The explanation of our rule is easy. Here is a gyrostat (Fig. 23) something like the earth in shape, and it is at rest. I am sorry to say that I am compelled to support this globe in a very visible manner by gymbal rings. If this globe were just floating in the air, if it had no tendency to fall, my explanation would be easier to understand, and I could illustrate it better experimentally. Observe the point P. If I move the globe slightly about the axis A, the point P moves to Q. But suppose instead of this that the globe and inner gymbal {51} ring had been moved about the axis B; the point P would have moved to R. Well, suppose both those rotations took place simultaneously. You all know that the point P would move neither to Q nor to R, but it would move to S; P S being the diagonal of the little parallelogram. The resultant motion then is neither about the axis O A in space, nor about the axis O B, but it is about some such axis as O C. To this globe I have given two rotations simultaneously. Suppose a little being to exist on this globe which could not see the gymbals, but was able to observe other objects in the room. It would say that the direction of rotation is neither about O A nor about O B, but that the real axis of its earth is some line intermediate, O C in fact. If then a ball is suddenly struck in two different directions at the same instant, to understand how it will spin we must first find how much spin each blow would produce if it acted alone, and about what axis. A spin of three turns per second about the axis O A (Fig. 24), and a spin of two turns per second about the axis O B, really mean that the ball will spin about the axis O C with a spin of three and a half turns per second. To arrive at this result, I made O A, 3 feet long (any other scale of representation would have been right) {52} and O B, 2 feet long, and I found the diagonal O C of the parallelogram shown on the figure to be 3½ feet long. Observe that if the rotation about the axis O A is _with_ the hands of a watch looking from O to A, the rotation about the axis O B looking from O to B, must also be with the hands of a watch, and the resultant rotation about the axis O C is also in a direction with the hands of a watch looking from O to C. Fig. 25 shows in two diagrams how necessary it is that on looking from O along either O A or O B, the rotation should be in the same direction as regards the hands of a watch. These constructions are well known to all who have studied elementary mechanical principles. Obviously if the rotation about O A is very much greater than the rotation about O B, then the position of the new axis O C must be much nearer O A than O B. [Illustration: FIG. 24.] [Illustration: FIG. 25.]