THE MOON’S MOTIONS.

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acting inwards on the moon when traversing herperigeal arc O s p 0 4 . For the former forces exceedthe latter (see fig. 35), and act daring a longer time,owing to the moon’s slower motion near her apogee.Hence, so far as the radial force is concerned, there isa balance of perigeal recession in a complete lunarrevolution when the orbit is placed as in fig. 35. Asregards the tangential force, there is retardation asthe moon moves from M 4 to a, and therefore (see fig. 33)the perigee recedes. In moving from a to M 3 , themoon is accelerated, and therefore (see fig. 32) theperigee still recedes > thence to p there is retardation,and (see fig. 33) the perigee advances; and, lastly, inmoving to M x , the moon is accelerated, and therefore(see fig. 32) the perigee still advances. But therecession over the apogeal arc a M s is greater thanthe advance over the perigeal arc M 3 p M x , for likereasons to those urged in the corresponding case inthe preceding position. Hence, on the whole, thereis a balance of perigeal recession due to the tangentialforce. But we have seen that there is also a balanceof perigeal recession due to the radial force. Hence,there is a total balance of perigeal recession when themoon’s orbit is placed as shown in fig. 35.

Obviously the same result would have been obtained!f the major axis had been placed as in fig. 35, butw ith the positions of the perigee and apogee inter-changed.

•Next, let us take such intermediate cases as are'Uustrated in figs. 36 and 37, where p and a, as in the