A NEW METHOD OF ESTIMATING STREAM-FLOW 
109 
EXAMPLE OF SUBSTITUTION IN OBSERVATION EQUATIONS FOR NORMAL STREAM-FLOW 
For the months used in Solution M, from which most of the constants in 
equation (76) were derived, the substitution in observation equations was made in 
that solution by using the derived values of the corrections from that solution 
shown in equation (70). In Solution N, the substitution in observation equations 
was made, using the smoothed-off values of the constants, (71), shown in (73). In 
Solution P, the substitution was made using the final constants of equation (76), 
except that R', a and R', b were used separately, to correspond with the method of 
computing r, a and r, b used in the months of these solutions. It should be evident 
that in each case of these substitutions in observation equations, a set of residuals 
was obtained like those in the right-hand members of the sample of observation 
equations shown on page 152. The smoothed-off values used in Solution N gave 
smaller residuals than were those of Solution M. Another reduction in the size of 
the residuals was effected by using the more correct values of the unknowns of 
Solution P. As it is the purpose here to show the final residuals obtained by use of 
the best derived normal-flow equation for Stream A, equation (76), the sample of 
observation equations shown on page 152 will be changed to the form of equation 
(58), which is a combination of equations (59) and (60). 
Substitution in 
observation equations for June 1913, Stream A 
The unit is 0.001 cubic foot per second 
Date 
Sc 
riR'i 
rzR'z 
r z R'i 
UR\ 
rsR'i 
r 6 R' 8 
nft'7 
r$R's 
rtaR 9a 
7"9&ft 96 
rioR io 
-D' 
V 
1913 
June 1 
+ 108 
- 4 
- 4 
-3 
-3 
- 3 
- 8 
-14 
+38 
— 2 
-4 
-4 
— 179 
- 82 
2 
+ 108 
- 5 
- 3 
-3 
-4 
— 2 
- 8 
-14 
+35 

-3 
-4 
-177 
- 80 
3 
+ 108 
- 6 
- 4 
-2 
-4 
- 2 
- 8 
-14 
+ 30 
+ 3 
-3 
-4 
-175 
- 82 
4 
+ 108 
— 4 
- 4 
-3 
-4 
- 3 
- 8 
-14 
+ 25 
+ 7 
-3 
-4 
-173 
— 80 
5 
+ 108 
- 3 
- 3 
-3 
-4 
- 5 
- 6 
-14 
+ 20 
+ 11 
-3 
-4 
-171 
- 77 
6 
+ 108 
- 3 
- 2 
-2 
-4 
- 6 
- 6 
-14 
+ 18 
+ 12 
-3 
-4 
-169 
- 75 
7 
+ 108 
+ 11 
- 2 
-2 
-4 
- 6 
- 6 
-14 
+ 17 
+ 13 
-3 
— 4 
-182 
- 74 
8 
+ 108 
+ 4 
+ 8 
-2 
-3 
- 6 
- 7 
-14 
+ 16 
+ 13 
-3 
-4 
-187 
- 77 
9 
+ 108 
+ 8 
+ 3 
+ 6 
-2 
- 6 
- 7 
-14 
+ 13 
+ 16 
-3 
-3 
— 181 
- 62 
10 
+ 108 
+ 16 
+ 6 
+ 2 
+ 3 
- 5 
- 6 
-15 
+ 8 
+ 20 
-3 
-3 
— 218 
- 87 
11 
+ 108 

+ 12 
+4 
+ 6 
- 5 
- 7 
-15 
+ 5 
+ 21 
-3 
-3 
— 224 
-101 
12 
+ 108 
+ 9 

+ 8 
+ 5 

- 7 
-16 
+ 4 
+ 21 
-3 
— 3 
-226 
-100 
13 
+ 108 
- 3 
+ 7 

+ 9 
+ 3 
- 9 
—13 
+ 3 
+ 21 
-3 
-3 
-218 
- 98 
14 
+ 108 
- 4 
- 2 
+4 
+ 6 
+ 6 
-10 
-12 
+ 2 
+ 21 
-3 
-3 
-206 
- 93 
15 
+ 108 
- 5 
- 3 
-2 
+ 3 
+ 12 
- 9 
-12 
+ 1 
+ 21 
-3 
-3 
-199 
- 91 
16 
+ 10S 
- 4 
- 4 
-2 
+ 2 
+ 8 
- 5 
-12 
— 2 
+ 23 
-3 
-3 
-197 
- 91 
17 
+ 108 
+ 2 
- 3 
-3 
-3 
+ 10 
- 3 
-13 
- 6 
+ 26 
— 2 
-3 
-202 
- 92 
18 
+ 108 
+ 4 
+ 1 
-2 
— 4 
+ 7 
+ 1 
-13 
-10 
+ 28 
-2 
-3 
-205 
- 90 
19 
+ 108 
- 3 
+ 3 
+ 1 
-4 

+ 6 
-13 
-13 
+31 
-3 
-4 
-197 
— 88 
20 
+ 108 
- 4 
- 2 
+ 2 
-1 
— 1 
+ 7 
-14 
-16 
+ 32 
-3 
-4 
— 187 
- 83 
21 
+ 108 
- 3 
- 3 
-1 
+ 2 
- 5 
+ 10 
-14 
-17 
+34 
-3 
-4 
-1S2 
- 78 
22 
+ 108 
- 3 
- 2 
— 2 

- 3 
+ 10 
-14 
-19 
+ 34 
-3 
-4 
-175 
- 73 
23 
+ 108 
+ 1 
2 
-2 
-3 
- 1 
+ 10 
-14 
-20 
+ 34 
-3 
-4 
-174 
- 70 
24 
+ 108 
- 3 
+ 1 
-1 
-3 

+ 6 
-11 
-21 
+ 34 
2 
-1 
-169 
- 65 
25 
+ 108 
- 6 
- 2 
+ 1 
2 

+ 4 
- 8 
-23 
+ 35 
2 
-4 
— 161 
- 60 
26 
+ 108 
- 7 
- 4 
-2 

- 2 
+ 2 
- 5 
— 23 
+ 35 
2 
-4 
-154 
- 58 
27 
+ 108 
- 4 
- 5 
-3 
-1 
- 3 
- 1 
- 1 
-24 
+ 34 
-2 
-4 
-150 
- 56 
28 
+ 108 
+ 3 
- 3 
-4 
-3 
- 3 
- 1 

-24 
+ 33 
-2 
-4 
-157 
- 57 
29 
+ 108 
- 3 
+ 2 
-2 
-5 
- 3 
- 4 
+ 1 
— 23 
+33 
2 
-4 
-147 
- 49 
30 
+ 108 
- 5 
- 2 
+ 1 
-4 
- 3 
- 4 
+ 1 
-23 
+33 
-1 
-4 
-140 
- 43 
In the above tabular arrangement of the substitution in observation equations, 
the heading on each column identifies the term of the observation equations as 
shown on page 152, except that, instead of using the headings there shown, viz, 
R, riRi } r*Rt, . . . ri Ri and —D, there has been used S c , rjl'i, r 2 R' 2 , . . . r lo R\o 
