44G Lord Kelvin on Graphic Solution 



gravity. Let (x, y) be coordinates of the moon relative to 

 OX in line with the sun, outwards, and OY perpendicular to 

 it in the direction of the earth's orbital motion. The well- 

 known equation of motion relatively to revolving coordinates 

 gives, for 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, 



d 2 x a dy or \ dV 



__ 2(B j_ (B8(a+ , )= __ J . . . (i) 



dh/ dx 2 dV 



dP+^dT-^ ="V • • • ( 2 ) 



where V is the potential of the attractions of the sun and 

 earth on the moon, and a> the angular velocity of the earth's 

 radius-vector. From this we find, for the relative-energy 

 equation 



i (¥ + ^) =E+|a,2( ^ +/)_v ' • • (3) 



where E denotes a constant ; and for the relative-curvature 



equation we find 



dxd l y — dyd*x _ dt Ndf 



(dtf + dy 2 )* ~~ (da? + dy*fi + dx' + dy 2 ' ' W 



where N denotes the component perpendicular to the path, 



of the resultant of (X, Y) with 



dV 



X = rf(x + a)- a ^, (5) 



dx v ' 



Y =^-f ( 6 >- 



Hence if q denote moon's velocity and p the radius of curva- 

 ture of her path, relatively to the revolving plane XOY, we 



i ? 2 = E + |a>V + </ 2 )-V, .... (7) 

 and 



i==^ + * (8). 



p q q 2 K > 



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

 earth, and supposing the earth's mass infinitely small in com- 

 parison with the sun's, we have 



j=<»\ (») 



and therefore 



where m denotes the earth's mass, and r= \/(x 2 +y 2 ) 



. . y _ a a m 



