346 Wisconsin Academy of Sciences , Arts , and Letters. 
D = Density of the water; D n , the density at any given tem¬ 
perature. At 4°, D is equal to; one at any other tempera¬ 
ture it is less than one. 
Z = distance from the surface of the lake expressed in centi¬ 
meters. 
W = work done in warming the lake as a whole or any given 
stratum. In general, this is stated in gram-centimeters 
per square centimeter of area of the lake; thus following 
the notation employed in stating the heat budget 
(Cf. Birge ’15). Other units may be employed if con¬ 
venient or desirable for special ends, as is stated later 
in this paper. Another datum plane may also be selected, 
but since the direct work of the wind is performed on 
the surface of the lake, the general reference will be to 
that surface. This reference will be understood if no 
other is expressed. 
The formula for the work done in warming a stratum, by 
mixture, from 4° to any given temperature n°, is, there¬ 
fore: 
W=RTXZX(1-D n ). (Formula I.) 
From this formula the value of W will be given in gram- 
centimeters per square centimeter of the surface of the 
lake. RT gives the weight in grams of a column of 
water whose base is one sq. cm. and whose height is the 
thickness of the stratum when its area is conceived as 
extended to that of the surface of the lake. The product 
RTxZ, therefore, states in gram-centimeters the work 
that would be done in warming the stratum by mixture, 
if D were reduced to zero, so that 1 — D = 1. The third 
factor (1 — D) states the loss of density as a fraction of 1, 
so that the final product is the measure of the work 
done, stated in gram-centimeters per square centimeter 
of the surface of the lake. 
In the expression (1 — D n ), 1 is the density of water at 4°, 
and therefore is equal to D 4 . If the lower limit of tem¬ 
perature considered is any other than 4°, its density, 
say D m , must be substituted for 1. 
