38] ON NEWTON'S SOLUTION OF KEPLER'S PROBLEM. 291 



If the error of x be of the order i, that of x' will now be of the 

 order 2i + 2, that of x" will be of the order 2 (2i + 2) + 2 = 4z + 6, and so 

 on, so that the degree of rapidity of the approximation is still greater than 

 before. 



If we chose to take the mean anomaly itself as the first approximate 

 value of the eccentric anomaly that is, if we put 



we should have z = z e sin z, 



and the value of 8x given by the first method would be 



^ e sin z 



~l-ecosz' 



while that given by the second and more accurate method would be 



e sin z 



1 e cos (z + \e sin z) ' 



and the error of x 1 = x a + 8x would be of the 3rd order in the former case, 

 and of the 4th order in the latter. 



In practice, however, a much nearer first approximate value of x may 

 be always found by inspection, and of course the smaller the error of this 

 value is, the more rapid will be the rate of the subsequent approximations. 



The methods above explained have been long known. The first method 

 is given at p. 41 of Thomas Simpson's Essays on Several Subjects in 

 Speculative and Mixed Mathematics, published in 1740; and Gauss' method 

 given at pp. 10 12 of the Tlieoria Motus, published in 1809, is essentially 

 the same. 



The second method, or rather the modification of the first, is given by 

 Cagnoli in his Trigonometric, at pp. 377, 378 of the first edition, published 

 in 1786, and at pp. 418 420 of the second edition, published in 1808. 



Now, my object in the present note is to point out that the first 

 method explained above is exactly equivalent to that given by Newton in 

 the Principia, at pp. 101, 102 of the second edition, and at pp. 109, 110 

 of the third edition, when Newton's expressions are put into the modern 



analytical form. 



372 



