496 



to n-\-v, so tliat [x/ai] = J>: xn-^j'itn^j ^ [■ün-\-j'(tn-^j]". Here and in 



1 



what follows [/,]' will denote a sum over h from 1 to n.and[„^j']" 



a sura over j from 1 to v. 



We maj tlierefore write 



or, because 



Kk = Ak + B,, 4- Cn + ... + Mu = rtj^A + /?/,B + y;,C + ... + iti;;, 



[«/,^]'^ + [«AA]'i>^ -I [«/,r/,]'r + . . + [«.^^1//-]' = [an+jK„+j']'\ 



[rA«/-]'-i + [nf^i^]'J3 + [vA^'C + .. 4- [yM]' = [7.+,ir„+/]", 

 » 



Putting 



we iiave 



[«/.^]'^ + [«/,/?/J'i? + [«Ay/JC+ ... + [«A^i//.]' + [«/Q,]" = 0, 

 [/3hm]'A + [/?A^]'i? + y9,Yi]'C + ... + [fikMn]' + [A-'Q,]" = 0, 



[na/.l'^'i f [y/./?/J's + [y/ri'c + ... + [r,,Mi,]' + [y/Q,]" =: o, 



Introducing 



«y'|/[i/^«/^] ,, , f^jVbi^n 

 we obtain, after multiplying successively by [^[gia;^], |/[^A'], 



y[gicn, ... 



Ci/ArtA'j'A' + [(/hdhi^hh/ + [^//.«/(C/J.- + ... + [ghakmh]' -f [a^'^_y]" = 0, 

 [9hhah]'''i^ -\- [(lhhh^]'y + iif/J'hChyz + ... + [p^/i^A^nA]' -f- [èj'^^]" = 0, 

 [f7//'/<^//]'.^- + [<jhChhi>]'y -\- [(jhc/rjz -f ... + [ghChmf]' + [cj'gy]" = 0, 



> 



^ equations, which together w^ith the v conditions 



a/x + bj'ij 4- c/z + ... + wj^-' = 

 serve to determine the N' variables x, y, z, ... and the v auxiliary 

 quantities qj . 



IV. In order to determine the weights of x, y, z, . . ., i. e. of 

 A, B, C, . . ., we must examine the influence undergone by vl from 

 a variation of 9)?, the vectors 35, ^, . . remaining unaltered. 



A variation of 3)ï only acts upon 51, 53, C^, . . when the foot of ^ on 

 the space ^'a_v of intersection moves. If the foot is fixed, iv may freely 



