A MOVING form or rigid mass, Under whate'er conditions, Along successive screws must pass Between each two positions. It turns around and slides along— This is the burden of my song. The pitch of screw, if multiplied By angle of rotation, Will give the distance it must glide In motion of translation. Infinite pitch means pure translation, And zero pitch means pure rotation. Two motions on two given screws, With amplitudes at pleasure, Into a third screw-motion fuse; Whose amplitude we measure By parallelogram construction (A very obvious deduction). Its axis cuts the nodal line Which to both screws is normal, And generates a form divine, Whose name, in language formal, Is “surface-ruled of third degree.” Cylindroid is the name for me. Rotation round a given line Is like a force along. If to say couple you incline, You're clearly in the wrong;— ’Tis obvious, upon reflection, A line is not a mere direction. So couples with translations too In ail respects agree; And thus there centres in the screw A wondrous harmony Of Kinematics and of Statics,– The sweetest thing in mathematics. The forces on one given screw, With motion on a second, In general some work will do, Whose magnitude is reckoned By angle, force, and what we call The coefficient virtual. Rotation now to force convert, And force into rotation; Unchanged the work, we can assert, In spite of transformation. And if two screws no work can claim, Reciprocal will be their name. Five numbers will a screw define, A screwing motion, six; For four will give the axial line, One more the pitch will fix; And hence we always can contrive One screw reciprocal to five. Screws-two, three, four, or five, combined (No question here of sex), Yield other screws which are confined Within one screw complex. Thus we obtain the clearest notion Of freedom and constraint of motion. In complex III. three several screws At every point you find, Or if you one direction choose, One screw is to your mind; And complexes of order III. Their own reciprocals may be. In IV., wherever you arrive, You find of screws a cone. On every line in complex V. There is precisely one; At each point of this complex rich, A plane of screws have given pitch. But time would fail me to discourse Of Order and Degree, Of Impulse, Energy, and Force, And Reciprocity. All these and more, for motions small, Have been discussed by Dr. Ball.