CHAPTER VI. 



The normal equations and the characteristics as referred to 

 the system of the equator. 



18. At the end of this memoir we in tables II, III and IV give the values of 

 the coefficients of the equations (58), (64) and (66) for each square. From these the 

 normal equations are computed. 



For the second characteristics we obtain the normal equations 



X i . 11.9312 -f ,y t . 2.3492 + ^. 1.6016 = 27.1655 2' — 3.0432 

 X x . 2.3492 -f- y l . 1 1.8124 -f ^ . 1.6600 = 53.5855 q — 9.2538 

 X l . 1.6016 4- i/ 1 . 1.6600 -f #1 • 13.0340 = 33.6440 q — 5.4571 



. 17.1058 = — 4.0998 q -f 0.5636 



^ . 14.7558 = — 0.5154 q -f- 6.6289 



1\ . 14.5844 = -f 3.1967 q' — 0.5963, 



which give 



V^' 2 'oo 





= 4- 1.2281 q 



— 0 0697 



vjn'.o 



= Vx 



= + 4.0236 q 



— 0.7249 







= 4- 1.9110 q' 



— 0.3177 





= lh 



= — 0.2397 q' 



+ 0.0330 





= ïi 



= — 0.0349 q' 



+ 0.4492 







= 4- 0.2191 <j' 



— 0.0409 



To obtain, from these values, the axes of the ellipsoid we must form the 

 cubic (61*). We get 



S 3 — .S 2 (7.1627 q — 1.1123) 4- s (14.8707 q ' 2 — 4.2923 q' + 0.0983) — 

 — (9.1421 q 3 — 3.6607 q 2 4- 0.2064 q' — 0.0017) = 0. 

 This equation we cannot solve without using numerical values of q . Hence 

 we give in table VI the roots for four different values of q . Computing then the 

 vertices from equations (63) and the axes from (61**) the table VII is obtained. 

 The unit is the class breadth which is very near equal to the velocity of the sun. 



