SECT. 1.] THREE COMPLETE OBSERVATIONS. 181 



and it readily appears, that if n, ri, n&quot;, are regarded as small quantities of the first 

 order, 77 1, rj 1, rf 1 are, generally speaking, quantities of the second 



order, and, therefore, 



e_ er_ 



6&quot; 6&quot; 



the approximate values of x, y, differ from the true ones only by quantities 

 of the second order. Nevertheless, upon a nearer examination of the sub 

 ject, this method is found to be wholly unsuitable ; the reason of this we 

 will explain in a few words. It is readily perceived that the quantity (0. 1. 2), 

 by which the distances in the formulas 9, 10, 11, of article 114 have been multi 

 plied, is at least of the third order, while, for example, in equation 9 the quan 

 tities (0. 1. 2), (I. 1. 2), (II. 1. 2), are, on the contrary, of the first order; hence, 

 it readily follows, that an error of the second order in the values of the quanti 

 ties ^, n -^ produces an error of the order zero in the values of the distances. 

 Wherefore, according to the common mode of speaking, the distances would be 

 affected by a finite error even when the intervals of the times were infinitely 

 small, and consequently it would not be admissible to consider either these dis 

 tances or the remaining quantities to be derived from them even as approximate ; 

 and the method would be opposed to the second condition of the preceding 

 article. 



. 132. 



Putting, for the sake of brevity, 



(0.1.2) = 0, (O.L2)1X = b, (0.0.2)Z&amp;gt;= + o, (O.IL Z)iy = + d, 

 so that the equation 10, article 114, may become 



ad 1 = b -4-c ^, -4- d ^-r, 



n n 



the coefficients c and d will, indeed, be of the first order, but it can be easily 

 shown that the difference c d is to be referred to the second order. Then it 

 follows, that the value of the quantity 



n+n&quot; 



