180 DETERMINATION OF AN ORBIT FROM [BOOK II. 



Thirdly and lastly, we require that the processes by which we pass from the quan 

 tities x, (/, to X, Y, successively, be not too complicated. 



These conditions will furnish the criterion by which to judge of the excellence 

 of any method : this will show itself more plainly by frequent applications. The 

 method which we are now prepared to explain, and which, in a measure, is to be 

 regarded as the most important part of this work, satisfies these conditions so that 

 it seems to leave nothing further to be desired. Before entering upon the ex 

 planation of this in the form most suited to practice, we will premise certain pre 

 liminary considerations, and we will illustrate and open, as it were, the way to it, 

 which might, perhaps, otherwise, seem more obscure and less obvious. 



131. 



It is shown in article 114, that if the ratio between the quantities denoted 

 there, and in article 128 by n, ri, n&quot;, were known, the distances of the heavenly 

 body from the earth could be determined by means of very simple formulas. 

 Now, therefore, if 



should be taken for z, y, 



L . 



6&quot; 0&quot; 



(the symbols 6, 6 , 6&quot;, being taken in the same -signification as in article 128) im 

 mediately present themselves as approximate values of these quantities in that 

 case where the heliocentric motion between the observations is not very great : 

 hence, accordingly, seems to flow an obvious solution of our problem, if two dis 

 tances from the earth are obtained from #, y, and after that we proceed agreeably 

 to some one of the five methods of articles 124-128. In fact, the symbols 17, if 

 being also taken with the meaning of article 128, and, analogously, the quotient 

 arising from the division of the sector contained between the two radii vectores 

 by the area of the triangle between the same being denoted by tf, we shall have, 



2L 



n 



