Birge—An Unregarded Factor in Lake Temperatures . 993 
trieb.” The number of liters stated is the reciprocal of the 
numbers in column III. I have repeated his results in col 
umn VI; the numbers being slightly changed as the values 
for density are not quite the same as those employed by 
Groll. 
In order to give a similar picture of the thermal resistance, 
I have stated in column V the amount of work in decimals 
of a erg, which would be required to mix a column of water 
1 sq. cm. in area, 1 m. high, in which the temperature gradient 
is uniform and whose upper and lower surfaces differ in 
temperature by 1°. 
The formulas from which these results have been computed 
have been worked out and furnished to me by Dr. H. C. Wolff 
of the department of mathematics, University of Wisconsin, 
whose valuable assistance I wish to acknowledge with thanks. 
The work done against gravity in mixing a column of water 
whose density varies with the depth, so that it shall become one 
of uniform density is 
(1) W (ergs) = A f(z) [ z — ^r] dz 
where A is the area of the cross-section of the column in sq. 
cm., C the height of the column in cm. and f (z) the function 
expressing the density in terms of z, the distance from the top' 
of the column. The density of water at 4°C. is to be taken as 
unity. 
If f(z) is a rational integral function of the second degree 
(1) reduces to the simple form 
AC's 
( 2 ) W (ergs) =; — [D 2 — DJ 
where D x and D 2 are respectively the density of the lower and up¬ 
per strata of the column. This condition is satisfied when the 
temperature gradient is uniform and when the relation between 
the density (D) and the temperature (T) is of the form 
B=aT 2 +/?T+y where a P and y are constants. If the tempera- 
atures at the surfaces of a column are assumed to be full degrees 
