m^-mKV vu ^,eia^in^ to the Moon's Orbit, '^ ^ -^'^ 29 



"But if we introduce the quantities usually denoted by c and^, 

 having assigned values slightly differing from unity, which 

 amounts to supposing the apse and node to have certain mean 

 motions, we find that the differential equations are satisfied by 

 adding to the first approximate expressions for the moon's coor- 

 dinates, terms, which always remain small ; and we thus know 

 that our first approximation was a good one, and that the true 

 and the only true solution of the differential equations has been 

 obtained. 



On the other hand, no solution can be a true one, which does 

 not contain the proper number of arbitrary constants ; and any 

 person who asserts that one of the constants usually considered 

 arbitrary is not so, is bound to show by what other really arbi- 

 trary constant the former is replaced. 



I will now proceed to consider Professor Challis's two theo- 

 rems, which are thus enunciated by him. 



Theorem I. All small quantities of the second order being 

 taken into account, the relation between the radius-vector and 

 the time in the moon's orbit is the same as that in an orbit 

 described by a body acted upon by a force tending to a fixed 

 centre. 



Theorem II. The eccentricity of the moon's orbit is a function 

 of the ratio of her periodic time to the earth's periodic time, and 

 the first approximation to its value is that ratio divided by the 

 square root of 2. '''^^ 



I will endeavour, in the first place, to show that these 

 theorems cannot possibly be true ; and secondly, to point out 

 the fallacies in the argument by which Professor Challis attempts 

 to establish them. 



^^ The problem will be simplified by supposing the moon to 

 iKbve in the plane of the ecliptic, and the earth's orbit to be a 

 circle. On these suppositions, Professor Challis's fundamental 

 equations become •: 



... , ,. 5 = _ ^ + ^, + 1»^ eos [e-^^hT-^) t "■ wS 



ffp -f ' dt^ r^ 2a'^ 2>a'^^ ^ ' -•■^'- yv M 



Multiply these equations by y and x respectively, and subtract 

 the results ; and again multiply by x and y, and add the results 

 together ; thus we obtain, after expressing x and y by means of 

 polar coordinates, *^ ^ >t*iMi 



d ( dO\ Sm'r^ ' ^** .-^o - 3idi:j/<im(m iub 



dtydt) "^ ~ 2^^^^ {20-%n't-^6'). viifuiic;40'jYiq,B(a) 



' d^r (dOy fj. ^ m^r ^ Sm'r ,c.a ' c^Tr^^ /m 



5? -^ U; = ~ 7^ + 2^ + ^3-cos(2^-.2n'^ + e'). (2) 



