382 Mr. E. 0. Snow on Restricted Lines and 



If Ix + my + nz = 



is the equation of required plane, the equations to find the 

 ratios of I, m, and n are 



763*1982 + 751-017m + 10617-960n = 0, 

 751-017Z + 1320-079m + 15421-230n = 0. 

 From these we find 



I to n 



5-4908 = 8 1*582 ~~ ^1* 



The equation of the plane is 



s = 5'4908# + 8-5582# 

 and 



T=1084-5-4908^-8'5582y (e) 



The temperatures given hy this formula are shown in 

 column 7 of the table, and the deviations from the observed 

 values in column 8. They do not differ greatly from the 

 results given by (7). The sum of the squares of the devia- 

 tions in the table is 1658' 97 18, which is, of course, greater 

 than the corresponding number given by the first method. 

 The " root mean square " is 6*44, not greatly different from 

 the first method value. Also l 2 + m 2 + n 2 becomes 104'3923. 

 The actual sum of the squares of the deviations perpendicular 

 to the plane is therefore 15*8917, which is less than the 

 value given by the fir.<t method, as it should be, but is not a 

 very great improvement on it. Thus in this example, as in 

 the last, the two methods lead to very similar results. 



III. For a third example we will take the case of a plane 

 in three dimensions to pass through two fixed points and to 

 be closest fitting to a series of other points. The data for 

 this case are taken from a railway time-table. The two fixed 

 points are two terminal stations, and the variables are x, the 

 distance (in miles) from one of these stations to some other 

 station ; y, the scheduled time (in minutes) allowed for a 

 train between those stations ; and 2, the first-class single 

 fare (in pence) between the stations. The figures are : — 



30 



49 



69 



52 



80 



117 



60 



97 



135 



69 



115 



156 



81 



136 



182 



100 



164 



224 



