A NEW METHOD OF ESTIMATING STREAM-FLOW 211 
RELATION OF S c TO MEAN, MEDIAN AND MODE OF FREQUENCY DISTRIBUTION 
In the formulas for normal stream-flow, S c expresses the flow which would 
occur at the end of a long period, 257 days, during which just enough rain fell each 
day to equal the evaporation plus the stream-flow on each day. During such a 
period the total amount of water in storage in the drainage area, underground, on 
the surface, and in the lakes and streams would not vary. In general on any day 
such an equality does not exist between rainfall on the one hand — the gain in 
storage — and the evaporation plus stream-flow on the other hand — the loss in 
storage. 
Defining S c in different words, it is that flow which would exist after a long 
period during which all of the influences tending to increase the stream-flow are 
exactly balanced by those tending to decrease the flow. It is then that normal 
stream-flow which should have the greatest frequency or the one which would be most 
probable. S c , then, corresponds to the mode of the frequency distribution of the 
stream-flows. 
According to the laws of stream-flow developed in this investigation, the 
stream-flow will always be the same under identical past meteorological conditions. 
Since the meteorological conditions, however, are accidental in nature, the improb- 
ability that they will repeat themselves according to fixed laws is great, hence the 
stream-flow which depends on the weather may be classed as accidental in nature. 
It follows, then, that the difference between any stream-flow greater or less than 
S c and S c takes on the nature of an accidental error, in which S c is the most fre- 
quent value of the measured quantity. 
It is well known that stream-flow plotted as a frequency distribution nearly 
always assumes the form of a skewed curve. Imagine such a curve superposed 
upon the symmetrical or Gaussian probability curve with their modes falling on the 
same vertical line. Since the symmetrical probability curve represents the law of 
distribution of errors which are purely accidental in nature, it follows from preced- 
ing statements that there should be a simple relation existing between the sym- 
metrical curve and the curve of actual stream-flow frequency. It was thought 
that the supposed relation existing between the two curves could be most simply 
expressed as a ratio. 
Omit from consideration stream-flows smaller than S c . It is desired to obtain 
the ratio of the curve of actual stream-flow frequency for discharges greater than S c 
to the normal probability curve. In order to derive this ratio, the abscissas to the 
two curves must be expressed in the same unit. The unit used is the probable 
error of a single observation, which is given by the formula 
p. e.= 0.6745 
fe o» 
in which v is the difference between any observed stream-flow greater than S c and 
S c , and N' is the total number of observations of stream-flow greater than S c . By 
means of equation (94) the unit of abscissas of the curve of actual frequency dis- 
tribution can be computed, and by actual count the frequency of the flows greater 
than S c which are between successive multiples of the p.e. can be obtained. If we 
now assume that the area under each curve to the right of S e is unity, the ordinates 
