Planes of Closest Fit to Systems of Points. 379 



therefore, the best fit to the system o£ points of a line 

 through by both methods. 







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II.* Iii this case we will take the figures of an alloy of 

 three metals — copper, aluminium, and manganese. The 

 percentage of manganese present varies from to 10*24, 

 and of aluminium from to 7*40. It is not possible to tell 

 from the mere figures if the distribution is approximately 

 coplanar or not, but the material seemed good enough to 

 work upon. The freezing-point of pure copper is 1084° C. ; 

 in this case, therefore, we require to find a relationship of 

 the form 



T-1084=^.^ + ^.y, 



where T is the temperature at solidification of an alloy con- 

 taining x°/ Q of" aluminium and ?/% of manganese. Taking 

 our origin at zero percentages of aluminium and manganese 

 and 1084° C, we require to obtain a plane through the 

 origin and fitting most closely the observations in the second, 

 third, and fourth columns of Table II. 



* The figures for this example were taken from Table 43 (p. 2'29) of 

 the "Ninth Report to the Alloys Research Committee" to the lust, 

 of Mechanical Engineers, by Dr. \V. Rosenhain and Mr. F. C. A. II. 

 Lantsberry. 



2 C i> 



