﻿OF MECHANICAL QUADRATURES. 197 



IV. To find by Quadratures the Partial Differentials of the Geocentric Bight Ascension 

 and Declination of a Comet or Planet relatively to the Elements of its Orbit. 



This is a problem of frequent occurrence where it is required to find the orbit which 

 best satisfies a large number of observations. Its nature is such that no solution can 

 be expected, for the attainment of which a very considerable amount of labor is not 

 necessary. 



After the approximate elements of the orbit are known, instead of varying these 

 elements themselves, as in the usual method of finding equations of condition for their 

 correction, it will be an equivalent process to use x u , y , z , Vx e , Vy , and Vz„ for the 

 assumed constants of the orbit, and to form equations of condition for determining their 

 corrections. The best epoch for x , y , &c, will usually be at nearly midway between 

 the extreme observations. The coefficients of the unknown quantities in these equa- 

 tions are the partial differentials of a and d, the geocentric right ascension and decli- 

 nation, relatively to x , y c , z , Vx , Vy , and Fz , and they must be known before the 

 equations can be solved. 



Having with the above constants found x, y, and z from / = to t= nr, by the 

 process of Sect. III., each constant is to be separately increased by the addition of a 

 quantity S x 6 , 8 y„ 8 z , 8 Vx , 8 Vy , and 8 Vz , small enough to allow of its square being 

 neglected. Commencing with Sx^ from the equations 



r 2 = I s + f + z\ iVr> = xVx + yVy + zVz, (24) 



|2?^ = i_ £=(F*)» + (7y)*+(F Z )«-f, Fx = — £*, Fy=-^y, Fz = -^z, (25) 

 we have 



dti = 2x Sx , V3r s = 2Vx Sx a , Fd r\ = — 2 Fx Sx = - S ( i ), (26) 



in which Vx and Fx correspond to the intervals between 8r* and Sr] which may 

 be considerably larger than when the total value of .r is computed. As Vx, Fy, and 

 Vz will vary from their previous values after the first interval, we have, excepting when 

 ^ = 0,^5^ = — ±8 r 2 — 8 (^); and since 2F8r\ = — 4,Fx Sx = — S(^) is a con- (27) 

 stant for all values of r, and — is known, FSr will be known when 8 r 1 is known. 

 And the process of finding 8r 2 from Z = to t = mt is very much the same as that em- 

 ployed in Sect. II. Neglecting at first in (10) Sd l and higher orders of the differ- 

 ences of FSr 1 , which will be very small, we obtain Sr 2 nearly, which affords an accurate 



value of F8r] = — — 3 8rl — 8 2 -^-. Again, neglecting Sd], we find Fdr\, and so on, 



r \ 

 correcting for the differences of FSr* as far as may be necessary to give sufficiently 



accurate values of 8r\ 8r*. For the corresponding equations for the changes of 



y and z, they are to be substituted for x in (26). 



