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



A uv being the first minor o£ the constituent o£ the (M-fl)th 

 column and (t;-|-l)th. row, and being positive or negative 

 according as [u + r) is even or odd. 



Thus the coefficients in the required formula can be 

 found at once by the evaluation of a number of determinants 

 of order (n -f k) *. 



Particular Cases. 



3. The simplification of the above results in a few par- 

 ticular cases will be useful. 



(A) k=0. — In this case we have a plane passing through 

 a single fixed point and closest fitting to a system of points. 

 Here all the A/s disappear, and the equation of the plane 

 becomes 



where 



A being 





Aoo 



Roo 



Roi 



R02 



Rio 

 Rn 



R s0 



R*i 



E 



0/i 



R«o 

 R»l 



R, 



E [!r being equal to R ru , and is the sum of the products 

 x u . %v taken throughout the system of points. 



If x s and <r s be the mean and standard deviation of z s , 

 and r uv the correlation between the coordinates x u and x v> we 

 have 



R U?J = 8(x u x v ) = N<7 w o-„r« c + !$x u a; v if u =£v, 



and R uv =■ No- tt 2 -f-N^ 2 if u = v. 



The analogy between A and the determinant used in the 



* It is not difficult to show that, by first finding the a's in terms of the 

 X's from the first n of equations ((5), and substituting these values in the 

 last k of the same set — thus giving k equations for the X ; s, — the a's can 

 be found in a form involving only deteiminants of order n, though the 

 number of determinants it is necessary to evaluate is increased. If k is 

 large, this increase is considerable. The general result in this form 



will not 

 (§ 3). 



be given, but it is exemplified in a particular case below 



