The turning about two different lines is impossible in three-dimensional space. To take another illustration, suppose A and B are two parallel lines in the xy plane, and let CD and EF be two rods crossing them. Now, in the space of xyz if the rods turn round the lines A and B in the same direction they will make two independent circles.
Fig. 9 (137).
When the end F is going down the end C will be coming up. They will meet and conflict.
But if we rotate the rods about the plane of AB by the z to w rotation these movements will not conflict. Suppose all the figure removed with the exception of the plane xz, and from this plane draw the axis of w, so that we are looking at the space of xzw.
Here, [fig. 10], we cannot see the lines A and B. We see the points G and H, in which A and B intercept the x axis, but we cannot see the lines themselves, for they run in the y direction, and that is not in our drawing.
Now, if the rods move with the z to w rotation they will turn in parallel planes, keeping their relative positions. The point D, for instance, will describe a circle. At one time it will be above the line A, at another time below it. Hence it rotates round A.
Fig. 10 (138).
Not only two rods but any number of rods crossing the plane will move round it harmoniously. We can think of this rotation by supposing the rods standing up from one line to move round that line and remembering that it is not inconsistent with this rotation for the rods standing up along another line also to move round it, the relative positions of all the rods being preserved. Now, if the rods are thick together, they may represent a disk of matter, and we see that a disk of matter can rotate round a central plane.