€50 BEPOKT— 1892. 



the equations of the moon's motion, if a denote the distance from (the earth) of 

 the centre of gravity of the sun and earth, 



^-2co^- a,-(« + a) = -^- .... (1), 



dJ^y dx ,-, dV ,«. 



^, + 2co^^^-co-y =-^— . . . . W, 



where V is the potential of the attractions of the sun and earth on the moon, 

 and 0) the angular velocity of the earth's radius vector. From this we find, for 

 the relative-energy equation 



where E denotes a constant, and for the relative-curvature equation 



dxd^y-dyd'x dt. ^dt- 



-^'^ ' „\l+d?T^ .... (4), 



(dx-' + dy'-^ (dx''^dy''\^ 



where N denotes the component perpendicular to the path, of the resultant of (X,Y) 

 with 



X=<o*(:?.+«)-'^J (5), 



dx 



Y=o,^y-5^ (6). 



dy 



Hence if q denote moon's velocity and p the radius of curvature of her path, rela- 

 tively to the revolving plane XOY, we have 



i2» = E-i-i(o-(a-» + y2)-V (7), 



and 



1 -20, N .r.. 



- = + -o . . • . • • . yPl- 



PIT 



Calling S the sun's mass, and a his distance from the earth, and supposing the 

 earth's mass infinitely small in comparison with the sun's, we have 



-^ = a)^a (9), 



a 



and therefore 



O)^*!? . TO 



- V = "- ^ + 



where m denotes the earth's mass, and r- v'C^'^ + y'). 

 Hence 



(10), 



?» 



-V = K(2«'-^«^ + 2.t"-y-) + - . . . (11). 



T 



With this and with <i) = l and m = lP', for simplicity in the numerical work which 

 follows, we have 



%*^t-''--y% <'»>' 



2» = 2E + 3ar«+- (14), 



r 



