Closest Fit to Systems of Points in Space. 561 



line going through the centroid. Hence : Hie best- fitting 

 straight line for a system of points in a space of any order goes 

 through the centroid of the system. 



Now let there be n points each fixed by q variables 

 #1, x 2 . . . ar , and let 



Sj = S [x l )/n, .r 2 = S fa) /n . . . i g = S fa) /n. . (i .) 



fix the centroid, or the mean values of the variables ; 



(t\ — S fa 2 ) /ft — x^, a 2 x 2 = S (x 2 2 )/n — x 2 2 , . . . 



0-% = S fa 2 ) In — a#*, . (ii . ) 



fix the standard-deviations (errors of mean square), or in- 

 directly the moments of inertia or second-moments about the 

 axes of coordinates, through the centroid parallel to the axes 

 of the variables x Y , x 2 . . . Xq. And, lastly, let 



S (x u Xv) — nx u x v .... , 



rxuxv = (111.) 



nax u <rx v 



for all pairs of values of u and v from 1,2, 3, . . . q, fix the 

 congelations of the variables, or indirectly the products of 

 inertia or product-moments about the axes. 



Now let li 9 l 2 , Is . . . lq be the generalized direction-cosines 

 of a plane at perpendicular distance p from the origin. We 

 shall have 



V+V+V+---+V= s i ( iv -) 



Further, if U be the sum of the squares of the perpendicular 

 distances of the system of n points from the plane 



l x X, +Z 2 #2 + '3#3 + « •• + 'y*f==l»j « • • • (v.) 



we require to make a minimum of 



U = S(/ 1 .r 1 + l 2 x 2 + kx z + . . . + IqXq—p) 2 , . . (vi.) 



by variation of l u l 2 , . . . lq, p subject to (iv.). Differentiate 

 first with regard to p and we have 



*iS fa) 4- / 2 S fa) + IS fa) + • • • + lq$ fa) -wp=0; 



.*. p = hxi + l 2 x 2 + . . . + l q x n , .... (vii.) 



which shows us from (v.) that : the best-fitting plane passes 

 through the centroid of the system. 



Now vary (vi.) and add to it Q times the variation of (iv.), 

 Q being an undetermined multiplier. We have, by equating 

 to zero the coefficient of dl u , 



hSiWu) 4- Z 2 Sfa#„) + . . . + kSfa 2 ) + . . . + ?,Sfaor«) 

 -^Sfa) + Q/«=0. 



