191 



Then 1 . Changing only the approximate perihelion distance by a 

 small quantity, he obtains new values of m and m'. 



2. Changing only the time of perihelion by a small quantity, he 

 obtains other values of m and mf. 



From the relation of these values, he obtains tveo equations, by the 

 solution of which the corrections of the perihelion distance, and of 

 the time of perihelion are obtained. 



If these be not sufficiently correct, as will generally be the case, 

 new corrections must be investigated. 



Every correction of the elements requires nine long operations, in 

 which, to obtain the necessary exactness, it is requisite to use 

 logarithms of seven places of decimals. 



By the alteration I am about to suggest, there will be required for 

 the first corrections only three operations, in which seven places of 

 figures will be necessary ,• in the other part of the process, five or even 

 four places of logarithms will be sufficient; and, in the subsequent cor- 

 rections, only three operations to seven places of figures will be neces- 

 sary. The repetition of the part of the process in which four or five 

 places only were used, may in general be dispensed with. 



Instead of two hypothesis as in M. Laplace's method, by which two 

 equations with two unknown quantities are obtained, I obtain with- 

 out any hypothesis, two equations in which the fluxion of the peri- 

 helion distance and of the time from perihelion are the unknovm 

 quantities, in the following manner: — 



Let the true values of U and V be U + KJ and V + SY. 



Then U 4- ^U = V + SV and U' +^U' = V + 5V'. 



Now we can compute 5U, SU', W, JV so as to obtain two equa- 

 tions of the form 



adp + bdt V— U 

 a'dp + Z>'dt'=V'- U'. 



