82 



C. V. L. Charlier 



We evidently hfive: 



A" + B" + C" = A' + 7?' 4- C. 



The right member of (56), put equal to a constant, gives the equation of the 

 velocity ellipsoid, referred to the astronomical system of coordinates . If it were 

 possible to determine from the observations the values of the 6 constants A", B" , 

 . . ., F'\ we should have the directions of the three axes of the eUipsoid, as well 

 as the magnitude of the axes, through the known theory of surfaces of the second 

 degree. 



21. If in (56) the velocities referred to are introduced, we get 

 (56) f= AU' + BV + CW + 2BVW ^ 2EWÜ 2FUV, 



where 



A = A"^i,,' + 5"t,,^ + C"'T3i^ + 2/)"t,,Tbi H- 2i;"T3xTn + 2J^"TnTn^ 

 C = A"'!j + + 6"'t33^ + 27)"t,3T33 -f 2E"t33Ïx.. + 2F"t,3Ï,3. 



^ = ^'1"Ti2Ti.S + ^"T22T23 + <^"T32Ï3.3 + ^"(ï22 TsS + T32 T23) + ^"(ïssTlS + Y33T12) + -^"(ïl2 Ï23 + T13 Tsä) 

 ^ = ^■'"TisTu + ^"T23Ï21 + <^"T33T31 + ^"(T23 T31 + T33 Ï21) + ^"(^33 Tu + T31 T13) + -f"'(Tl3 Tsi + Til T23) 

 ^ = ^"TuTi2 + i^"T2lT22 + <^"Ï31Ï32 + ^"(ï21 Ï32 + Tsi TS2) + ^"(ïsi T12 + T32 Til) + ^"(ïn Ï22 + Tl2 T21) 



Here U and V denote the components of velocity at right angles to the line 

 of sight, whereas W is the velocity parallel to this line. If for any square the 

 disti'ibution of these linear velocities were known, we should have the values of the 

 6 quantities A, B, . . ., F; and the values of A", B"\ . . ., F" could then be found 

 from the six linear equations (58*). The problem to determine the three axes of 

 the velocity ellipsoid then were completely solved. 



The spectroscopical observations of the motions of the stars in the line of 

 sight will give us, once, a means to solve the problem more indirecti}^ as then at 

 least three square are wanted. 



For the present we have to consider the problem to determine A", B", . . ., F" 

 from observations of the proper motions perpendicular to the hne of sight. If a 

 group of stars, in a certain square, is considered, and only the motions perpendicular 

 to the line of sight are taken into consideration, the third component W may have 

 any value. We, hence, have to integrate 



over all values of W from — cc to -)- • ■'^) 



Performing this integration / takes the form 



(59) f=A,U-' + B,r'^2F,UF, 



') Compare Schwabzschild: Nachrichten von der Kg).. Ges. der Wiss. zu Göttingen. 1907. 



