The general Characteristics of the. frequencyfunction of stellar movements 



29 



By the way, it may be remarked already here that on account of the symme- 

 try of the squares the normal equations take very simple forms. Thus it will never 

 be necessary to solve systems of normal equations with more than 6 unknowns and 

 even that only in one single case. 



14. We will first develop the method to find the characteristics of the second 

 order in system II and to compute the vertices and the three axes of the velocity 

 ellipsoid. The method was worked out by the author in January last year and a 

 preliminary notice was published shortly afterwards l . 



Simultaneously a method using only two of the three characteristics available in 

 each square was published by Gyllenberg in connection with his studies of the 

 radial motions 2 . 



Up to that time only a method assuming an ellipsoid of revolution had been 

 worked out by Charlier. Indeed, Charlier makes some general remarks concer- 

 ning the solution of the problem of the three-axial ellipsoid, but on the whole he 

 seems to have regarded it as too complicate 3 . 



From the equations (36) Charlier (p. 83) has developed the formulae 



A 1 C= AC-E 2 = {B"C"— D"*y ( 12 2 + {C"A"—JE" s )i 22 * + [A"B"-F" å )^ 2 2 



+ 2{D"E"-C"F"y lr2 -i, 2 + 2{E''F''-A''D")i 22 \ 32 + 2{F'D''-B"E') Wl2 



B y C= BC-& = (B"C"-/)" 2 ) Tu 2 + (C"A"-E"*y !n * + {A"B"-F"*ft 31 * 



+ 2(D"E"-C"F")^ 21 + 2(E"F" — A"D")'( 21 y„ + 2(F"D"-B"E"Y( 9l Tll 

 - CF, = -(FG-DE) = (B"C"-B"^ n Yl2 + (G"A"-E"^ 2lha + (A"B"-F" s ) Wä2 



Theoretically, if the quantities A t , B t , F i are known for two squares, andas C by 

 equation (36) is a linear function of A", B" , C", 7)", E", F" it seems as if the 

 solution of those last quantities would lead to equations of the twelfth degree. 



The whole difficulty consequently lies in the factor C of the left membrum. 

 But we already have derived a property of C that will help us from this difficulty. 



We have written 



and in the § 6 we found the equation 



(32) 



1 S. D. Wicksell: A general Method to determine the three axes of the velocity ellipsoid 

 from the proper motions of the stars. Medd. f. Lunds Observ. N:o GO. 



- K. A. W. Gyllenberg: On the three axial distribution of the velocities of the stars. 

 Medd. fr. Lunds Observatorium N:o 59. 



8 Charlier: Stellar statistics II, chapter II. 



+ (D"E"-C"F")(i ll i S2 + W21 ) 

 + (E"F"-A"D"){ Ws2 + Wax ) 

 + (F'D"-B" J Er')(T 8 i7i, + T„T 11 )- 



A = 



A, F, E 

 F, B, D 

 E, D, C 



»i = I A, Ft 

 I F v B y 



