994 Wisconsin Academy of Sciences, Arts, and Letters . 
centigrade, the density can be taken directly from the table and 
the ergs computed from the equation given above. If A=1 sq. 
cm. C=100 cm. and and D 2 are two of the numbers in column 
II, then the number of ergs is 833 times the difference between 
X) 2 and Di. These results are shown in column V. 
If it is desired to compute the thermal resistance directly 
from the temperatures observed, the following formulas will 
yield approximate results. An empirical relation between the 
density and temperature of water at temperatures above 4° is 
<3) 
D = 1 
93 (T — 4) 1 - 983 
10 * 
A very close approximation to this is 
14) 1V =1 — r)T2 - 3 Q 6 6 T + 47 
which gives as an approximate value for work 
(5) W(ergs) = ^ |t» - 3 J (T, - T,). 
.Below are given the differences between D and D 1 ', showing 
the degree of approximation reached by formula (4). 
4 ° 
5 °, 
10 ° 
15 °, 
20 °. 
25 °. 
30 °, 
ir 
D 
D-D 
1.00000 
0.99998 
0.99971 
0.99914 
0.99827 
0.99710 
0.99563 
1.00000 
0.99999 
0.99973 
0.99913 
0 99824 
0.99707 
0.99567 
+0.00000 
+0.00001 
+0.00002 
— 0.00001 
— 0.00003 
— 0.00003 
+ 0.00004 
The values of W, as computed by formula 5 differ in the third 
decimal place from those derived directly from the tables of 
density and computed according to formula 2. 
Formula 5 also shows that the approximate value of the 
work done in mixing a column of water is proportional to the 
temperature gradient • — ^provided that the mean tem¬ 
perature, Tm, remains constant. That is to say, if the tem¬ 
perature gradient of a stratum of water is uniform and the 
