Planes of Closest Fit to Systems of Points. 383 



the corresponding figures up to the other terminus being 

 114, 187, and 244 respectively. The figures should be 

 expected to be fairly coplanar, and any formula obtained to 

 represent them ought to give a good " fit." Four cases can 

 be worked out here, viz. those obtained by making the sum 

 of the square of the deviations in the directions of x, y, z 

 and perpendicular to the plane respectively a minimum. We 

 find : 



S(> 2 ) = 28526, S(y*) = 105264, 



S(y 2 ) = 76827, 8(*a-} = 64160, 



S(e 2 ) =144311, SOy)= 46801. 



For the best fit in the direction of z, the equation of the 

 plane will be 



z = ax ■+- by, 

 with the condition 



224 = 114a + 1876 (rj) 



In this case we have 



A = 



1 64160-114 X 105264-187 X 



64160-114 X 2Sb26 46801 



105264-187 X 46801 76827 



Using; the relations 



Aio 7 A 2 o 



« = -=-», b = 



we obtain 



a= 2-2376 + -00526 X, 



b = -90706--00077X. 



Substituting in (97), X becomes —27*2290. and therefore 



a = 2-0943, 



b = -0281, 



and the best fitting plane in the direction of z is 



z= 2-0943.T + -0281y (0) 



In a similar manner we find the best fitting plane in the 

 direction of y is 



y = 1-6327^ -f'0036^, (f) 



and in the direction of x it is 



a = -5031y + -Q817* (f) 



