Astronomical and Nautical Collections. 379 



[ Note of the Editor. The formulas used by Laplace con- 

 siderably resemble these. If the f* and v, of §. 67, be made for 

 the three hypotheses ^', fx,", /a'", and /, /', v"', we have 

 y (ft' — jA.") + X (lA.' — y.'") = f*', and 

 y (/ — /) + z (/ — /") = *' ; the resolution and 

 use of these equations being perfectly similar to those of §. 75, 

 and y being the factor by which the alteration of the perihelium 

 distance is multiplied, and x the factor of the alteration of the 

 time of the passage through the perihelium, by means of which 

 the true alterations of these elements are determined. Some- 

 times, however, it may become necessary to take the second 

 differences into consideration, and the editor has employed 

 Laplace's formulas with advantage in the calculations of se- 

 veral comets : he therefore inserts them as applied to Laplace's 

 own method ; but they may be accommodated, without difficulty, 

 to every other. We are to compute the values of (a and ► for 

 the five following hypotheses; (1) with the elements found 

 by the first approximation ; (2) with a small alteration of the 

 perihelium distance ; (3) with twice as great an alteration ; (4) 

 with the approximate distance unaltered, and with a small al- 

 teration of the time ; (5) with twice as great an alteration : now 

 if we distinguish the five values of //t and v by the number of 

 accents, and make x and y the factors of the alterations in the 

 2d and 4th hypotheses, which are required to find the true alter- 

 atibns of the elements, we may find the values of x and y by the 

 following equations. = (4 /a" — 3 ^' — f*'") y + (ft"' — 2 /*" 



+ /*') y' + (4 /" - 3 f.' - p.'"") X + (^ - 2 f*"" + y!) X' + 2 



y.', and = (4 /' - 3 .' - /") y + ( » " - 2 ►" -|- /) f + (4 /'" - 

 3/ _ ,'"") X + (>■""' — 2 »"" + /) x" -h 2 1. We may also ob- 

 serve that we may resolve these equations by extermination, but 

 that we are led by a troublesome operation to an equation of the 

 fourth degree ; and that it is therefore more convenient to find 

 approximate values with the omission of the quadratic terms x 

 and y-, and then to compute these terms with the values so found 

 and to renew the operation, so that x and y may be deduced 

 from the linear equations thus obtained.] 



