168 DETERMINATION Ot AX ORBIT FROM [BOOK II. 



The equations -X&quot;=0, Y= will be exactly satisfied if for x and y their 

 true values are taken ; if, on the contrary, values different from the true ones are 

 substituted for x and y, then X and Y will acquire values differing from 0. The 

 more nearly x and y approach their true values, the smaller should be the result 

 ing values of X and Y, and when their differences from the true values are very 

 small, it will be admissible to assume that the variations in the values of X and Y 

 are nearly proportional to the variation of x, if y is not changed, or to the varia 

 tion of y, if x is not changed. Accordingly, if the true values of x and y are 

 denoted by , ^, the values of X and Y corresponding to the assumption that 

 # = -[&quot; ^j y = t] -j- fi, will be expressed in the form 



in which the coefficients a, ft, y, d can be regarded as constant, as long as A and p 

 remain very small. Hence we conclude that, if for three systems of values of 

 x, y, differing but little from the true values, corresponding values of X, Y have 

 been determined, it will be possible to obtain from them correct values of x, y so 

 far, at least, as the above assumption is admissible. Let us suppose that, 

 for x = a, y = b we have X = A, Y = B, 

 x = tt,y = V X=A Y=ff, 



x =a&quot;,y = l&quot; X = A&quot; Y= B&quot;, 



and we shall have 



A = - 



From these we obtain, by eliminating a, ft, y, d, 



t __a(A B&quot; A&quot;B )-}-a (A&quot;BAB&quot;)+oi (AB ! A B) 

 A B&quot; A&quot;Bt -f- A&quot;B A B&quot; -\-AH- A B 



_ b(AB&quot; A&quot;B i ) + V(A&quot;BAB&quot;) -f- 1&quot; (A B? A B) 

 V ~ A B&quot; A&quot;B + A&quot;B A B&quot; -f A B&quot; A B 



or, in a form more convenient for computation, 



,( a)(A BA B- ) + (a&quot; a)(A B A B) 

 A B&quot;A&quot;B -}-A&quot;BAB&quot;+AB A B 



_ , , (y 1) (A&quot;B A B&quot;) -f (V b)(AB r A B) 

 ~ A B r ^ i1 ~ r T? A B 



