376 Mr. E. G. Snow on Restricted Lines and 



The equation of the line, therefore, is 



where 



x sm -x + y cos - = 0, 



2&12 



tan0r=^ ^- 



-^11 ^22 



2S(«/) 



and the notation is altered to agree with the usual form. 



This value of tan gives rise to two values of -, each less 



than 180°. In a particular numerical example, however, it 

 is not difficult to pick out the value required ; while it can 

 be verified that the other value corresponds to the " worst- 

 fitting " line. 



In cases of n>2 it is better to substitute the values of the 

 R's direct in the determinant above, and to find V m by 

 the usual methods of approximating to the roots of an 

 equation. 



(B') If & = 1, the equation to determine Y m is 



d == 



Rn— V OT 



Rl2 



R 2 i R»i» pi 



(R 22 — Vm) Bn2, Pi 



Bit* 



pi 



R 2?1 (R,m— V m ), p n 



p 2 p n , 



o, 



the origin and the point (p x p 2 . . . . p n ) being the fixed 

 points. 



In this case, equations (12) take the form 



^i(Rn — V m ) + Z2R21+ + Inf&n 



= —*Pi> 



h^-ln 



+ h^2n + + hQ&nn ~ V TO ) = — \p n . 



Hence l t is proportional to d t , the first minor of the con- 

 stituent in the tth column and bottom row of d, and V m is 

 given the value which is the least root of d = 0. Using (10), 

 the actual values of the Z's can be found. 



6. It will be seen from the foregoing analysis that the 

 work involved in determining the " closest fitting " plane by 



