Secular Acceleration of the Moon's Mean Motion. 329 



the mean of the reciprocals of the cubes of the sun's distance 

 throughout the year, is to the reciprocal of the cube of the mean 

 distance, as the actual diminution in the moon's gravity from the 

 sun's disturbing influence, is to the diminution if the earth re- 

 volved in a circle at the same mean distance. To find the value 

 of the first term of this proportion, let ADBE (fig. 2) represent 

 the earth's elliptical orbit, and S the sun placed in one of the foci. 

 Draw any two conjugate diameters ED and HN, and join SH, 

 SE, SN and SD. Also from the other focus draw FE and FN. 



Let the semi-transverse axis Fig. 2. 



=«. 



Let the semi-conjugate axis 

 = h. 

 Let the eccentricity =e. 



" SE=a + ar. 



" SDorEF=a-a:. 



" SN=a4-y. 



" SHorNF=a-y. 



1 1 



Then 



And 



SE^ 

 1 



'{a+x) 

 1 



SD^~(a — a;)^ a-' ■ «" ■ a" ' a' 

 By adding the two series together, and omitting all terms ex- 

 cept those written down on account of their smallness, we have 

 112 12a:2 



+ 



In like manner 



SE = 

 1 



SD 

 1 



a^ 



2 



a' 



Hence 



7+^ 



SN = 

 1 



SH = 



4 Vlix-'-Vy^) 



SE3'SD=» 

 value of the four is 



1 3(ar^-fy^) 



-+ 



and the mean 



«•* a" 



Now by the properties of the ellipse, CH^=SExEF=(a-fa;) 

 y.{a — x)=a'^ — X- . 

 And in like manner, CE-=a'' —y"^. 

 Therefore CH 2 -fCE2=2a2 -(ar^-^y'-). 

 Or by transposition a;2+y2 =2a2 _(CH- +CE=). 

 But by properties of the ellipse, CH^-fCE- =a- +6% and 6^ 



Therefore x^ \y^- ^la"^ —{1a^ -e^)=e'^ . 



