are known. Thus on investigation we see that the motion of the figure about a fixed point reduces itself simply to that of a circular arc, PQ, upon a sphere, and, generally, that we may determine the motion of any spheric figure by that of any arc lying within it, just as we have already found that in con-plane motions we could replace a figure by a line.
Every spheric figure, as PQ in Fig. 30, which moves upon the surface of a sphere, can be moved from one position PQ into another P1Q1 through spheric turning about some point on the surface of the sphere; and this point can be determined by finding the intersection of two great circles passing through the centres of the lines PP1 and QQ1, and perpendicular to those lines. The point of intersection is then the required pole O, because the spherical triangles OPQ and OP1Q1 are similar and equal-sided. The point O is the temporary centre for the supposed spherical turning. The two great circles cut each other twice, once at O, and once at the other end of the diameter passing through O.
But by hypothesis the distance of the figure PQ from the fixed point A is constant, and therefore the diameter passing through O and A is stationary during the turning relatively to the figure, and so becomes the temporary axis of the assumed motion.
A new turning supplies a second pole O1, another, a third pole O2, and so on, and by joining these with arcs of great circles we have a spheric polar polygon. A second spheric polar polygon, rigidly connected with the moving figure, corresponds to the first. If a series of straight lines be drawn passing through the corners of these polygons and the fixed point A, − that is, a series of diameters of the sphere passing through these corners, − we obtain two pyramids, about the angles of which the separate turnings take place.
It will be seen that the method we have here used bears the most complete analogy to that employed in the consideration of motions in a plane. If we continue the process further by