564 Prof. K. Pearson on Lines and Planes of 



detain us long. Let its equation be 



X, — x{ X 9 — X 2 X % — xJ Xn — Xn , ... N 



Draw the plane perpendicular to this line through x{ y x 2 , 

 x^ ... xj\ i. e., 



l l X l + l 2 X 2 + l 3 X 3 + . . . + IqXq = H, 



where H = \x x + l 2 x 2 + l 3 x/ + . . . 4- IqXq. 



Then if p be the perpendicular from any point in space on 

 the line (xiii.): 



p* = fo-^4- (*,-**')*+ , . . + [Xq-Xq'Y 



- { ?! {X X - «/) + l 2 {X 2 ~ X 2 ') + l 3 (#3 - «»') + ... + l q {Xq- .?'/) } 2 . 



Now Xi, x 2 . . . Xq and l l9 l 2 . . . Iq, subject to the relation 

 li+l 2 +l 2 + . . .-\-lq 2 = l, are the constants at our disposal. 

 Sum p 2 and differentiate to find when U = S(p 2 ) is a minimum. 

 We have for type equation 



whence we see : 



— ^-^4 — = symmetrical function of x's. 



Or, we must have 



M\—®\ _ X 2 — X 2 _ _Xq — Xq 



'l '2 iq 



which show us that the straight line passes (as we have already 

 noted) through the centroid of the system. We can accord- 

 ingly take Xi,x 2 . . . Xq to be that centroid, and we find : 



<*,« U S(p2) 2,9. 1 2 



n n v 



+ 2^ 0^0-^ ^ x j? 2 + . . . + 27 9 _i/ 9 o-^_i<r a . 9 r^.ir Jr? ] . 



But the expression in square brackets is precisely the square 

 of the mean square residual with regard to the plane, 



?lOl — #l) + h( x 2 — ^2) + • • • + lq (Xq — Xq) = 0, 



or % 2 . Thus we have : 



Now clearly <r 2 x -f <r 2 Xo + . . • + o 2 x q is a constant. Hence 

 5/ 2 will be a minimum when 2 2 is a maximum, or when the 



