164 



[(/ibiüi] X + [gihi^] y + [i/,6,cj ^ + . . . + [gihinu] + [qj -^ ]' = O, 



y 



[,^/Ciai] .V + [///<V^>/] .'/ + [.'//■ c/M e + . . . + [/7ic/wj ] + \qj ~-~ ] = O, 



These N^ equations serve, together with the v conditions *Pj = O, to 

 determine the jV variables .r, y, c, . . . and the v auxiliary quantities cjj'. 



Now the solution of the problem is not represented by the centre 

 of the hjperellipsoids i2, but b}^ the point, iji which the intersection 

 space <i> (space of conditions) is touched by an individual of the set 

 of the hyperellipsoids 52. 



The analytical treatment of the problem is simplified by taking 

 the coordinates so small, that in the expressions <Pj homogeneous 

 linear forms suffice. The geometrical meaning of this is that a new 

 origin (>' (.^'^,, ?/„, z„, . . .) is chosen in the space of conditions *P near 

 the probable position of the required point. So the sj)aces 0j are 

 replaced by their tangent spaces Rj, and the space of conditions by 

 its tangent space R of jS' — r diinensions, intersection of the tangent 

 spaces Rj. 



Denoting the coordinates obtained by ti'anslation to O' by $, »i, S, .., 

 so that X = .i'o +§'••• '^"^^ putting 



(tix^ + /?/y„ + yiz, 4- ... /!/=: m , aix^ + hiy^ + c,-r„ + ... + mi = mi, 

 we find 



2/7= [p^icax + ^,y 4- yiz .. + {liY] = [pi{ai^ + /?,'»; f y^'? + •• + 1^)'] 

 or, putting 



«is + ^in + V'b -]-'■' i^i= Vi, 



2U=[piVi^]. 

 The equations fPj {x, y, z, . .) = may now be ^vritten : 



or, since 0' is assumed in *Pj:=zO, and higher powers of ^,n,^,--- 

 are to be neglected, 



d<Pj dfpj d<Pj ^ . , s 



-^è + ~^n + ^^ + '-' = ^ = 1,....) 



Ox oy oz 



or 



Wj = «y § -1- ^/ .i + yy g + . . . = ^ aj' ê = 0. (i = 1 , . . . V). 

 So the normal equations appear in the following foi'm 

 [a/a/M ê + [flioif^i] n + [f/iaici] C + . . + [y;aim{] + \qj aj']' = 0, 

 [y, hi a; J 5 + l^.' hi '\ i] + [yi 6. c/ ] ? -f . . + [en 6/ m; ] + Vqj l'*j\ = 0, 



