OF ASPECT, .ROTATION, AND LIBRATION. 177

the order indicated by the numbers on the centralshort line.

But now let us take into consideration the effect ofthe want of perfect accordance between the moon’smotions of revolution and rotation. She rotates uni-formly on her axis, or very nearly So, while she moveswith varying velocity round the earth. But fig. 56,Plate XIII., shows, that in order that the same faceshould always be shown, there should be perfectagreement between the motions of rotation andrevolution.

Let MjMjMgM* (fig. 70, Plate XV.) be the lunarorbit about E, the earth, being the perigee ofthe orbit and C its centre, the shape of the orbit (seenote, p. 27) being appreciably circular. Then themoon moves more rapidly over the arc Mj B than overthe arc B M s ; and therefore, at the end of a quarterof a revolution she is not at B, but in advance of thatpoint. To find how much, we have only to considerthat in a quarter of a revolution a line from the earthto the moon sweeps over a quarter of the areaMjMgMgMg. Hence, if M 2 be the moon’s place atthe end of a quarter of a revolution, the area M 2 E Mjis equal to the quarter B C M, of the complete orbit.So that the triangle C D E must be equal to the spaceB D M 2 . This will obviously be very nearly the case ifD is the middle point of the line C B (see further thenote on p. 107). And in like manner, we obtain thepoint M 4 , reached by the moon after three-quarters ofa revolution, by drawing E D'M 4 through the middle