Planes of Closest Fit to Systems of Points. 



369 



Multiply equations (4) by \s (s = l, 2, . . . . k), and add 

 to (3). Then, by the ordinary theory o£ maximum and 

 minimum values, we know that the coefficient of da* 

 (* = lj 2, . . . .n) in the equation so obtained must vanish* 

 This gives the n equations 



\Pti + \pt2+ +^kPtk 



= B. 0t — a i R lt —a 2 R2t— —ajint, . . (5) 



if R ttl . = S(a? tt a7»)=R lJlt for all positive integral values of u 

 and r. 



(5) gives n equations connecting the (n + k) unknowns, 

 a l5 a 2 , . . . . a n , Ai, X 2 , . . . . Afc. (2) gives k other equations 

 between the a's, in which, however, the X's are absent. 

 Thus we have the following set of equations from which to 

 determine the a's and the X's : — 



a 1 R lt + a 2 R 2t + . 



Let A denote 



+ a n R nr \-\ i pti + ^2Pt2+ +Xkptk = Rot- 



0=1, 2,.... o 



+ a n p f 



(* = 1, 2, ....*.) 



POs 



K 6 ) 



RoO RlO R«0 i ? 01 P02 fOk 



R 01 R n R nl pn pi2 ^i/fc 



R()» Rl>i • • • • R»ra 2 7 "1 P"2 • • « • P«i 



jOul ^11 pn\ 



^02 ^12 #13 



P0k Plk Pnh 



a determinant of the order (n -f h -f 1) . 

 Ihen the solutions of equations (6) are : 



an( 





\= - 



Ago 



A( tt + ff ),o 



(« = 1, 2, ....») 

 (.=1,2,.. ..A) 



• (7) 

 . (8) 



