OF ARTS AND SCIENCES. 69 



" The above places are referred to the mean equinox of January 1st, 

 1847, and mean obliquity. The planet's place is corrected for stellar, 

 but not for planetary aberration. It is also corrected for planetary par- 

 allax. The residual errors, though small, show in the course of six 

 months a sensible deviation of the orbit from a circular form. They 

 show at the same time that if the eccentricity is greater than 0.06, the 

 true anomaly must be nearly ± 90°, a possible, but, as was said be- 

 fore, an improbable case. 



" The next step in the investigation was to make equations of condi- 

 tion of the form, 0=ax-{-by-{-cz-\-n/m which x is 50 X z' ^, 

 Tj = 10 X J V, z =^ J A300, a, h, and c are computed coefficients, v the 

 daily increase of X, and A300 the heliocentric longitude of Neptune on 

 the 300th day of the year. Finally, n is the equivalent heliocentric 

 value of z/ a above with sign changed. The number of equations 

 was reduced to nine, by taking for the first the one third of Nos. one 

 and two, above. Then follow the next five unchanged, then the mean 

 of Nos. eight, nine, and ten. No. eleven is rejected, then the mean 

 of Nos. twelve and thirteen. Lastly, the mean of Nos. fourteen, 

 fifteen, and sixteen. The nine conditional equations have then equal 

 weight, and stand thus : — 



° Residual Error. 



£ 



= — 0.303 X X — 2.700 X y + i X z -{- 3.8S , — d!o8 



-_|. 3.016 —3.000 +1 +1.00, + 0.49 



-+3.363 —2.400 +1 — 0.27 , + 0.19 



= + 3.6S5 -1.800 +1 -1.10, +0.22 



— + 4.038 —1.000 +1 —3.07,-1.03 



— + 4.268 — 0.200 + 1 — 4.12 , — 1.31 



— + 4.594 + 1.267 + 1 — 2.44 , + 1.03 

 = + 4.248 + 3.950 +1 — 1.73 , — 0.13 

 zz: + 3.332 +6.133 +1 +1.81,-0.16 



The solution by least squares gives, 



= 118.879 XX+ 7.477 X 2/ + 30.443 X z — 45!'629. 

 = 7.477 X a; + 85.149 X 2/ + ^-^^O X z + 1-687. 

 0= 30.443 X a;+ 0.250 X 2/ + 8.111 X z— 8.627. 



II 



Whence, 2 ee = 4.21. 



x =1 + 3.255712. 

 y=: — 0.272963. 

 z = — 11.1475. 



