The motion of tlie stars 



81 



The direction cosines of relative to K.^ may, similarly, be defined by the 

 scheme 



so that 



(54) 



and 



(54*) 





X" 



Y" 



Z" 



X' 





hl 



hl 



r 



'^12 





c 



-32 



z 



c 



-.3 



S 



^33 



ü" 



V" 

 W" 



hl v + s,, r + w\ 



Hl U' + h2 v + ^23 



r 



'33 



w 



f/' = s„ f7" + E,, F" + s3,Tr", 



y' = h2 C^" + ^22 F" + £3,Pf", 



The direction cosines of relative to can now be expressed by and ay. 

 We evidently have 



(55) 



cos X'X 

 cos X' Y : 



cos X' Z ■ 



'^11 111 "^21 hi "t" '^31 Ts I 1 

 Ti2 "I" ^21 T22 ~l~ ^31 Ï32 > 

 -1 1 'fis ~1~ -21 T23 ^31 T33 ■ 



(55*^ 



cos FX = Tu + e22T21 + S32T31' 



COS Y Y = =i2Ti2 + =22T22 ^32T32' 

 cos r'Z= £j2 7i3 + Y23 + S32Y33. 



^13 Til ^23 T21 4~ =33T31i 



(55==^*) 



COS Z'X 



cos Z Y — S^gTj^a + ®23T22 ""I" ^33T32' 



COS Z' Z ■■ 



= 13 T13 ~l~ ^23 T23 "t" ^33 T33 ■ 



20. The system is the usual astronomical system of coordinates. The 

 position of K.^, for each square, referred to is known. The values of Yy, for 

 each square, have, indeed, been given in tab. VII. The problem is to determine 

 Sjj through observations of the projected proper motions in our 48 squares. 



Using the relations (54*), / accepts the form 



(56) /= A"U"'^ + B"V"'- + C"W"-' + 2D"V"W" + 2E"W"U" -f 2F"IT"V'\ 

 where 



-11 



C" = A'b^^^ + B' 



'11 -21 



Lnnds Univ:s Årsskrift. N. F. Afd. 2. Bd 8. 



B" = A' 

 C" = A' 

 D" = A's^^ _ 



F" = A's,, So, + i^'£i2^22 + C'£,o£ 



21^ + -ß'^22^ + C"£23^ 

 -(- C ^33% 



^~ ^ =22=32 ~i~ ^ ^23 -33' 



21 -31 



13 '23 • 



11 



