124 DETERMINATION OF ELLIPTIC ORBITS. 



The quadratic equation (24) gives two values of the correction to be 

 applied to the position of the body. When they are not too large, 

 they will belong to two different solutions of the problem, generally 

 to the two least removed from the values assumed. But a very large 

 value of Ag 2 must not be regarded as affording any trustworthy 

 indication of a solution of the problem. In the majority of cases we 

 only care for one of the roots of the equation, which is distinguished 

 by being very small, and which will be most easily calculated by a 

 small correction to the value which we get by neglecting the quadratic 

 term.* 



When a comet is somewhat near the earth we may make use of the 

 fact that the earth's orbit is one solution of the problem, i.e., that p 2 

 is one value of Ag 2 , to save the trifling labor of computing the value 

 of (St"'"'). For it is evident from the theory of equations that 

 if p 2 and z are the two roots, 



~ 



Eliminating ("'"'), we have 



(*- 

 whence 



('"') 



Now ,-, tL is the value of Ag 2 , which we obtain if we neglect 



the quadratic term in equation (24). If we call this value [A<? 2 ], 

 have for the more exact value t 



The quantities Ag t and Ag 3 might be calculated by the equations 



(27) 



\ 

 J 



* In the case of Swift's comet (V, 1880), the writer found by the quadratic equation 

 - '247 and *116 for corrections of the assumed geocentric distance *250. The first of 

 these numbers gives an approximation to the position of the earth ; the second to that 

 of the comet, viz., the geocentric distance '134 instead of the true value "1333. The 

 coefficient -&- was used in the quadratic equation ; with the coefficient the approxi- 

 mations would not be quite so good. The value of the correction obtained by neglecting 

 the quadratic term was '079, which indicates that the approximations (in this very 

 critical case) would be quite tedious without the use of the quadratic term. 



t In the case mentioned in the preceding footnote, from [Ag 2 ] = - '079 and p. 2 = "25, 

 we get Ag 2 = - '1155, which is sensibly the same value as that obtained by calculating 

 the quadratic term. 



