ïlie motion of the stara 83 



where 



(59*) GB^ = BG—D\ 



CF^ = FC - DE. 



The observations give the value of A^, and F^. The results from tivo 

 squares give us six equations, which — theoretically — suffice for determining tlie 

 values of A", B" , . . ., F" and then of the constants of the velocity ellipsoid. 

 Actually the general problem seems to be rather too complicate for an analytical 

 solution ^). I give however some developments, useful for the general problem, 

 before passing on to the more special solution. 



22. Substituting the values (58) of A, B, . . , F v/e get, after some reductions, 



AC—E^ = {B"C"—D"y!,,' + {C"A"—E"%/- + (A'' B"—F" 



+ 2{D"E"-G"F"yi,,'!,, + 2{E"E'-A"D"yi,,-(,, + 2(F"D"- B"E"yi, 

 BC — I)'' = {B"G"—D"'Y(,,' + {G'A"—E"')^,J + {A"ir'—F"-'Y; 



2 



(60) + 2(i)"^"-C"i^")YnTn + 2(£"F'-^"/>")T,, T,,i + 2(F" D"-B" EJi,,';,,, 



FG - DE = - (B" G"~D"'yi,, y,, - [G" A" -E'^Yi,, - {A"B"-F"'Yi,, 'i,, 



- {D"E" - G"F") (Tj, + Ï12 - (E"F"-A"D") (y,, Yss+T^^- Tsi) 



-(F'D"-B"i;")(YBiYi2+T3äTu)- 



Using the formulae (57) which express the velocities in as functions of the 

 velocities in K^, we get 



B'C"—D'' -= B'C's^^' + C'^'s,^' + ^'B'e^,\ 

 G"A"—E"' = B'G'e,,- + G'A's.,,' + A'B's^^^^ 

 . ,Q^^ A"B"-F'' - 5'C'£3,^ + G'A',,,^ + A'B's,,\ 



^ > B"E"-G"F' = B'G's^^s,^ + G'A's^.B,, + A'B'.,,e,,, 



E"F"—A"U' = B'G'^^e^^^ Ga's,,b^, A'B'b,,^,,, 



Substituting these relations in (60) we finally get for the coefficients in (59) the 

 following form, explicitly expressed as functions of tlie half-axes [A' = \ : a,- etc.) 

 of the velocity ellipsoid and their cosines of direction {sy) referred to K^. 



A^G== i?'ü'(£,, Y12 + + ^31 Ï32)' 



(02) + C'A'(£,2Yi2 + ^22T22 + S33Y32)' 



-\- A B (ei3Yi2 + ^23T22 + ^ssTss)^' 



B,C= ^'f^'KiYn + S21Y21 + S31T31)' 

 (62*) + C'A'{b,, Yn + ^22 T21 + ^32 T31)' 



+ ^'B'ihsln + 223T2I + =33T3l)'' 



^) Having already ended this research I found a possibility to solve the general problem, 

 which suggestion I communicated to Magister Gyllenberg, second assistant of the observatory. 

 He has also, I think, succeeded to work out the complete solution for a velocity ellipsoid with 

 three unequal axes. 



