arc PQ of a circle having its centre on the line OB, and thus be brought to the position Q. Again, by a rotation about OC, Q may be made to describe the arc QR of a circle, having its centre on the line OC, and be brought to the position R. Thus, by two properly chosen steps, P to Q and Q to R, P is brought to R; and by giving the rotations about OB, OC, proper degrees of rapidity, each of the two steps may be taken in the same time as was the single original step, P to R. Suppose, now, that the two rotations, which we have hitherto supposed to be made one after the other, are made simultaneously: then the steps will, as it were, be taken together and in the same time as the single step; and P will, under the combined influence of the two rotations, be brought to the same position as it would be brought to under the influence of the single rotation. Thus, two rotations, about OB, OC, of proper degrees of rapidity, may be regarded as producing the same effect on every point of the body as does the single rotation about OA.
Now, if Q, Figure 66, be a place on the Equator, the earth has no motion of rotation about A'QE; and consequently the plane of vibration, having itself no motion of rotation, will appear to preserve a fixed position relatively to an observer at Q. Again, let NOS, Figure 68, represent the earth's axis; let O be the centre of the earth and P a place on its surface; and let OR be a line perpendicular to OP and in the plane NPS, or the plane of P's meridian. Then, instead of supposing the earth's motion to consist of a single rotation about NOS, we may regard it as arising from two properly chosen rotations about OP and OR; and we may consider separately what would be the apparent
