SECT. 1.] TO POSITION IN THE ORBIT. 13 



that the changes of the logarithm of the sine from e to E -j- x, and the changes of 

 the logarithm of the number from e sin e to e sin (e -|- x\ can be regarded as 

 uniform, we may evidently put 



e sin (e -f- x) = e sin e + - , 



the upper sign belonging to the first and fourth quadrants, and the lower to the 

 second and third. Whence, since 



-\-z = M-\- e sin (e -\- x), we have x =. -^*y (M-\- e sin c e), 

 and the correct value of 2?, or 



e -j- x = M -j- c sin + ^ry (M-\- esms E), 



the signs being determined by the above-mentioned condition. 



Finally, it is readily perceived that we have, without regard to the signs, 

 /x : X 1 : e cos e, and therefore always p &amp;gt; 1, whence we infer that in the first and 

 last quadrant M-\- e sin e lies between and f, -\- x, and in the second and third, 

 e-\-x between t and M-\- e sin e, which rule dispenses with paying attention to the 

 signs. If the assumed value e differs too much from the truth to render the fore 

 going considerations admissible, at least a much more suitable value will be found 

 by this method, with which the same operation can be repeated, once, or several 

 times if it should appear necessary. It is very apparent, that if the difference 

 of the first value from the truth is regarded as a quantity of the first order, the 

 error of the new value would be referred to the second order, and if the operation 

 were further repeated, it would be reduced to the fourth order, the eighth order, 

 etc. Moreover, the less the eccentricity, the more rapidly will the successive 

 corrections converge. 



12. 



The approximate value of E, with which to begin the calculation, will, in most 

 cases, be obvious enough, particularly where the problem is to be solved for 

 several values of M of which some have been already found. In the absence 

 of other helps, it is at least evident that E must fall between M and M e, (the 

 eccentricity e being expressed in seconds, and the upper sign being used in the 



