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point D' of E B'. Now, it will be readily conceivedthat since the moon when at B is at her mean distance,she is travelling nearly at her mean rate in theneighbourhood of this point (her orbit being nearlycircular in shape), so that at M 2 she is no longergetting in advance of her mean place, and has, there-fore, attained her maximum displacement in advance.In like manner, when she is at M 4 , she has attained(approximately) her maximum displacement behindher mean place. And it is very easy to find theeffects (necessarily maximum effects, at the pointsM 2 and MJ of the non-accordance between the motionsof rotation and revolution. If the moon swept at auniform rate round the point E, she would be at P andP' at the times when, in reality, she is at M 2 and M,(P E P' being drawn at right angles to C E). This isobvious, since the four angles P E M„ P E M 3 , P'E M 3 ,and P' E M : are all equal, and the moon occupiesequal times in going from M 4 to M 2 , thence to M s ,thence to M 4 , and thence, finally, to M x again.* So
* In fact, we are assuming that P E M 2 , P' E M„ represent themaximum values of the difference between the true and the meananomaly, or, with ordinary notation, the maximum values of {9-nfyNow it is obvious that the circular measure of the angle P E M, is
2 0 E
very nearly represented by or ^y 2 e. Hence we are assuming
that the maximum value of 6 - n t is very nearly equal to 2 e. Inreality, this value is represented by an infinite series, beginning
2
11c 3
e+ 3^ +
599 c 35.2 : »
&c.
For the mean value of the lunar eccentricity, the term involving damounts only to 0'00003792, or less than the 2,895th part of 2 c.