SECT. 3.] ANY NUMBER OF OBSERVATIONS. 271 



a new condition : he requires, namely, that the sum of the differences, the signs 

 remaining unchanged, be equal to zero. Hence it follows, that the number of 

 equations exactly represented may be less by unity than the number of unknown 

 quantities ; but what we have before said will still hold good if there are only 

 two unknown quantities. 



187. 



From these general discussions we return to our special subject for the sake 

 of which they were undertaken. Before the most accurate determination of 

 the orbit from more observations than are absolutely requisite can be com 

 menced, there should be an approximate determination which will nearly satisfy 

 all the given observations. The corrections to be applied to these approximate 

 elements, in order to obtain the most exact agreement, will be regarded as the 

 objects of the problem. And when it can be assumed that these are so small 

 that their squares and products may be neglected, the corresponding changes, 

 produced in the computed geocentric places of a heavenly body, can be obtained 

 by means of the differential formulas given in the Second Section of the First 

 Book. The computed places, therefore, which we obtain from the corrected ele 

 ments, will be expressed by linear functions of the corrections of the elements, 

 and their comparison with the observed places according to the principles before 

 explained, will lead to the determination of the most probable values. These 

 processes are so simple that they require no further illustration, and it appears at 

 once that any number of observations, however remote from each other, can 

 be employed. The same method may also be used in the correction of the parcir 

 lolic orbits of comets, should we have a long series of observations and the best 

 agreement be required. 



188. 



The preceding method is adapted principally to those cases in which the 

 greatest accuracy is desired: but cases very frequently occur where we may, 

 without hesitation, depart from it a little, provided that by so doing the calcula- 



