﻿OF MECHANICAL QUADRATURES. 



195 



III. Computation of Heliocentric Coordinates x, y, and z, by Quadratures. 

 The method of proceeding so nearly resembles that for finding r, that a k\v words 

 will serve to introduce it. 



According as x, y, or z is to be found, we have 



F» = -5«, -Fy=-£y, *** = -£*. ( i 5) 



With the constants x u , y a , z„ J 1 x , J 1 y„, and J 1 z , and the series for r 5 as already com- 

 puted, we obtain F a and F t , and thence x.,, y 2 , and z. 2 ; Fx 2 , Fy„ and Fz t give x 3 , 

 y 3 , and z 3 , and these again x„ y t , and z t , and so on ; noticing the higher orders of dif- 

 ferences when necessary. 



A very useful test of the accuracy of the whole series for r, x, y, and z is afforded by 

 putting 



2 2 I 2 I 2 



r n = X n + y„ + \- (16) 



For if this condition is satisfied, it indicates that every preceding value of r, x, y, and z, 

 from t = to t = n r, is correct. 



In calculating x, y, and z to seven places of decimals, with logarithms of five figures, 

 the number of days in each interval should not exceed 2 r. For the maximum value 

 of F is = — £ = — k 2 (lJ. Since Ar = 0.0003 about, if z. > 2 , F will not be given with 

 accuracy beyond seven places of decimals with five-figure logarithms. 



In the following example, x, y, and z are the coordinates of Halley's comet, referred 

 to the equator, for Greenwich mean midnight, from August 1.5 to August 9.5, 1835. 

 The necessary constants, determined from the elements in the Nautical Almanac for 

 1 839, are as follows : — 



August 1.5, x = + 0.9934592 j\ x = -f 0.0007486 5 

 Vo = _L. 1.6196067 A\y— — 0.01G3206 

 z = -f- 0.6400825 a\ z = — 0.0035933 7 



The corrections for second differences of F are insensible. 



1835, 

 Aug. 1.5 

 2.5 

 3.5 

 4.5 

 5.5 

 6.5 

 7.5 

 8.5 

 9.5 



-(-0.9934592 

 .9942078 

 .9949192 

 .99.V.924 

 .9962267 

 .9968211 

 .9973749 

 .9978870 



-(-0.9983565 



J l x 



-f-7486 5 

 7113 6 

 6732 4 

 6342 8 

 5944 6 

 5537 4 

 5120 9 



-(-4694 8 



-372 

 381 

 389 

 398 

 407 

 416 



-426 



-4-1.6196067 

 .6032861 

 .5869053 8 

 .5704638 8 

 .5539609 3 

 .5373958 7 

 .5207680 

 .5040766 2 



-(-1.4873210 2 



J l y 



-163206 

 163807 2 

 164415 

 165029 5 

 165650 6 

 166278 7 

 166913 8 



-167556 



-f-0.6400825 

 .6364891 3 

 .6328718 9 

 .6292304 1 

 .6255643 ] 

 .6218732 

 .618)566 8 

 .6144143 5 



4-0.6106457 9 



J'z 



-35933 7 

 36172 4 

 36414 8 

 36661 

 36911 1 

 37165 2 

 37423 3 



-37685 6 



238 

 242 

 246 

 250 

 254 

 258 

 —262 



When x, y, and z are wanted for a long period, it will be well to guard against an 

 accumulation of errors in the series, by computing a new set of constants for another 

 epoch, directly from the elements of the orbit. 



