196 A NEW METHOD OF ESTIMATING STREAM-FLOW 
The most accurately determined of all the constants is M of which the value is 
+5.19, with a probable error of ±0.07, only l/74th or 1.3 per cent of itself. This 
means that on any day on which the mean air temperature on the watershed is 
greater than 28° F., and there is an abundance of snow and ice available for melting, 
there is one chance in two that the computed rate of addition to the stored water 
in the ground due to melting is correct within 1.3 per cent, if one regard the 
observed mean temperature as exact and assume that all the errors are accidental 
in character. 
Suppose one assumed that on a given day there is a rain or net melting suffi- 
cient to produce a gain in storage, and that thereafter for 256 days there is no 
change of storage on any day, the rainfall or net melting on each day being just 
sufficient to equal the evaporation plus stream-flow. For Stream A, referring to 
Table 51, the maximum part of the change in storage on the current day which 
reaches the stream subsequently in any time interval involved in the definitions of 
the r's is 3.0 per cent for the 64 days included in the 65th to 128th days, inclusive, 
after the change in storage takes place (0.047X64 = 3.0). This is represented by 
the R', constant of which the probable error is ±0.016 or ±34 per cent, hence it 
may be said that the total computed extra stream-flow during that time interval 
is in error ±34 per cent. This would be true if the error persisted with one sign 
through that 64-day interval. It is extremely unlikely that it would. It would 
tend to obey the law of accidental distribution of errors, hence the error would be 
much less than ±34 per cent. The phrase "extra stream-flow" is intended to 
mean that part of the flow which is an excess above the constant part of the flow, S c . 
On Stream B, still assuming a gain in storage followed by no change for 256 
days, the maximum part of the gain in storage on the current day which reaches 
the stream subsequently in any time interval involved in the definitions of the r's 
is 1.0 per cent for the 32 days included in the 33d to 64th days, inclusive, after the 
change in storage takes place (0.032X32 = 1.0). This is represented by the R\ 
constant of which the probable error may be assumed the same as for Stream A, viz, 
±0.003, or ±9 per cent, which would be the error in the total computed excess 
stream-flow during this time interval under the assumed conditions if the error 
persisted throughout the interval with the same sign, which would be unlikely, 
causing the actual error to be less. 
In the actual case in nature in which a gain or loss in storage is not followed 
by constant conditions thereafter, the probable error of the computed excess flow 
on any day would be a complicated function of the probable errors of the separate 
constants. As a measure of the over-all accuracy of the total computed flow, the 
excess flow plus the constant part of the flow, the probable error of a single observa- 
tion may be used, which for Stream A is ±8.6, and for Stream B, ±8.4. These 
errors are ± 8.0 per cent and ± 8.7 per cent of the constant part of the flow for each 
stream, respectively (— L -= 0.080. — = 0.087 ). Therefore, according to this 
\108 96 ) 
test, on any day on which the total stream-flow is at or near the constant part of its 
flow, there is one chance in two that the error in the total computed flow of Stream 
A is less than 8 per cent and of Stream B 9 per cent. 
The most convincing test of the accuracy of the computed normal stream-flow 
is given subsequently in tabular form and by graphs. 
