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seen that each loop springs from one of the points Mi,M 2 , M 3 , M 4 (where the tangential force vanishes, andthe radial forces have their unequal maxima), andbends round so as to end at another of those fourpoints; and we see that at the four points m v m 2 , m s ,m 4 (midway between the former, and not far fromthose where the radial force vanishes) the tangentialforce has its maximum value.*
Now as the moon is passing over the arc M 4 M 2 , thetangential force, acting in the direction shown by thecurves, is retardative, and most effectually so when themoon is in the middle of this arc, or at the point %.As the moon passes from M 2 to M s , the tangentialforce is accelerative, and most effectually so when themoon is at the middle of the arc M 2 M 3 , or at the pointm 2 . As the moon passes over the arc M S M 4 , thetangential force is again retardative; and it is againaccelerative as the moon traverses the arc M 4 M uattaining its greatest value when the moon is at themiddle of these respective arcs, or at m % and m 4 .Since, then, retardation ceases to act when the moonis at M 2 , the moon is moving there with minimumvelocity, so far as this cause of disturbance is con-cerned. In like manner the moon is moving withmaximum velocity at M s , with minimum velocity at
* The tangential force attains its maximum midway betweenthe points M„ M 2 , M„, M 4 , and not, as is sometimes stated, at thepoints where the radial force vanishes. It will be obvious fromfig. 20 that if we call the angle H E M, 0, we have H B=3 cos 9sin B = £ sin 2 9 ; and this expression has its greatest value whensin 20 = 1, or 0=45°.