﻿OF MECHANICAL QUADRATURES. 199 



V. Calculation of Perturbations by Quadratures. 



The usual application of quadratures to the calculation of perturbations is in the 

 summation of the momentary variations of the elements of the disturbed orbit, which; 

 it has been found, may be expressed with much simplicity, considering the general 

 intricacy of the problem, in terms of the coordinates x, y, and z, and their first differ- 

 ential coefficients, combined with the disturbing forces. For tins purpose the positions 

 of all the disturbing bodies of the system are computed for the middle of each inter- 

 val, and also, either directly, or indirectly by the use of auxiliary angles, the coordinates 

 of the disturbed body x, y, and z, with their first differential coefficients for tin s;unc 

 epochs. From these, combined with the disturbing forces from t = i to t = n — £, are 

 derived the momentary variations of each clement for the same times, and thence, by 

 using (5) to sum up their values, will result the alteration of each of the six elements 

 at the end of each interval. 



After the perturbations of the elements for each interval have been ascertained, 

 the changes which they introduce into x, y, and z are investigated, leading finally to the 

 perturbations in geocentric right ascension and declination. Where the change of the 

 elements is large, the process is sometimes repeated with corrected values of x, y, and z, 

 in order to include the squares of the disturbing forces. 



If we commence with the constants .r , y , z„, Vx„, Fy„, and Vz for < = 0, we have 



Fx = -£x-i t A, Fy = -* i y- li B, F* = -£-pC; (36) 



in which 



fl A = m' {-^- + p%)+ m" (-;p- + pri) + &c, 



,, B = w! (^ + gi) + m" (*=£' + fy + &c, ( 37 ) 



,z — z' :' > iz — z" z" k 



fiC^m 1 \-yT- + ^3) + m" {-jiT + ^-3) + &C, 



are denominated the disturbing forces, ft being the sum of the masses of the sun and 

 the disturbed body ; ni, m", &c., the masses of the disturbing bodies ; x, x", &.c, their 

 coordinates from the sun ; r, g, r", g", &c, their distances from the sun and from the 

 disturbed body, of which the heliocentric coordinates and distance are x, y, z, and r. m', 

 m", &c, it will be noticed, being referred to the same unit with ft, arc always very small 

 compared with it, and therefore the second terms in (36) are much less than the first. 



Making always r 1 = x 2 + f + z 2 , instead of finding it by Sect. II., because the pro- (38) 

 cess there given cannot here be adopted, the semi-axis being no longer constant, we 

 may compute Fx u , Fx„ &c, and thence find //'x , J' y„, i'z„ x„ x a , Sic, as in Sect. III., 

 with this difference, that Fx, Fy, and Fz contain the terms fi A, {.i B, and fi C, and 

 that r 2 is to be found from (38). 



