90 A NEW METHOD OF ESTIMATING STREAM-FLOW 
The agreement between the actual and theoretical distribution of residuals is 
reasonably good. The evidence indicates that the residuals on the whole are largely 
accidental in character, a characteristic common to all good observations. The 
computation of the probable error of Ei and E t on the assumption that the residuals 
obey the law of accidental error is therefore justified, and those probable errors 
should be increased but little to allow for the discrepancy between the actual and 
theoretical distribution of the residuals. 
RESULTS FROM SOLUTION BB t , LAKE SUPERIOR 
The observations on Lake Superior contributed very little to the final evapora- 
tion formula. They served to verify the value of x = 2.6, and modified very slightly 
the value of E 2 . The value of E 2 from Solution V 6 is +1.27. The final adopted 
value of Et is + 1.24, differing by only 0.03, or 2.4 per cent from the value 1.27. The 
negative value of E l in Solution BB iy —0.154, is obviously absurd. It is only 3 
times its own probable error, however. The probable error of E 3 in Solution BB t 
is nearly 5 times its own probable error. 
The relative importance of the two sets of values is also shown in Table 26 
in the relative assigned weights of the unknowns. The weight of Ei from Solution V t 
is 2.8 times its weight from Solution BB t . The weight of E 2 from Solution V t is 
2.5 times its weight from Solution BB 4 . 
The poor results obtained on Lake Superior are probably not due to errors in 
the evaluation of the mean surface of the whole lake by corrections for wind and 
barometric effects. In fact the mean elevation of the whole surface of Lake 
Superior, corrected for wind and barometric effects only, can be determined with 
greater accuracy from the Marquette gage alone than the mean elevation of the 
whole surface of Lake Michigan-Huron, corrected for wind and barometric effects 
only, can be determined from observations at the Mackinaw gage alone. The 
probable error of the mean elevation of the lake surface in the first case is ±0.014 
foot, and in the second case ±0.016 foot. 
The reasons for the absurd negative value of E x and the poor determination of 
E t from observations on Lake Superior are probably to be found in the compara- 
tively meager data for obtaining the mean rainfall on the land and lake, wind 
velocity and saturation deficit. Contrast Plates 1 with 4, and Tables 1 with 2, 3 
with 4, and 5 with 6, relating to Lakes Michigan-Huron and Superior, respectively, 
with reference to the comparative amount of data available on the two watersheds. 
THE BEST VALUE OF x IN THE EXPRESSION 
(lOO X ) 
It was mentioned on page 81 that 7 least-square solutions, dealing with 
evaporation only, were made. The results of the two final solutions have been 
presented, as representing the final results of this investigation on evaporation. 
It is now proposed to give briefly the facts from all seven of the final least- 
square solutions for determining the evaporation, with the view of presenting 
principally the evidence leading to the adoption of the value 2.6 as the best value 
for x. The question of the proper exponent of the wind velocity, and a discussion, 
besides the one to be here given, of the proper number of terms — two or three — to 
evaluate properly the evaporation, will be reserved for a later place. 1 
The principal facts from the final 7 least-square solutions for determining the 
best form of the evaporation equation, as well as for evaluating the constants in it, 
1 See page 98. 
