492 



Denoting the projecting vector by ^ (tensor K, direction cosines 

 xi, components Ki) we have really 



2i + ^ + g + . . . + ^D2 = if . 

 As ^ is perpendicular to '^, $, S, . . . , we have 



(•21, il) = 0, ('$, 5?) = 0, (^, .^) =: 0, etc 



or 



[«, Ki ] =z 0, [/?,• isT,- ] = 0, [y,- TT,- ] == 0, etc. 



or because 



Ki=Ai-{-Bi-\- Ci+... + Mi=aiA + /?,-^ + y-C + . . . + J/,-, 

 [a,^ A + [«,/:?,] ^ + [«;y,] C + . . . + [aiMi] = 0, 

 [/?,«,] ^ -f [/?,-^] i? -}- [/?,y,] C + ... + [/^iMi] = 0, 



[y.«/] A + [y,/9,] 5-1- [y-] c 4- . . . + [y.^//] = 0, 



By multiplying these equations by V^lgiai"], }/[(/ilH''], ylgiCi"^, 



. resp., we obtain the "normal equations" : 



[gici''] X + [giaihi] y -\- [giüiCi] ; + ... + [giaimi] = 0, 

 [gibiüi] X + [gibi'] y + [gihici] z -{-... ^ [gihimi] — 0, 

 [giCiüi] X + \()iCibi] y -f- [^,c,-'] ^ -f . . . + [(7/ c,- 7n/ ] = 0, 



II. After these developments which also are given by von Schrutka 

 and Rodriguez we proceed to determine the weights of the variables. 



For this we notice that all the quantities Afi have the weight 'J, 

 and therefore have an equal mean error e. From this ensues, that 

 the projection of ?)i in any direction has the same mean error s. 



We have to investigate the influence on ti due to the variation 

 of ^, if the other variables ^, (^, . . . do not undergo that influence. 



A variation of ^l which does not displace the foot on AVy of the 

 projecting vector ^, does not act upon any vector '^1, ^, <i, ... So we 

 have only to do with a variation of the projection W of i53ï on i?^v ■ 

 In order to leave the vectors 03, 5, . . . intact, the foot is to be moved 

 in a direction i> perpendicular to $, ^, . . ., and, because it lies in 

 i^AS also perpendicular to -S". 



Denoting by Gi the direction cosines of é, we may put the equation 



(•2l,6) + (T)?,e).= 0, 



obtained by multiplying the equation of observation scalarly with 

 é, in the form 



AlaiOi]=z — J\Js 

 Ms designating the projection of 5)ï on é. 



As 31s has the mean error s, the mean error sa of A equals 



