﻿OF MECHANICAL QUADRATURES. 201 



the summation of F8x F8x„ , Fdy, F8y n , FSz F8z n by (5) will 



give 8 Vx, 8Vy, and 8 Vz at the epoch t = (n — *) x \ for this time we also know 

 8x, 8y, and 8 z, and these give, added to Vx, Vy, Vz,x,y, and z computed from the 

 elements at t = 0, the actual values of Vx, Vy, Vz, x, y, and z for the epoch t = 

 (n — £) x or t ' = ; we may then proceed to find 8 x, Sy, and 8 z from t' = to f = 

 precisely as between / = and i = 0. 



There seems to be no ready mode of testing Sx, Sy, and Sz without computing 8r" 

 independently; we might then put \ 8r > = x8x + y8y + z8z. But the expression for< 41 ) 

 FSr 2 under disturbing forces is rather complicated. It may be well to notice, how- 

 ever, that in an undisturbed orbit the quantity q = xVy — y Vx is constant. Under 

 the influence of disturbing forces it is shown in most treatises on perturbations that 

 the momentary variation of q is 



Vq=y(iA — x /.i B (42) 



at t = 0, 8 q = ; then using (5) we may sum up the known values of Vq from ( = ir 

 to t = (n — z)t, whence we find 8q at t = (n — i) x, and we ought to have 



q-\-Sq — xVy~yVx, (4:!) 



,r, y, Vx, and Vy being the actual values of x, y, &c, found as above att = (n — i) x. 



Assuming for the elements of the moon's orbit at Greenwich, midnight, June 

 15th, 1846, 



Longitude of Descending Node, . . . 36° 04' 47".0 ( _ _ . 1B t0j ,„ 

 t. • , t^ ,• »t , ,«» or, ««„ ^ ( True En., June 15, 1846. 



Perigee to the Descending Node, . . 107° 35' 07".6 ) l ' 



Inclination, 5° OS' 35".4 



Mean Longitude, 57° 45' 40".7 



Mean Daily Motion, 13° 24' 30".66 



Log. Eccentricity, 8.6487211 



Mean Distance = ( fo of the sun's, 



8 x, 8y, and 8z may be found for a few days by the method explained above. 

 The exact values of F8x, F8y, and F8z are 



F3x=M(^^-^)-MA;FSy = M(^ 3 -f 3 )-MB-,FSz = M(<^ r - z ? )-MC-, 



(2r — Sr)Sr=(2x — dx)Sx-{-(2y — Sy)Sy-{-{2z — 3z)5z; 



M being the sum of the masses of the earth and moon, and MA, MB, and MC the sun's 

 disturbing forces, those of the planets being neglected. Taking the ecliptic for the plane 

 of x and y at t = = June 15.5, we have 



<Kr = 0, Sy o = 0, <5- =0; Vdx o =0, 73^ = 0, 73* =0i 



F3x = — MA, Fdy = — MB, F8z = —MC; 



27 



