NORMAL GEOTHERMAL GRADIENT 93 
However, the mean difference of only 2.44 feet per degree Fahrenheit 
is small and indicates that the total error introduced into the mean 
by the use of either method is not particularly serious. 
The calculations carried out to obtain the results in the last 
column of the table are the same as those by means of which the Com- 
mittee on Underground Temperature of the British Association for 
the Advancement of Science (Report 1882, page 88) obtained the 
value of 64 feet per degree Fahrenheit. The calculations (Formula 
8) were based on 36 observations in Europe, chiefly in mines and ar- 
tesian wells. As a result of correcting the gradients in Mont Cenis 
and St. Gothard tunnels, the Report of the Committee for the year 
1883 (page 49) contains the value of 1°F. in 60 feet. The fact that two 
corrections in the British data change the rate of temperature increase 
by 4 feet per degree is an indication that averages taken in this way 
are not altogether reliable. The defect in this method of averaging is 
due to the fact that the observations are not sufficiently representa- 
tive of the different geological conditions. With a few observations 
on each of a-very great number of different geological units distributed 
over a large area, the values would tend to be consistent. This def- 
inition of a normal gradient corresponds to the mathematical defini- 
tion in which the number of gradients in a large area is infinite, or, 
from a practical standpoint, very large. 
Strict compliance with the mathematical requirements is im- 
possible. The next best method of procedure is to obtain as much 
information as possible from a limited number of observations by 
means of the mathematical theory of probability and least-square 
adjustments of the data of observation. 
The curvature of the depth-temperature curves greatly compli- 
cates the problem of evaluating a theoretically correct average or 
normal gradient. If the depth-temperature curves were straight lines, 
the weight of 6 would be given correctly by the well known relation 
that the weights are inversely proportional to the squares of the prob- 
able errors, the latter being obtained rigorously from the usual least- 
square adjustment of the straight line, y=a+dx, to the observed 
temperatures at the given series of depths. The curves in Figure 2 
show that the residuals (v=/) are very large in comparison with the 
errors of observation. This method, therefore, assigns minimum 
weights to curves of maximum curvature, disregarding, for the most 
part, the accuracy of the observations. Furthermore, the value of the 
gradient itself varies rapidly with the depth. When the range of 
depths is large, as in the last two columns of tabular values of 6, the 
resulting average gradient is difficult of correct interpretation. For 
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