Let us now suppose a system to receive at the game time two 'notions, round two different axes of repose : that is to say, given two different motions, required the motion which will result from the two motions impressed on the system at once. There will be at the first instant an instantaneous axis of repose, which it is required to find. First let the two axes pass through the same point A (Fig. 1), and choose the angle n A c out of the four angles made by the two axes. in such manner that points of the system hying in the angle n A would be elevated by the rotation round B a, and depressed by that round c A, or rice versa. On the axes take A 13 and a c, lines propor tional to the angular velocities about those axes, complete the paral lelogram A n, and draw the diagonal A O. Then A n is the axis of repose at starting (which however it may not continue to be), and A n represents the angular velocity round that axis at starting, in the same manner as A n and A o represent the impressed angular velocities about AB and A C. [CoMroSMON.] Next let the axes be parallel to one another, say perpendicular to the plane of the paper, passing through A and n (Fig. 2). If the rotations be such that A and n would both rise, or both fall, on the paper, each by the rotation about the other, take a point c in ma A produced, nearest to the axis about which the angular velocity is greatest (say that of A), and such that c A is to c 13 as the angular velocity about 13 to the angular velocity about A. Then the axis of repose at starting is a line passing through c parallel to thelormer axes, and the angular velocity is the difference of the angular velocities about A and 13, and in the direction of the greater. In this ease the directions of the rotations about A and 13 [Dineceiost OF MOTION] are different. There is ono remarkable case, namely, when the rotations about A and B are equal. In this case the rule would lead us to a rotation equal to nothing made about a point at an infinite distance—one of extreme con clusions which require interpretation. The fact is that these two rotations give only a shnple motion of translation = A 13 X Angular Velocity per second, and such as to make the system move upwards or downwards on the paper according as the separate rotations would make the points A and 13 move upwards or downwards. This parti
cular case will be more intelligible when looked at with the help of the THEORY OF CourLEs.
But if the rotations be in the same direction, so that A will be lowered and B raised, or rice versa, each by the rotation about the other :—Take a point n, dividing A B so that A n is to n B as the angular velocity about n is to that about A. Then will the axis of repose at starting be a parallel drawn through n to the axes passing through A and B, and the angular velocity will be the sum of the angular velocities about A and n, its direction being that which lowers A on the paper and raises B, or eke versa, according as is done by the given angular velocities.
Lastly, let the axes be neither wand nor intersecting (Fig. 3), as n and c D :—Through the point m, in which c n meets the common perpendicular, me, draw R P parallel to A n, and at the instant at which the rotations round A n and c D commence, impress two equal and contrary rotations about r F, each equal to that about A B. These produce no effect, so that the composition of the four rotations given the same result as that of the two. Now, as above stated, the rotation round A s, and its equal and contrary round E 1', produce nothing but a motion of translation, while the remaining rotation about E F, com pounded by the first rule with that about c D, gives what would be an axis of repose, if it were not for that translation. The whole result then is, that the system begins to move about an axis, which axis begins to undergo a translation in space.
For higher theoretical investigations in this subject, the reader is referred to the writings of N. Poinsot. The best practical view will be derived from experiments with the GYROSCOPE. See also an article on the Gyroscope by Mr. John Bridge in the 'Philosophical Magazine' for November, 1557.