80 Mr. J. C. Adams on Professor Challis's new Theorems 



Now these equations^ which are equivalent to the former, are 

 satisfied to terms of the second order inclusive by putting 



r=a-s 1— -^-\-^e^—ecos{cni-\-€-''sr) — -^e^C0B2{cnt-\-€—'ur) 



— m« cos (2n^-fe— 2n7+ e') 



15 ] 



-'-^mecos(2nt'\-€-2n't-\-^—cnt-^e—m) > 



5 



6 =:nt + € + 2e sin {cnt -\- €—v) -h-T^e'^ sin 2 (cw/ + e— to-) 



11 



+ -g- m^sin (2nt + €—2n't + ^) 



15 , 



-\- -j-mesixi(2nt -i- €^2n't + ^ — cnt -^ €—'a)f 



where 2 1^ ,0 m' rl i 3 ^ 



AT fl'^ w ' 4 ' 



and fl, €, e, and w are the four arbitrary constants required by 

 the complete solution. 



The fact that the differential equations are satisfied by these 

 expressions for r and Q, whatever be the value of e, is quite suf- 

 ficient to show that Professor Challis is mistaken in restricting 

 e to one particular value. 



The terms of the second order in the value of r, which depend 

 on the arguments 



271/4-6— 2w7H-e and 2w^H-€— 2»7 + 6'--C7i^ + €— w, 



and which constitute the well-known inequalities called the 

 '' variation '^ and the " evection," prove the incorrectness of Pro- 

 fessor Challis's Theorem I. ; since in an orbit described by a 

 body acted on by a force tending to a fixed centre, and varying, 

 as Professor Challis supposes, as some function of the distance, 

 the expression for the radius-vector in terms of the time cannot 

 possibly contain any terms dependent on the sun's longitude, 



I now come to consider the reasoning by which Professor 

 Challis arrives at his theorems. All this reasoning is based on 

 bis equation 



U; +;^ ~ 2?5 + ^-0, •••(C) 



the truth of which, he says, cannot be contested. In speaking of 

 the truth of this equation, Professor Challis cannot mean that it 

 is anything more than an approximation to the truth, since in 

 forming it he avowedly neglects all quantities of orders superior 

 to the second. 



Now what I assert is, first, that the degree of approximation 

 attained by the equation (C) is not sufficient to justify Professor 



