184 DETERMINATION OF AN ORBIT FROM [BOOK II. 



obvious approximate values of the quantities P, Q, of which the true values are 

 precisely 



__ 

 6 &quot; rS qtf cos/cos/ cos/&quot; 



from which hypothesis will result errors of the first order in the determination of 

 (f, and therefore of ff, d&quot;, or of the second order in the special case several times 

 mentioned. Although we may rely with safety upon these conclusions, generally 

 speaking, yet in a particular case they can lose their force, as when the quantity 

 (0. 1. 2), which in general is of the third order, happens to be equal to zero, or so 

 small that it must be referred to a higher order. This occurs when the geocentric 

 path in the celestial sphere has a point of contrary flexure near the middle place. 

 Lastly, it appears to be required, for the use of our method, that the heliocentric 

 motion between the three observations be not too great : but this restriction, by 

 the nature of the very complicated problem, cannot be avoided in any way; 

 neither is it to be regarded as a disadvantage, since it will always be desired to 

 begin at the earliest possible moment the first determination of the unknown 

 orbit of a new heavenly body. Besides, the restriction itself can be taken in a 

 sufficiently broad sense, as the example to be given below will show. 



135. 



The preceding discussions have been introduced, in order that the principles 

 on which our method rests, and its true force, as it were, may be more clearly 

 seen : the practical treatment, however, will present the method in an entirely 

 different form which, after very numerous applications, we can recommend as 

 the most convenient of many tried by us. Since in determining an unknown 

 orbit from three observations the whole subject may always be reduced to 

 certain hypotheses, or rather successive approximations, it will be regarded as a 

 great advantage to have succeeded in so arranging the calculation, as, at the 

 beginning, to separate from these hypotheses as many as possible of the compu 

 tations which depend, not on P and Q, but only on a combination of the known 

 quantities. Then, evidently, these preliminary processes, common to each hypoth 

 esis, can be gone through once for all, and the hypotheses themselves are reduced 



