384 Wisconsin Academy of Sciences, Arts, and Letters. 
Plate VIII. Summation and subtraction curves of calories and 
work; summer heat-income of lake Mendota. Horizontal 
scale for calories ten times that for work. Summation curves in 
broken lines; subtraction curves in full lines. The summation 
curve for calories is AA: that for work is AA'. The value at the 
intersection of any meter-line with the work-curve shows the 
amount of energy expended in distributing through the water above 
the meter-line, the amount of heat indicated at the intersection of 
the same meter-line with the heat-curve. For instance, a depth 
of 5 m. about 745 g. cm. of work has been done to distribute in 
the water above that depth about 8150 cal. In both cases the 
numbers refer to sq. cm. of the surface of the lake. At 10 m. the 
amount of work expended is increased to nearly 1065 g. cm. and 
13800 cal. have been left behind. Thus 5650 cal. have been brought 
from the surface into the 5 m.-lO m. stratum and distributed 
through it by an expenditure of 325 g. cm. 
It is noticeable that in the upper meters heat and work increase 
together and at about the same rate; but as depth increases the 
curves diverge and in the lower and cooler water the number of 
calories increases much more rapidly than the number of gram- 
centimeters of work needed to distribute it. This means that the 
decline of thermal resistance decreases work more rapidly than 
the increase of depth adds to it. The deep lake therefore calls for 
less work in proportion to gains of heat, than does the shallower 
lake. 
The subtraction curve for work is BB; that for calories is DD. 
These curves show at the surface, for example, that about 1210 
g. cm. of work per sq. cm. of the lake’s surface were required to 
distribute 18370 cal. per sq. cm. into the water of the lake; that 
at the depth of 5 m. about 465 g. cm. were left to distribute about 
10300 cal. through the water below 5 m. 
The dotted lines GG and EE show the amount of work and heat 
at any given depth as referred, not to the area of the lake’s surface, 
but to the area at that depth. Thus the main curves BB, DD, 
show that at the depth of 10 m. there are left about 145 g. cm. 
and 4600 cal. per sq. cm. of the lake’s surface. But since the area 
of the lake at 10 m. is 61% of that at the surface, these results are 
divided by 0.61. The quotients show that there were brought to a 
depth of 10 m. nearly 7600 cal. for each sq. cm. of the area of the 
lake at that depth and there were similarly about 240 g. cm. of 
work available for its distribution. Similar computations are made 
for various points along the curves; the results are platted and 
connected by the dotted lines. This change in the datum plane is 
necessary if similar strata in different lakes are to be compared. 
See p. 357. 
