leaf 1 of the pinion will again engage the tooth 1 of the wheel. Thus it is evident, that in the case here supposed the leaf 1 of the pinion will continually be engaged with the teeth 1, 11, 21, 31, 41, and 51 of the wheel, and no others. The like may be said of every leaf of the pinion. Thus the leaf 2 of the pinion will be successively engaged with the teeth 2, 12, 22, 32, 42, and 52 of the wheel, and no others. Any accidental inequalities of these teeth will therefore continually act upon each other, until the circumference of the wheel be divided into parts of ten teeth each, unequally worn. This effect would be avoided by giving either the wheel or pinion one tooth more or one tooth less. Thus, suppose the wheel, instead of having sixty teeth, had sixty-one, then after six revolutions of the pinion the leaf 1 of the pinion would be engaged with the tooth 61 of the wheel; and after one revolution of the wheel, the leaf 2 of the pinion would be engaged with the tooth 1 of the wheel. Thus, during the first revolution of the wheel the leaf 1 of the pinion would be successively engaged with the teeth 1, 11, 21, 31, 41, 51, and 61 of the wheel: at the commencement of the second revolution of the wheel the leaf 2 of the pinion would be engaged with the tooth 1 of the wheel; and during the second revolution of the wheel the leaf 1 of the pinion would be successively engaged with the teeth 10, 20, 30, 40, 50, and 60 of the wheel. In the same manner it may be shown, that in the third revolution of the wheel the leaf 1 of the pinion would be successively engaged with the teeth 9, 19, 29, 39, 49, and 59 of the wheel: during the fourth revolution of the wheel the leaf 1 of the pinion would be successively engaged with the teeth 8, 18, 28, 38, 48, and 58 of the wheel. By continuing this reasoning it will appear, that during the tenth revolution of the wheel the leaf 1 of the pinion will be engaged successively with the teeth 2, 12, 22, 32, 42, and 52 of the wheel. At the commencement of the eleventh revolution of the wheel the leaf 1 of the pinion will be engaged with the tooth 1 of the wheel, as at the beginning of the motion. It is evident, therefore, that during the first ten revolutions of the wheel each leaf of the pinion has been successively engaged with every tooth of the wheel, and that during these ten revolutions the pinion has revolved sixty-one times. Thus the leaves of the pinion have acted six hundred and ten times upon the teeth of the wheel, before two teeth can have acted twice upon each other. The odd tooth which produces this effect is called by millwrights the _hunting cog_.