Planes of Closest Fit to Systems of Points, 



£77 



the second Criterion is much greater, at any rate in all cases 

 of n > 2, than that necessitated hy the first criterion. In the 

 second method, if (n — Jc) is three or more, the smallest root 

 o£ an equation of the third or higher degree has to be 

 approximated to. This in itself is no light task, and is 

 not necessary in the first method. The methods do not 

 necessarily lead to results at all alike (see Example 3, below), 

 and only the terms of the particular question in hand can 

 decide which method is to be used. The second gives the 

 best geometrical fit, considered in a direction perpendicular 

 to the plane. The first gives the " regression " plane — i. e., 

 the most probable value of one variable in terms of the 

 others. This is the most frequently needed in practical 

 cases, as is exemplified in Examples 1 and 2. 



7. Illustrations. 



I. The second column in Table L* gives the temperatures 

 (Centigrade) at solidification of a series of alloys of iron and 



Table I. 



X. 



Percentage 



of Carbon 



present. 



m y ' 

 Temperature 



at end of 



solidification. 



Temperature by 

 1st method to 

 nearest degree. 



Temperature by 

 2nd method to 

 nearest degree. 



•02 



1470 



1501 



1501 



•12 



1470 



1483 



1483 



•16 



1465 



1476 



1476 



•17 



1450 



1474 



1474 



•24 



1448 



1461 



146 L 



•38 



1416 



1436 



1436 .. 



•53 



1404 



1409 



1409 



•61 



1394 



1395 



1394 



•80 



1351 



1360 



1359 



•81 



1351 



1358 



1357 



1-31 



1286 



1268 



1267 



1-51 



1244 



1231 



1230 



1-85 



1179 



1171 



1169 



2-12 



1 1 10 



1122 



1120 



221 



1107 



1105 



1103 



carbon. The percentage of carbon in the various alloys is 

 given in the first column. The solidifying temperature of 

 pure iron is 1505° C. Any curve, therefore, which attempts 



* The table is taken from a paper on " The Range of Solidification 

 and the Critical Ranges of Iron-Carbon Alloys," by II. C. H. Carpenter, 

 M.A., and B. F. E. Keeling, B.A., in the Journal of the Iron and Steel 

 Institute, No. 1, for 1904. 



Phil. Mag. S. 6. Vol. 21. No. 123. March 1911. 2 C 



