32 Walter Gyllenberg 



Substituting in equation (15) U', V, W from the formulae (1 7*) the exponential 

 expression gets the form: 



(18) e- lhf , 

 where 



(19) f= A"V" 2 + £"F' 2 -f C"W" 2 + 2 D" V" W" + 2 E" W" V" + 2F"U"V". 

 Here is 



J" = A' Sll » + 6l8 » + C e 18 «, 

 B" = A ' e 21 2 -f B' e 22 2 + C's 23 2 , 



(20) ^"j> i+ fy + l0 ^ 



0 = A S 21 E 31 + B £ 22 £ 32 + C S 23 £ 33 ' 



E" = 4' e 31 s u + B' e 32 e 12 + C ? 33 e 13 , 

 F" = ^' s n 8„ 4- B' e lg s 22 + 6" e 13 s 23 . 



Our problem is now to determine the coefficients A", B" , C", D", E" , F" 

 from the observations. 



Turning the system of coordinates so that it coincides with the i£" 3 -system or 

 in other words substituting the values (16) in (19) we get: 



(21) f=AU s + B V s + CW 2 + 2 DVW + 2 EWU + 2 *W, 

 where 



A = A" Tll 2 + B" y 21 2 + C" Ï31 » + 2 0" y 21 T;u + 2 0" Tsi Yll + 

 + 2*"' TliT211 



B = yl" Tl2 2 + ii" T 22 2 + G" t 32 2 + 2 /)" T „ Y 3 2 + 2 JS?" Ï32 ïl2 + 



0 = a" ïl3 2 + B" Ï23 2 + c" + 2 zr Ï23 y 83 + 2 *r Ï33 Tl3 + 



(22) D = A" t 12 y 13 + ß" Y 22 Y 23 + T 32 Tan + D " (T22 T33 + Ï32 T23) + 



+ (T32 Y13 + Ï33 T12) + (T 12 Y23 + Yia T22). 

 E = A " Tib Tu + Y 23 T21 + C " T33 T31 + D " (Y23 Tai + T33 T21) + 



+ ^" (T33 Tu + T31 Tis) + F " (Tis T21 + Tu Tas). 

 F - A " 'In T12 + ß " T21 T22 + C " T31 T32 + 7> " (Tm T32 + T 8 i T 22 ) + 



+ E " (T31 Y12 + T 32 Ï11) + F " ('In T22 + Y12 T 2 J- 



29. In order to get a solution of the problem exclusively from observations 

 in the line of sight, i. e. the Z-axes, we have to integrate the function (18) over all 

 values for U and V, f having the form (21). 



Performing this integration we get 



(23) <t> 0 = K.e- i/iBWt , 

 wliere 



ABC — AD- — BE 2 - CF 2 -f 2DEF 



(24) H = 



AB — F 2 



