﻿202 . ON THE USE OF EQUIVALENT FACTORS 



We will now proceed to compare the three methods of solution (I.), (II.), (HI-); ^y 

 applying them to the following series of equations of six indeterminates, taken from a 

 memoir, by Gauss, on the elliptic elements of the orbit of the planet Pallas.* 



Original Equations.'\ 



1 = — 183!934- 0.79363 fZl,+ 143.66 d 7 + 0.39493 i^tt + 0.95920 (Zg. — 0.18856 da + 0.17387 di 



2 = — 6.81 — 0.02658 dl,+ 46.71 (/y + 0.02658 <Z7r — 0.20858 (Z 9) + 0.1 5946 do + 1.25782 di 



3 0=— 0.06 + 0.58880 (ZL+ 358.12dy + 0.26208(Z7r — 0.85234d9 + 0.14912(ZQ + 0.17775(Zt 



4 = — 3.09 + 0.0I318dL+ 28.39(Zy — 0.01318(Z 77 — 0.07861 (Zq; + 0.91704<ZQ + 0.54365tZz 



5 = — 0.02+ 1.73436 dL-\- 1846.17 (Zy — 0.54603 d tt- 2.05662 dtp — 0.18833 d Q — 0.17445 d i 



6 = — 8.98 — 0.12606 (ZZ— 227.42(Zy + 0.12606(Z7r — 0.38939 fZ9 + 0.17176(ZQ— 1.35441 rfi 



7 = — 2.31+0.99584(ZL + 1579.03rfy + 0.06456(Zff+1.99545d(j) — 0.06040 (Zq — 0.33750 fZi 



8 0=+ 2.47 — 0.08089 (ZL— 67.22d7 + 0.08089<Z77 — 0.09970^9 — 0.46359 cZ a + 1.22803 (Zt 



9 = + 0.01 + 0.6531 1 tZ L + 1329.09 <Z 7 + 0.38994 d-n — 0.08439 dcf — 0.04305 d Q + 0.34268 d i 

 9, 0=+ 38.12 — 0.00218(ZL+ 38.47(Z7 + 0.00218dn- — 0.18710fZ9i + 0.47301 da- 1.14371 (Z i 



10 = — 317.73 + 0.69957 (ZL+ 1719.32 (Zy + 0.12913 <Z 77-1.38787 (Z 9 + 0.17130 (Z a — 0.08360 <Zi 



11 = + 117.97 — 0.01315(ZL— 43.84(Z7 + 0.01315cZ7r + 0.02929(Z9)+ 1.02138(Za — 0.27187 cZi 



The probable error of one of these equations is /a ^ + 90", the weights being 

 equal, excepting for O^, which has been excluded from each of the solutions. 



As an illustration of the mode of applpng the limits (43), we will make in (41) and 

 (42) g = — ■, we shall then have 



(x =. d L)< ± 100 " , da<± 0.013, 

 (3,= dyX ±0".07, cZJ<± 18. 



C, D, E, and F, being of the same order of magnitude with A, we may conclude from 

 (42), (43), and the above values oida and dh, that, if we reject in all the equations the 

 two right-hand figures from the values of m, and the three right-hand, figures from all 

 the other numbers, writing, for instance, the first equation 



= — 184" + 0.79 rZ L + 140 (Z y + 0.39 dTt-\- 0.96 dq, — 0.19 (Z a + 0.17 d i, 

 and the second 



= — 7" — 0.03 (ZL+ 50<Zy + 0.03d7r — 0.21 (Z9 + O.I6 (Za + 1.26 <Zi, 



the probable error of the value of either of the quantities of dL, dy , when the 



equations thus written are solved by the method of least squares, will exceed the prob- 



* Disquisitio de Elementis EUipticis Palladis. Comment. Soc. Reg. Gottingensis Recent. Vol. I. 

 t The computations necessary for the solution of these equations have been executed by Messrs. J. F. 

 Flagg and T. H. Safford, jr. 



