106 RELATIONS BETWEEN SEVERAL [BOOK I. 



Hence it will readily be perceived, that in the new formula, 



&quot; N) sin^ (N r N). 



the denominator is double the area of the triangle contained between the ex 

 tremities of the three radii vectores, that is, between the three places of the 

 heavenly body in space. When these places are little distant from each other, 

 this area will always be a very small quantity, and, indeed, of the third order, 

 if N N, N&quot; N are regarded as small quantities of the first order. Hence 

 it is readily inferred, that if one or more of the quantities r, r, r&quot;, N, N , N&quot;, are 

 affected by errors never so slight, a very great error may thence arise in the de 

 termination of p ; on which account, this manner of obtaining the dimensions of 

 the orbit can never admit of great accuracy, except the three heliocentric places 

 are distant from each other by considerable intervals. 



As soon as the semi-parameter p is found, e and II will be determined by the 

 combination of any two whatever of the equations I. by the method of article 79. 



83. 



If we prefer to commence the solution of this problem by the computation 

 of the angle IT, we make use of the following method. From the second of 

 equations I. we subtract the third, from the first the third, from the first the sec 

 ond, in which manner we obtain the three following new equations : 



Any two of these equations, according to lemma II., article 78, will give 77 and -, 

 whence by either of the equations (I.) will be obtained likewise e and p. If we 

 select the third solution given in article 78, II., the combination of the first equa- 



