374 Mr. E. C. Snow on Restricted Lines and 



The conditions for a minimum give 



~&\ + ph + p\P\\ + P2P12 + .... +flkPiJc = 0,-v 



(12) 



Xn+tlln+flip n i + fl2Pn2+- + P<k Pnk = 0,J 



where fi, fi u . . . . fik are undetermined constants, and 



X s = S(Z 1 a? 1 + +lnXn)&a 



= tirli s + • • • • + Inllnsi 



where, as before, 



Multiply equations (12) by Z l9 Z 2 , 4 respectively, and 



add. Remembering the conditions (10) and (11), we at 

 once obtain 



^-f/ 1 X 1 + / 2 X 2 + .... +4X a = 0, 



where l t . . . . l n have the values which make V a minimum. 



But then IJL 1 -f Z 2 X 2 + + Z»X„ becomes V m , the minimum 



value of V. 



It follows that Tr 



fi — — V m . 



Substituting this value of /a in (12), we shall have with (11) 

 (n -f k) equations between l x . . . . Z„, yu, x ... . /aa. and V wl . Hence 

 we can eliminate the Z's and the fi's and obtain an equation 

 to determine Y m . 



Since X s = l^+ .... + /Jt„ s , 



this eliminant can be written in the determinantal form 



Ru — Ym R 2 i ••- R»i Pn Pu P\h 



^12 R22 — Y TO .... R„ 2 P21 P22 • • • • P2A 



D 



Rm 



^2n 



H nn — Y J m pm p n 2 Pnk 



Pn 



P21 



.... Pnl .... 



P12 



P22 



.... p n2 .... 



P\k 



P2k 



Pnk 











= 0. 



This is an equation of the (n — k)\h degree in Y"\ Its 

 roots are necessarily positive (being the sum of a number of 



