212 A NEW METHOD OF ESTIMATING STREAM-FLOW 
to the symmetrical probability curve for various multiples of the p.e. can be ob- 
tained from the well-known formula 
a 
2 s*K - x * 
*(x) = 4= I e dx ( 95 ) 
V7T I 
or 
in which a is the error, r the probable error of a single observation, p = 0.476936, 
and $(x) the probability that the error will be less numerically than the limit which 
is expressed in terms of the probable error. 
In order to avoid irregularities in the data it is desirable to smooth out the 
actual histogram of observed stream-flow by some form of regular curve. In this 
investigation the Pearsonian curves were used, and it has been found that with the 
size of class interval used in these studies the data can be represented by the Type 
III curve of which the general equation is 
y = y e-y* (1+?) *• (96) 
in which y is the computed frequency of the stream-flow, x, measured in terms of 
p.e. of equation (94), y , y and a are constants to be derived by the method of 
moments from the observed frequency distribution, and e is the base of Naperian 
logarithms. 
On Streams A and B, the S c discharge was known, as determined by the least- 
square computations. With this known, the p.e. could be computed by equation 
(94). This p.e. divided into D' measured from S c as origin is the - of equation (95) 
r 
and divided into D' measured from the mode of (96) as origin is the x of equation 
(96). From the computations yielding equation (96), the mean, median and mode 
of the actual frequency distribution could be estimated. Let the ratio of S. to 
mean, median and mode be defined as follows: 
(97) 
s. 
Mean 
= K X 
S, 
=K t 
Median 
s e 
= K t 
In which K x , K 2 and K» are assumed to be constants determinable from the 
data on Streams A and B. With these constants known, three values of S c can 
be computed for any other stream in a moist climate from the mean, median and 
mode of its frequency distribution. The most probable value of S c for the stream 
is taken to be the arithmetic mean of the three separately determined values. 
In order to make the foregoing specific, an example of the computations on 
Stream A will be presented, and subsequently a summary will be given of the 
results obtained on other streams. 
The order of procedure and the results derived on Stream A are those given 
in the following numbered paragraphs: 
