13] STUDIES ON NEWTON'S LUNAR THEORY. 229 



Again, the velocities at A and C are the values of H/r ; that is to 

 say, they are in the proportion 



_l_/i+ 3 J?ll . JL_/i- J5LI. 



1 x \ 4 1 mj 1 + x \ 4 1 mj 



Hence the curvatures of the orbit at A and C are as 



This ratio must be equal to the ratio of the curvatures already found, viz. 



: (l+x)[(l+ x)- (1 - m) 2 + (1 - x) 2 (2m - m 2 )]. 



Hence we have a proportion, from which, multiplying extremes and 

 means, we obtain the equation 



- 2m 2 (1 -m) 2 (l-x)- (l+xY-2mr (2m -in") (1 -x) 4 ] 

 3 m" 



m 



+ m 2 (1 - m) 2 (1 - x)~ (I + x) 2 + m 2 (2m - m 2 ) (1 + x) 4 ], 

 which is Newton's equation of p. 401, expressed symbolically. 

 This equation may be developed into the form 



f 9 ?n 4 1 



V + Tfi/TI -^l\_-^^ + x{6-^(2-m 1 )(2m-m 2 )} 



1 iv(l in) ) 



+ x 2 {6m 2 - 24m 2 (2m - m 2 )} + x 3 {2 + 4m 2 (2m - m 2 )} - x 4 3m 2 ] 



3 m 2 



= [2 - m 2 + x {12m 2 (2m - m 2 )} 



2 1 m L 



+ x 2 {6 - 8 (2m - m 2 ) + 2m 2 - 8m 2 (2m - m 2 )} + x 3 {1 2m 2 (2m - m 2 )} - x 4 m 2 ]. 



Neglecting at first all terms of the fourth order or of the order m 2 a,, 



we get 



3 2-m 



iy VYl 



~2 (1 -m) (3 -2m) (1 -2m)* 

 Taking as Newton does, m 2 =- i.e. m= -0748011, this gives 



1 / O/ Zid 



x= -00720475. 



In the first edition Newton gives '0072036, thus apparently taking 

 account of the first power of x only. The complete equation for x is 

 = -03487783 -4'851179x+-02978676x 2 -2'( 



