DETERMINATION OF ELLIPTIC ORBITS. 127 



and therefore from r lt r 2 , r 3 and the given intervals. For our funda- 

 mental equation, which may be written 



indicates that we may form a triangle in which the lengths of the 

 sides shall be n^, n 2 r z , and n 3 r 3 (let us say for brevity, 8 1 , 8 2 , s 3 ), and 

 the directions of the sides parallel with the three heliocentric directions 

 of the body. The angles opposite ^ and s 3 will be respectively v 9 v 2 

 and v 2 v v We have, therefore, by a well-known formula, 



As soon, therefore, as the solution of our fundamental equation 

 has given a sufficient approximation to the values of r lt r 2 , r s (say 

 five- or six-figure values, if our final result is to be as exact as 

 seven-figure logarithms can make it), we calculate n lt n 2 , n 3 with 

 seven-figure logarithms by equations (2), and the heliocentric angles 

 by equations (34). 



The semi-parameter corresponding to these values of the heliocentric 

 distances and angles is given by the equation 



i*s (35) 



The expression n^ n 2 -f n z , which occurs in the value of the semi- 

 parameter, and the expression n^ n 2 r 2 + n z r 3 , or s 1 s 2 -fs 3 , which 

 occurs both in the value of the semi-parameter and in the formulae for 

 determining the heliocentric angles, represent small quantities of the 

 second order (if we call the heliocentric angles small quantities of the 

 first order), and cannot be very accurately determined from approxi- 

 mate numerical values of their separate terms. The first of these 

 quantities may, however, be determined accurately by the formula 



+ ~+ 3 - (36) 



With respect to the quantity s x s 2 +s 3 , a little consideration will show 

 that if we are careful to use the same value wherever the expression 

 occurs, both in the formulae for the heliocentric angles and for the 

 semi-parameter, the inaccuracy of the determination of this value from 

 the cause mentioned will be of no consequence in the process of 

 correcting the fundamental equation. For although the logarithm 

 of s l s 2 +s 3 as calculated by seven-figure logarithms from r lt r 2 , r 3 

 may be accurate only to four or five figures, we may regard it as 

 absolutely correct if we make a very small change in the value of one 





