Planes of Closest Fit to Systems of Points. 381 



Using equation (/3) above, we quickly reach 



ar= 5-4605 « + 8'4732y, 



and therefore 



T = 1084-5-4605^-8-4732?/. . . . (7) 



The values of T obtained by this formula are given in 

 column 5 of Table IT. The differences between these values 

 and the experimental results are shown in the next column. 

 It will be seen from the figures that the fit is a good one 

 except at the ends of the range. Had the last seven obser- 

 vations been omitted, i. e. had the amount of aluminium 

 present in the alloy been less than 6 °/ , a linear law such as 

 the above one would have agreed quite well with the observed 

 results. * As the authors of the original paper state that 

 " the precise temperatures given in the table possess no very 

 great significance," it seems quite reasonable to assume that 

 the observations, up to 6 % °£ aluminium, follow a linear 

 law. 



The sum of the squares of the deviations from the observed 

 temperatures in this case is 1641*0210, and the "root mean 

 square " is 6*41 *. The sum of the squares of the deviations 

 measured perpendicular to the plane can be obtained from 

 the above figure by dividing by {(5'186) 2 + (8'392) 2 + l}, 

 i. e. 102-6120. It is found to be 15*9924. 



When the second method is used, the equation in Y m is 



779'134-V m 751-017 10617-96 



751-017 1336-015- V m 15421-23 



10617-96 15421-23 190296 -V, 



= 0. 



This when expanded becomes 



VL-192411Vf„ + 52425892V wl ~786607941 = 0. 



We want the least root of this cubic. It is quickly seen 

 to be in the neighbourhood of 15, and by successive approxi- 

 mations is found to be 



V m = 15-9362, 

 very nearly. 



* The second decimal place was taken into account in finding this 

 figure. This was done in order to compare with the results of [t) t which 

 do not greatly differ from (y). 



