The general Characteristics of the frequencyfunction of stellar movements 31 

 Further from (11) 



£i _ ft. 2 V 1 g l~ Q-l ft" _ Q.^2 x l ~l ~ ' r i" £f" A i X \ H\ ~ P i 



(60) 



*3 



A'., 



7)" = * t 



ft 2 Pi - 3j gl 



7?" = » 



7c. 



when 



* 0 = #, 6 ilf = 



= *i ?/i *i — «i <7i 



F „ { s ^ - P t 

 1 



Pi 2 + 2p, <i x r x . 



Pi. yi. ?i 



Having found iL", B", C", /)", F" we obtain the axes of the ellipsoid 

 from the equation 



A"— F", E" =0, 

 F", B"—k, D" 

 E", D", C"—k 

 where, the roots being £ t , £ 2 , £ 3 , we have 



= A' i, — B g 8 = C, 

 and the equation of the ellipsoid 



A'U'* + B' V"- + C'W'-= 1. 

 However, to obtain the axes of the ellipsoid we need never make the com- 

 putation (60). Indeed, on account of the formula (10), we have 



V 



and it follows that the roots s L , s 2 , s 3 of the equation 



= 0 



(61) 



are equal to 



x x — .9, p t , r, 



Pi- Vi— *, 'ii 



V 

 il' 



B 



or, calling the axes of the ellipsoid o 1 , o 2 , o 3 , we have 



(61**) s^V 0 ! 2 



Writing the cubic (61) 

 (61*) s 3 - 

 we have for the coefficients 



K = x i + V\ + 



s s = V a 2 2 



\ s 2 + h t s — Jc 3 = 0 



(62) 



k 2 = x 1 y ± + y x e t — q* — r* 



k 3 = », ^ g t — »J g, 2 - «/ t >', 2 — Pl « _j_ 2p, g, r, 



