290 ON NEWTON'S SOLUTION OF KEPLER'S PROBLEM. [38 



and x" = xf + Sx', 



x" will be a much more approximate value of x than a?; and so on, to 

 any required degree of approximation. 



If the error of the assumed value x be supposed to be of the order 

 i, when e is taken as a small quantity of the first order, then the error 

 of the value x' will be of the order 2i + 1 = i' suppose, similarly the error 

 of the value x" will be of the order 2i' + I = 4i + 3, and so on, so that 

 the order of the error is more than doubled at each successive approximation. 



The above explains the immense advantage of this process over the 

 use of series proceeding according to powers of e, when great precision is 

 required in the result; since, in this latter method, the addition of a new 

 term only increases the order of the error by unity. 



The degree of rapidity of the approximation may be still further increased 

 by the following slight modification of the above process. 



Starting, as before, with the value x , and calling z z = Sz , we should 

 obtain a much more accurate value than before of the correction Sx to 

 be applied to x a , by putting 



-e cos : + x 1 -e cos 



Now, e being supposed to be small, Sz is an approximate value of 8x 

 and may be written for it in the small term in the denominator. 



Hence, if we put 



x' will be a nearer approximation to the true value of x than was obtained 

 before by the corresponding operation. 



Similarly, if x' e sin x' = z', 



and z z' = 8z', 



and if 8x' = - .. ~ ,. , 



1 - e cos (x + f Sz') 



then x" = x 1 + Sx 1 



will be the next approximate value of x, and the process may be continued 

 as far as we please. 



