214 A NEW METHOD OF ESTIMATING STREAM-FLOW 
These moments give a mean discharge of 153.6, and a value of K, the Pearsonian 
criterion by which to judge the type of equation which best represents the data, of 
— 16.4, which indicates a Type III curve. 
The constants of equation (96) evaluated from the first to third moments 
inclusive of Table 60 are 
(98) 
y Q = 619.84 
7 = +0.544346 
a= -1.426997 
ya= -0.776781 
which, substituted in equation (96) gives 
y = 4-619.84e- 05 « 3 ^ ( 1 ~T426W ) "°' 776m (99) 
in which y is the frequency of the discharge, x, of Stream A, the unit f x being 185. 
The origin of x is the mode, which, calculated from the moments of Table 60, is at 
-186. 
(8) By means of equation (99), the frequency of occurrence of any discharge 
may be computed, and it is desired to find the ratio of this discharge frequency for 
discharges greater than S e to the normal frequency of those discharges. In column 
1 of Table 61 are shown the discharges of Stream A greater than S c in increments of 
0.5 the p.e. or 92.5 (shown alternately as 92 and 93). In column 2 is shown the 
increments, a/r, of equation (95) or (D' — *S C )/185. In column 3 is shown the x of 
Z)'-(-186) 
equation (99), or r^r . Any figure in column 4 is the total number of 
observed stream-flows between the limits shown in column 1 from the discharge 
preceding the one opposite it to one less than the discharge following the 
one opposite it; thus there were 527 discharges between 0.108 and 0.292 cubic 
foot per second, inclusive. In column 5 are shown the frequencies of dis- 
charges between integral increments of a/r as computed by equation (99) by the 
method of areas, using three ordinates in each area, and computing the area by 
Simpson's formula. As 185 was used in the unit of x in deriving equation (99), the 
sum obtained by adding every other value in column 5, beginning with the first, 
should equal the total number of observed flows greater than S c = 108, or 700. In 
column 6 is shown the products of p = 0.476936 into column 2. These are the upper 
limits of the probability integral, equation (95). Column 7 shows the values of 
equation (95) for the limits shown in column 6. These represent the total area 
under the symmetrical probability curve from the origin, or S c , to the limit shown 
in column 6 on both sides of the origin. In column 8 are shown the increments of 
column 7 between alternate limits shown in column 2; thus, the probability 
of an accidental error occurring between the limits one and two times the probable 
error is 0.32265. In column 9 is shown the total number of accidental errors in 
700 which will occur between alternate limits shown in column 2 ; thus 350 + and — 
errors will occur between and 1 time the probable error. The total obtained by 
adding every other value in column 9, beginning with the first, should equal 700. 
The ratio of calculated actual frequency to the symmetrical frequency is given by 
dividing the values in column 5 by those in column 9, and is shown in column 10. 
The frequency curve calculated from equation (99) and shown in column 5, 
Table 61, is shown on Plate 20, as curve 2. On this plate the vertical scale on the 
