Closest Fit to Systems of Points in Space. 563 



be as small as possible. Substitute this value of S 2 in the 

 type equations (viii.), and we find the required values of 



hi h-" k, usi »g ( iv -)- 



This is the complete analytical solution of the problem of 

 drawing the best-fitting plane through n non-coplanar points. 

 We see that it depends only on a knowledge of the means, 

 standard-deviations, and correlations of the (j variables. 



Whenever we may suppose that variation is due to errors of 

 observation or measurement, — i. e., is not organic, but there 

 exists a unique functional relation between the true values 

 of the variables, — then, assuming it of the first degree, we may 

 determine the best values of the constants in the manner given 

 above. 



(3) A geometrical interpretation is of course to be found 

 from (viii.) bis. Consider the quadric 



G 2 X X X* + <T 2 X 2 -V 2 ~ + • • • + a 2 x q X q 2 + tGxfxyx^X^CL 



+ . . . + '2a Xfj ^a X(j r Xq _ yXq .'c fi _ l :r q = e\ . . (xi.) 



where e is any line. Then this quadric will be " ellipsoidal u 

 since the coefficients of x 2 . . . Xq 2 are all positive quantities. 

 Let K be its radius- vector measured in the direction l u l 2 , . . . l, h 

 or perpendicular to the plane from which we are measuring 

 the residual- ; then clearly: 



U=neVR 2 , 

 M,- S 2 =e*/R 2 . 



(xii. 



Thus the inverse square of the radius o£ this " ellipsoid " 



measures the square of the mean square residual. We shall 

 speak of the ellipsoid as the ellipsoid of r<'.<i<ln<il<. Since 2 is 

 to be a minimum, R must be a maximum ; or we conclude : 

 tliat the best-fitting plane is perpendicular to the greatest axis 

 of the ellipsoid of residuals and the minimum mean square 

 residual varies inversely as the length of this axis. 



A case of failure can only arise if the ellipsoid of residuals 

 degenerates into an " oblate spheroid," i.e., when every plane 

 through its shorter axis is one of ei best fit," or into a sphere, 

 when every plane through the centroid of the system of points 

 is an equally good fit. Tin- sphericitv of distribution of 

 points in space involves the vanishing of all the correlations 

 between the variables and the equality of all their standard- 

 deviations. It corresponds to isotropic inertia in the theory 

 of moments in dynamic-. 



(4) The theory of the best-fitting straight line need not 



