560 Prof. K. Pearson on Lines and Planes of 



of leg-lengths ; but a point at a given time will have 

 one position only, although our observations of both time 

 and position may be in error, and vary from experiment to 

 experiment. In the case we are about to deal with, we sup- 

 pose the observed variables — all subject to error — to be plotted 

 in plane, three-dimensioned or higher space, and we endeavour 

 to take a line (or plane) which will be the " best fit " to such 

 a system of points. 



Of course the term " best fit " is really arbitrary ; but a 

 good fit will clearly be obtained if we make the sum of the 

 squares of the perpendiculars from the system of points upon 

 the line or plane a minimum. 



For example : — Let P 1? P 2 , . . . P n be the system of points 

 with coordinates «j, «/, ; # 2 , ,y 2 \ • • • x n y n i and perpendicular 

 distances p u p 2 , . . . p n from a line A B. Then we shall make 



U = S(jt> 9 )=a minimum. 



If y were the dependent variable/ we should have made 



S(y'— y) 2 =a minimum 



(y being the ordinate of the theoretical line at the point 

 x which corresponds to y), had we wanted to determine the 

 best-fitting line in the usual manner. 



J 4/ 



B 



Now clearly U = S(jt? 2 ) is the moment of momentum, the 

 second moment of the system of points, supposed equally 

 loaded, about the line A B* But the second moment of a 

 system about a series of parallel lines is always least for the 



