192 
A NEW METHOD OF ESTIMATING STREAM-FLOW 
THE TWO FINAL SOLUTIONS FOR DETERMINING C, F, M AND T" 
In this stream-flow investigation, made by methods as outlined in steps (1) to 
(6) on pages 146 and 147, several separate least-square solutions were made on the 
basis of each group of observational data. The form of each successive solution 
from one group of data was based upon all the information available up to the time 
that that form was adopted, including the information from earlier solutions based 
upon the same data. 
There was a total of 34 least-square solutions made for determining the values 
of C, F, M and T", 18 on Stream A and 16 on Stream B. The two final solutions 
for determining numerical values of these unknowns were those designated as 
Solution A A on Stream A, and Solution X on Stream B. Each of these solutions 
was the culminating one on each stream based upon the same, or nearly the same, 
group of observational data in each case. The principal facts from these two 
solutions are shown in Table 49. 
Table 49 — Principal facts from the two final freezing-melting solutions 
Nq. of days of observation used in the solution 
No. of observation equations 
Computed value of C 
Computed probable error of C 
Computed value of F 
Computed probable error of F 
Computed value of M 
Computed probable error of M 
Assumed value of T" 
Probable error of a single observation 
Solution AA — 
Stream A 
+ 
± 
+ 
zfc 
+ 
137 
137 
8.59 
1.16 
0.647 
0.117 
5.49 
0.123 
2S° F 
12.4 
Solution X— 
Stream B 
151 
151 
+ 3.97 
± 0.71 
0.403 
0.062 
5.04 
0.083 
28° F 
6.3 
+ 
+ 
The probable errors of C, F and M as shown were computed rigorously from the 
normal equations and the residuals of the observation equations. 
The probable error of a single observation as shown above is the probable 
error, computed rigorously from the normal equations and the residuals, of the 
computed average stream-flow of the stream at the point of measurement during 
one day. 
Table 50 shows the best values of C, F and M determinable from the observa- 
tions on Streams A and B in the winters of 1912 and 1913. The values of C, F and 
M and their probable errors in this table are the same as in Table 49. From these 
are computed the weights assigned to each unknown, the weighted mean value and 
its probable error, and the residuals of each value from its own weighted mean. 
The assigned weights are inversely proportional to the squares of the respective 
probable errors, corresponding to the assumption that all errors in the computed 
values of these constants are of the accidental character. Unit weight corresponds 
to a probable error squared of 0.1. 
The residual of the value of C in Solution A A, viz, —3.36, is over five times the 
probable error of the weighted mean C from both solutions, (5X0.61=3.0), there- 
fore that value of C, +8.59 ± 1.16, is rejected. 
With the rejection stated in the preceding paragraph, the following adopted 
values of C, F, and M and T" are the best that can be derived from the present 
investigation: 
