant rotation, viz. ACA´, is 2[pi] - 2C. This is equivalent to a negative rotation 2C about OC, whence the theorem that the effect of three successive positive rotations 2A, 2B, 2C about OA, OB, OC, respectively, is to leave the body in its original position, provided the circuit ABC is left-handed as seen from O. This theorem is due to O. Rodrigues (1840). The composition of finite rotations about parallel axes is a particular case of the preceding; the radius of the sphere is now infinite, and the triangles are plane. In any continuous motion of a solid about a fixed point O, the limiting position of the axis of the rotation by which the body can be brought from any one of its positions to a consecutive one is called the _instantaneous axis_. This axis traces out a certain cone in the body, and a certain cone in space, and the continuous motion in question may be represented as consisting in a rolling of the former cone on the latter. The proof is similar to that of the corresponding theorem of plane kinematics (§ 3).