202 DETERMINATION OF AN ORBIT FROM [BOOK II. 



the intervals of the times ; and hence new values of P and Q , which commonly 

 differ much less from those furnished by the first hypothesis, than the latter from 

 the original values themselves of P and Q. The second values of X and Y will, 

 therefore, be much smaller than the first, and the second values of P, Q , will be 

 adopted as the third values of P, Q, and with these the computation will be 

 resumed anew. In this manner, then, as from the second hypothesis more exact 

 numbers had resulted than from the first, so from the third more exact numbers 

 will again result than from the second, and the third values of P , Q can be taken 

 a.s the fourth of P, Q, and thus the calculation be repeated until an hypothesis 

 is arrived at in which X and Y may be regarded as vanishing ; but when the 

 third hypothesis appears to be insufficient, it will be preferable to deduce the val 

 ues of P, Q, assumed in the fourth hypothesis from the first three, in accordance 

 with the method explained in articles 120, 121, by which means a more rapid 

 approximation will be obtained, and it will rarely be requisite to go forward to 

 the fifth hypothesis. 



148. 



When the elements to be derived from the three observations are as yet 



wholly unknown (to which case our method is especially adapted), in the first 



ff 1 



hypothesis, as we have already observed, , 6 6&quot;, are to be taken for approximate 



values of P and Q, where & and to&quot; are derived for the present from the interv.als 

 of the times not corrected. If the ratio of these to the corrected intervals is 

 expressed by /n : 1 and u&quot; : 1, respectively, we shall have in the first hypothesis, 



X==\og[i log u&quot; -f log r\ log if, 



Y= log u -(- log fi&quot; - - log t] log r&quot; -j- Comp. log cos/-|- Comp. log cos / 

 -|- Comp. log cos/&quot; -|- 2 log r log r log r&quot;. 



The logarithms of the quantities p, u&quot;, are of no importance in respect to the re 

 maining terms ; log ; and log r&quot;, which are both positive, in X cancel each other 

 in some measure, whence X possesses a small value, sometimes positive, some 

 times negative ; on the other hand, in Y some compensation of the positive terms 

 Comp. log cos/, Comp. log cos/ , Comp. log cos/&quot; arises also from the negative 



