38] ON ZENGER'S SOLUTION OF KEPLER'S PROBLEM. 295 



If we determine f so that the error in the determination of x shall 

 vanish when 



we shall have 



7T 



X =2> 



f=e i_-e s + e 4 -etc. = sine, 



and the approximate equation for finding x z becomes 



, . sin e sin z 



1 sin e cos z ' 



The error still remains in general of the 3rd order in e, but the maximum 

 error will be smaller than when f is taken e. 



The value of x given by this equation is readily seen to be equivalent 

 to that given by Professor Zenger's equation, 



e, cosec z 



cot x cot z 



1 + - sin 2 e + ~ sin 4 e + etc. 

 6 40 



where we may remark that the quantity 



1 



1+ - sin 2 e + sin 4 e + etc. 

 6 40 



is equivalent to 



a series which converges much more rapidly than the series for its reciprocal, 

 employed by Professor Zenger. 



A still more advantageous result may, however, be obtained by deter- 

 mining / so that the error may vanish both when 



IT 



2TT 



and when x = -~ , 



73 

 that is when sin x = , 



