A NEW METHOD OF ESTIMATING STREAM-FLOW 107 
Using the values shown in column 10, Table 44, a substitution in observation 
equations was made, using the r's from Solution M and leaving the S, column 
blank. The equations had the following form: 
0.28r,+0.21r,+0.14r 1 +0.109r«+0.84r i +0.69r i +0.62r,+0.58r,-|-0.50r, a +0.38r» 
+2.17r 10 -D' = v (72) 
The right-hand member, v, of equation (72), in addition to containing the 
discrepancies between computed and observed stream-flow due to all the various 
causes, also contained the omitted constant part of the stream-flow, S c . Hence the 
mean v of Solution N with its sign changed became the adjusted value of S c of that 
solution. This value was +116; the change from +127, the S c of (71), being 
attributable to the smoothing-off process. By this adjustment the probable error 
of a single observation was reduced from ="= 12.4 in Solution M to =*= 8.6 in Solution N. 
With the new value of S c , +116, and equation (72) as written above, a sub- 
stitution in observation equations was made having the form 
+ 116+0.28r,+0.21r 2 +0.14r,+0.109r4+0.84r,+0.69r l +0.62r 7 +0.58r,+0.50r, a 
+0.38r )b +2.17r 10 -D' = v (73) 
for the three-year period 1911 to 1913 inclusively. This computation gave a set 
of residuals, v, of the nature of those of equation (58), from which, during times of 
flood, F"s of equation (57) as well as values of G for various D"s could be obtained 
for determining the flood-flow formula. After the derivation of the flood-flow 
formula, the illustration of which will be shown later, and taken from the computa- 
tions on Stream B, 1 a final adjustment was made to the normal flow formula for 
Stream A. This final adjustment became advisable from a study of the residuals 
of equation (73), in which it was found that large systematic errors existed princi- 
pally because of the +2.17ri — term. The adjustment, in this case, was designated 
as Solution P and it had for its form of observation equation the following: 
R+r l0 R l9 -L = v (74) 
in which R is the desired correction to the S c of equation (73) and R 10 is the desired 
correction to the R\ of equation (73). One observation equation of the form of 
equation (74) was written for each month. r x in equation (74) was the mean value 
of r :0 of Solution M for the month. — L was the mean residual, v, of equation (73) 
for each month. In selecting the months to be used in Solution P, all of the months 
in 1911 to 1913, to which equation (73) had been applied, were selected which were 
complete as to the r X9 value and apparently free from flood-flow. A month was 
not considered free from flood-flow unless every negative residual in it was numeri- 
cally less than 3.5 times the probable error of a single observation of Solution N, 
or (3.5X8.6 = ) 30, which, in the adopted units, is 0.030 c.f.s. 
EXAMPLE OF OBSERVATION EQUATIONS FOR ADJUSTING 5c AND R\ a 
The following observation equations of the form of equation (74) were used in 
Solution P, from which to derive corrections to S c and R'iq. 
See page 172. 
