4. _Study of the crab-like behaviour of a spinning body_, 30-49. Illustrations: Spinning top, 31; cased gyrostat, 32; balanced gyrostat, 33-36; windage of projectiles from {134} rifled guns, 36-38; tilting a hoop or bicycle, turning quickly on horseback, 38; bowls, 39; how to simplify one's observations, 39, 40; the illustration which gives us our simple universal rule, 40-42; testing the rule, 42-44; explanation of precession of gyrostat, 44, 45; precession of common top, 46; precession of overhung top, 46; list of our results given in a wall sheet, 48, 49. 5. _Proof or explanation of our simple universal rule_, 50-54. Giving two independent rotations to a body, 50, 51; composition of rotations, 52, 53. 6. _Warning that the rule is not, after all, so simple_, 54-66. Two independent spins given to the earth, 54; centrifugal force, 55; balancing of quick speed machinery, 56, 57; the possible wobbling of the earth, 58; the three principal axes of a body, 59; the free spinning of discs, cones, rods, rings of chain, 60; nodding motion of a gyrostat, 62; of a top, 63; parenthesis about inaccuracy of statement and Rankine's rhyme, 63, 64; further complications in gyrostatic behaviour, 64; strange elastic, jelly-like behaviour, 65; gyrostat on stilts, 66. 7. _Why a gyrostat falls_, 66, 67. 8. _Why a top rises_, 67-74. General ignorance, 67; Thomson preparing for the mathematical tripos, 68; behaviour of a water-worn stone when spun on a table, 68, 69; parenthesis on technical education, 70; simple explanation of why a top rises, 70-73; behaviour of heterogeneous sphere when spun, 74. 9. _Precessional motion of the earth_, 74-91. Its nature and effects on climate, 75-80; resemblance of the precessing earth to certain models, 80-82; tilting forces exerted by the sun and moon on the {135} earth, 82-84; how the earth's precessional motion is always altering, 85-88; the retrogression of the moon's nodes is itself another example, 88, 89; an exact statement made and a sort of apology for making it, 90, 91. 10. _Influence of possible internal fluidity of the earth on its precessional motion_, 91-98. Effect of fluids and sand in tumblers, 91-93; three tests of the internal rigidity of an egg, that is, of its being a boiled egg, 93, 94; quasi-rigidity of fluids due to rapid motion, forgotten in original argument, 95; beautiful behaviour of hollow top filled with water, 95; striking contrasts in the behaviour of two tops which are very much alike, 97, 98; fourth test of a boiled egg, 98. 11. Apology for dwelling further upon astronomical matters, and impertinent remarks about astronomers, 99-101.