A NEW METHOD OF ESTIMATING STREAM-FLOW 175 
Normal equations for Oct. 4~Nov. 6, 1911. Solution K, Stream B 
+39,0347?/,+ 2,955/2 /2 - 53,494 = 
+ 2,955/?/, +39,034/2 „+ 2,955/2,, - 52,739 = 
+ 2,955/2 /2 +39,034/2/,+ 2,955/2 /« - 40,938 = 
+ 2,955/2 /,+83,978/2/«+ 394/2,, -114,890 = 0^(82) 
+ 394/2/4+1,727/2/5+ 40/2 „ - 10,104 = 
+ 40/2/ 6 +3,491/2/,+ 40/2/7 - 7,215 = 
+ 40/2/,+7,019/2/t + 464 = 0. 
Attention is called here to the particular group of days embodied in the normal 
equations (82). The rain on October 4 started a flood-flow which was accentuated 
by the heavy rain on October 5. In the observation equation (78), seven r ,R t 
terms only are included which cover a total time interval of 33 days. Since there 
was no other r JX later than October 5 in this time interval, the last r /7 which can be 
computed is on November 6 — 33 days after October 5, inclusive. 
The final normal equations for Solution K, formed by combining four such 
sets of normal equations as are shown in (82), are as follows: 
Final normal equations for Solution K, Stream B 
final normal equations J or ooiutii 
+44,070/2/,+ 3,501/2 /2 + 660/2/,+ 792/2 /4 + 18/2. 
+ 3,501/2 ^+44,070/2/2+ 3,501/2/,+ 1,452/2 /4 + 18/2 
+ 660/2/1+ 3,501/2/2+44,070/2/,+ 4,161/2 /4 + 78/2 
+ 792/2/,+ 1,452/2/2+ 4,161/2 /3 +95, 142/2 /4 + 726/2 
+ 18/2/,+ 18/2/ 3 + 78/2/,+ 726/2 /4 +2,003/2 
+ 0/2/, 6/2/3+ 24/2 ,«+ 77/2 
18/2/,+ 0/2/,+ 0/2/7 - 64,680 = 0' 
18/2/6+ 0/2/7 - 62,866 = 
78/2/6+ 6/2/ 6 + 0/2/7 - 49,113 = 
'26/2/5+ 24/2/ 6 + 0/2,7 -128,568 = 
D/6+ 77/2/,+ 0/2/7 - 18,239 = 
0/2/, 6/2/3+ 24/2 /4 + 77/2/6+4,157/2/6+ 78/2 /7 - 10,981 = 
+ 0/2/, 0/2/2+ 0/2/3+ 0/2 /4 + 0/2/6+ 78/2/,+8,469/2/ 7 - 3,150 = 0j 
The final set of normal equations for Solution K (83), depends upon 166 obser- 
vation equations covering the flood-flows produced by the rains during the summer 
months of 1911, 1912 and 1913. 
The solution of the final normal equations for Solution K (83) gives the follow- 
ing values for the unknowns : 
(83) 
(84) 
/2/, = +1.34 /2 /6 =+8.50 
/2 /2 =+1.21 Z2/,= +2.48 
i2/ 3 = +0.87 Z2/ 7 =+0.35 
/2 /4 =+1.23 
The probable error of a single observation for Solution K, computed rigorously 
from the 166 residuals and from the normal equations, was ±20.0. 
Combining (84) with (81) according to (80), there is obtained for the values 
of the unknowns : 
(85) 
• R',t=+1A1 R' tl = +15.31 
/2' /2 =+1.45 R',t=+ 7.02 
Z2'/ 3 =+1.44 Z2'/7=+ 2.37 
Z2'/4=+1.95 
In interpreting the constants (85), it is to be remembered that in the computa- 
tion of r/i, r /2 , r/,, . . . r /7 , the values after r /4 were divided by 10, hence to get the 
above constants into their true relationship with reference to each other, R' fi , R' /t 
and R' n should be divided by 10. 
