SECT. 1.] THREE COMPLETE OBSERVATIONS. 167 



as impossible,* and, therefore, still less will it be possible to obtain a complete 

 solution of the problem by direct processes alone. 



But our problem may at least be reduced, and that too in various ways, to the 

 solution of two equations X=Q, F= 0, in which only two unknown quantities 

 x, i/, remain. It is by no means necessary that x, y, should be two of the ele 

 ments : they may be quantities connected with the elements in any manner 

 whatever, if, only, the elements can be conveniently deduced from them when 

 found. Moreover, it is evidently not requisite that X, Y, be expressed in explicit 

 functions of x, y : it is sufficient if they are connected with them by a system of 

 equations in such manner that we can proceed from given values of x, y, to the 

 corresponding values of X, Y. 



120. 



Since, therefore, the nature of the problem does not allow of a further reduc 

 tion than to two equations, embracing indiscriminately two unknown quantities, 

 the principal point will consist, first, in the suitable selection of these unknown 

 quantities and armnr/cment of the equations, so that both X and Y may depend 

 in the simplest manner upon x, y, and that the elements themselves may follow 

 most conveniently from the values of the former when known : and then, it will 

 be a subject for careful consideration, how values of the unknown quantities satis 

 fying the equations may be obtained by processes not too laborious. If this should 

 be practicable only by blind trials, as it were, very great and indeed almost intol 

 erable labor would be required, such as astronomers who have determined the 

 orbits of comets by what is called the indirect method have, nevertheless, often 

 undertaken : at any rate, the labor in such a case is very greatly lessened, if, in 

 the first trials, rougher calculations suffice until approximate values of the un 

 known quantities are found. But as soon as an approximate determination is 

 made, the solution of the problem can be completed by safe and easy methods, 

 which, before we proceed further, it will be well to explain in this place. 



* When the observations are so near to each other, that the intervals of the times may be treated as 

 infinitely small quantities, a separation of this kind is obtained, and the whole problem is reduced to the 

 solution of an algebraic equation of the seventh or eighth degree. 



