VECTOR METHOD IN THE DETERMINATION OF ORBITS. 153 



It will be observed that all the formulae I, Ha, 116, lie, III, may be 

 expressed in the general form 



except that the letters B I} B 2 , B 3 have different values in the different 

 cases, some vanishing in the more simple formulae. Moreover, if the 

 values of B I} B^, B 3 have been calculated for I, the values for Ha, 116, 

 or lie are found simply by subtraction of one of the numbers from the 

 three. It is evident that 116 will hardly be useful except in special 

 cases, as in the determination of a parabolic orbit in the failing case of 

 Gibers' method, and then it would be a question whether it would not 

 be better to determine the orbit from p 2 and p 3 , or p 2 and p v using Ha 

 or lie. 



Equations Ha and lie are very appropriate for the determination 

 of an elliptic orbit when the observed motion is nearly in the ecliptic, 

 by means of four observations with intervals nearly in the ratio 

 5:8:5. 



It is evident that the solution of (7) given above may be varied, in 

 ways too numerous to mention, by the use of the simpler forms Ila, 

 He, or III for I in the earlier stages of the work. This only involves 

 changing the values of B lt B 2 , B s , in (a), (6) and (c). 



It is not correct to say that in my expressions for the ratios of the 

 triangles the error is of the fifth order in general, or for equal intervals, 

 of the sixth. If we write Pi,p 2 ,pz> for the coefficients of 9^, $R 2 , 

 in I, and 5 for the error of the equation, we have exactly 



which gives p 



^x^ 3= p[ 

 9^x9*3 p 2 



Now *3 is my expression for the ratio of the triangles, and 



is its error. This is of the fourth order in general (since the denomin- 

 ator is of the first), and for equal intervals, of the fifth. The same is 

 true of the two other ratios. Thus we have 



Adding these equations and subtracting 1 [from both sides] we have 



i>2 (ffls 



Here the last term, which represents the error, is of the fifth 

 order in general, or for equal intervals, of the sixth. But the 



