166 



1 n' . ?' 



1 -' C' 



n- Ui.. 0-^ 



m 



yis 



1 li 



<7? Vil 



Oir> 





"^-^-w^V 1 + = 0, 



+ 



-7. • "^i 



+ = 0, 



the conditions 



1 7i' C 



o 



(i 



i')- 



9i 9d' 



also being satisfied. 



From the above JV -\- v equations with the ^V unknown quantities 



1 rt' ^' sj 1 



— , — , — ^ , . . . and the v unknown quantities — ;:; , -^ can be solved. 

 9i 9d' 9i^ 9ir>' 9( 



The method of solution of Hansen is found again by introducing 





whence 



§' _ 1 _ 



Av ^0/ 5 



9i 



^f* 





2oiA. 



= B 



91^' 



c. 



{i=\,...n) 



Then the modified normal equations furnish 



[giar^ A 4- [giaihi^ B + [^/a/o] C + . . . -(- [kjaj]; = 1, 

 [r,,-6,a/] ^ + [gihr^ B + [^;6,c/] C + . . . 4" [V»^/]' = 0, 

 [a.r.a/] A + [r/,.vi,] 5 + [fiic,'\ C H- . . . + [XvVyj' = 0, 



or 



or 



\jgiCi{^aiA)] + ik;Yj'M = ^^ etc. 



[^v«/] + [V«jT = 1' L^-' 6/] + [/.yii;]' = o, [hc,-\ -4 [^/y/]' = 0, etc., 



and the (rigorous) equations of condition run 



^fij' A = {j=\,,..v). 



From the set of equations 



^aiA = — (i r= 1 , . . . n) 



9i 



[Ava;]+[V«/J'=l , [^7^/]+[^.//^i']'=0 , [^Vo]+[Av'y,/J'=0 . etc. (A' in number) 

 ihe .V variables A, B, C, . . ., llie n unknown quantities ki and the 

 V anxiJiarv quantities IJ can now be solved. 



