Planes of Closest Fit to Systems of Points. 385 



(6) (?) (?) (+) 



Deviation Deviation Deviation Deviation 



in direction in direction in direction perpendicular 

 of z *. of y. of x. to plane. 



-4-795 + *227 + '285 - -181 



-5-850 +5-319 -2-200 +2-122 



-6-619 +1-445 - -178 + -275 



-8-265 -1-787 +1-592 -1-325 



-8-544 -3-] 01 +2-281 -1-952 



—9-966 + -070 + -796 - -563 



The sum of the squares of these deviations are 



(0) 341-6570, 



(?) 43-2458, 



(f) 13-3240, 



(0) 10*4942 (the exact value here should be.10'4966, 

 the value o£ V m above). 



The sum of the squares of the deviations given by (0), (f), 

 and (£) in directions perpendicular to those planes are found 

 to be (by dividing the above values by the sum of the squares 

 of the coefficients of the various equations) 63*4239, 11-7969, 

 and 10*5767 respectively, all these, of course, being greater 

 than the corresponding value given by (</>). 



Equation (</>) can be written in the three forms : 



z = 15*2048^-7-9644?/ .... ($') 

 ?j= 1-9091^- -1256* .... (£') 

 a?= -5238.Z/+ *0658^ .... (f) 



These equations should be compared with (#), (?), and (f) 

 respectively. The sum of the squares of the deviations 



* At first sight it seems remarkable that all the deviations given by 

 (9) are of the same sign, but a moment's consideration will show that 

 this is quite possible. For a line in the plane is fixed, and the plane can 

 only swing about this line. All the points may be on one side of the 

 plane, but on either side of the line. Swinging the plane about the line 

 to become closer (measured in a particular direction) to some of the 

 points, therefore, may take it farther from some others. To verity (6) 

 the results given by the planes s = 2 # 1404.r and s = 2.r+'0856, one on 

 either side of (6), were found. The sum of the squares of the deviations 

 given by these formulas were 341"6991 and 341'8112, both greater than 

 the corresponding number for. (6). . Thus (6) gives a true minimum. 



