Planes of Closest Fit to Systems of Points. 375 



squares of real quantities), and 'the smallest of these must be 

 taken. When this value of V m is substituted in (11) and (12) , 

 (n-\- k) homogeneous equations in l x l 2 . . . . In, /^ (jl 2 . . . . n k 

 are obtained, (n + k — 1) of which, together with (10), suffice 

 to determine all the Vs and the fis. As before, this can be 

 done in determinant form, the order of the determinants 

 involved being Qi-tk — 1). 



Particular Cases, 



5. Useful particular cases of the general formula are 

 obtained by taking k — and k=l. 



(A') If k = } we derive the case analogous to (A) above. 

 The equation to determine Y m takes the well-known form 



R n — V w R 21 



Ri 



\\ 



R/i2 

 R — V 



= 0. 



Putting n = 2, we have the two-dimensional case, and V^ 

 is the least root of the quadratic 



Bn-V, 

 R 12 



J\ 2 1 



= 0; 



so that 



and 



1/2 



2V ; „ = (E 11 +R 22 )-{(I1 11 -R 2 2) 2 + 4R 1 /}^ 



since R 12 = R 21 , 



2(R n -VJ - (H 11 -R 22 )-f{(R 11 -R 22 )2 + 4R 1 /} 1/2 . 

 Rn — R 22 



Put 



Then 

 and 



cos 6 



2R 12 



2(K n -V m ) = p(l + cos^) = 2 P 



1R 12 — 2p sin ^ cos ^. 



cos' 







The first of equations (12) now gi 



ves 



7 • r, 



n cos t } + '2 sin 9 = 0, 



since the j/s vanish when & = 0. 



