132 



ON PROFESSOR CHALLIS'S NEW THEOREMS [20 



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

 to terms of the second order inclusive by putting 



- 

 2 



- m' cos ( 2 nt + e -2n't + <?) 



15 

 --me 







- nt + e + 2e sin (cut + e CT) + - e ! sin 2 (cut + e w) 



-5- m'sin 

 8 



15 . , . , x 



+ -T- me sin (Int -f e 2n t + e cnt + c rs), 

 4 



a in' n' 3 



where w2 = ^' n * = ~<yf 2 ' m= ~^' c = l ~^ m ^' 



and a, e, e, and CT are the four arbitrary constants required by the complete 

 solution. 



The fact that the differential equations are satisfied by these expressions 

 for r and 6, whatever be the value of e, is quite sufficient to shew 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 



e' and 2nt + e 2n't + <? cnt + e OT, 



and which constitute the well-known inequalities called the "variation" and 

 " evection," prove the incorrectness of Professor 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 his equation 



c=o (O, 



