112 
A NEW METHOD OF ESTIMATING STREAM-FLOW 
The probable errors were computed rigorously from the normal equations, 
(31), and the residuals, v, of the 787 equations. 
If the theory is correct, if the constants E x , E 2 , R h R 2 , R h . . . R t were 
assumed correctly and if there were no errors in the absolute term /, the derived 
values of the unknowns, (32), would be the same as the assumed values. The 
following is a comparison between the assumed and derived values. 
(1) 
Assumed 
(2) 
Derived 
(3) 
Difference 
(D-(2) 
(4) 
Difference -f- p.e. 
= (3)/p.e. 
Mean 
Ei = +0.581 
+0.493 ===0.035 
+0.088 
2.5*1 
2.5 
E 2 = + .624 
+ .466± .065 
+ .158 
2.4 
Ri = + .01 
- .291± .039 
+ .301 
7.7* 
R 2 = + .07 
+ .253="= .049 
- .183 
3.7 
R 3 = + .17 
- .035="= .044 
+ .205 
4.7 
■ 
4.3 
Ri=+ .80 
+ .121± .188 
+ .679 
3.6 
R b =+ .20 
- .086=fc .117 
+ .286 
2.4 
ff 6 = + .27 
- .081=t .089 
+ .351 
3.9 
It is apparent from the above tabulation that the assumed values of the run-off 
into the lake were farther from the truth than the assumed values of the evapora- 
tion constants. The derived values of the latter differ by an average of only 2.5 
times their own p.e.'s from the assumed values, whereas the derived values of the 
run-off constants differ by amounts varying from 2.4 times its own p.e. for R s to 
7.7 times its own p.e. for R h with an average difference for all six derived values 
of 4.3 times their own p.e.'s. 
From the derived values of the R's, the total excess (or defect) of storage on 
the current day, r h which is delivered to the lake by the end of the sixteenth day 
thereafter, if during those 16 days the storage is held constant, is ( — 0.291) (100) -f- 
(0.253) (100) + (-0.035) (100) + (0.121) (5)(2)-t-(-0.086)(2.5)(4) + (-0.081) (1.25) 
(8) = —7.8 per cent, as compared with the assumed value of +37 per cent (page 106), 
a strong indication that the assumed values of the R's were too large. 
VARIABLE PART OF RUN-OFF INTO LAKE SUPERIOR 
On Lake Superior a least-square solution designated as Solution BB X was 
made in which the form of observation equation was identical with equation (28) 
'w-24ffsf 
except that an additional wind term, 
100 /_ 
Int 
E 3 , was introduced for 
lis solution a much better 
winds below 10 miles per hour, as in Solution BB 2 
basis was obtainable for estimating the values of R h R 2 , R 3 , . . . Ri. Previous 
to starting this solution, approximate values for the percentage of change in storage 
on any day in the drainage area, which reaches the stream on the current or later 
days, assuming the storage to be held constant after the current day, had been 
obtained on Streams A and B, Wagon Wheel Gap, Colorado, the detailed exposition 
of which appears in Part II of this publication. These values furnished a much 
more sound basis for estimating the R's than had been available up to that time on 
the Great Lakes. For example, on Lake Michigan-Huron, the values first esti- 
mated by the methods at first available were much too high, as previously men- 
tioned, and were reduced successively in various least-square solutions until the 
