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 P_{1}Q_{1} 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 PP_{1} and QQ_{1}, and perpendicular to those lines.
The point of intersection is then the required pole O, because the
spherical triangles OPQ and OP_{1}Q_{1} 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 O_{1}, another, a third pole
O_{2}, 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.

### §11.

Conic Rolling.

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