A NEW METHOD OF ESTIMATING STREAM-FLOW 145 
(56) served to determine less than 10 R"s. The last W determinable is indicated in 
equation (56) by the subscript n. 
On each of Streams A and B, equation (56) was applied and as many R"s 
derived as possible from simultaneous observations of the weather elements and 
stream discharges in the summers of 1911, 1912 and 1913. An R' was considered 
real when it came out greater than 3.5 times its own probable error. 
F 1 
The laws of normal flow so derived were first approximations, because — - was 
E a 
assumed and only part of the R"s were evaluated. 
These first approximations were made more exact by the step-by-step process 
outlined on pages 146 and 147. 
FORM OF OBSERVATION EQUATIONS FOR DETERMINING FLOOD-FLOW 
From any least-square solution in which equation (56) was used, a set of v's 
was obtained in the substitution in observation equations. From these v's the prob- 
able error of a single observation was computed. Normally, any residual greater 
than 3.5 times the probable error of a single observation was considered "suspi- 
cious," and a candidate for careful scrutiny. A residual greater than five times the 
probable error of a single observation was considered a candidate for rejection in 
the next least-square solution of the series. 
If a residual in equation (56) came out negative, the meaning was that the 
computed stream-flow, computed according to the assumed law of normal flow, was 
less than the observed stream-flow, D' . If it was negative and greater than 3.5 
times the probable error of a single observation, this was considered an indication 
that the flow concerned was a flood-flow. According to the law of probability, 
only 18 equations per thousand should have residuals greater than the suspicion 
limit if the law of flow as expressed by equation (56) is complete; that is, if there 
were no approximations involved in the theory, or if the residuals followed the 
normal law of error. The appearance of many more v's beyond the suspicion limit 
than are accountable for on the basis of accidental errors formed a clear indication, 
proof, that the law of flow as expressed by equation (56) is not complete; that in 
times of large flows, one must account for that part of the flow which equation (56) 
fails to account for by another expression. The interpretation of the large number 
of — v's during heavy rain-falls when equation (56) alone is used is that, as already 
stated, equation (56) accounts for only that part of the stream-flow which is pro- 
duced by water which travels largely through the ground to the stream. When the 
ground becomes glutted with water, when the water-table has risen to the level of 
the ground surface, a surface flow is started. It has been found that this surface 
flow is expressible in the form of equation (37). 
The internal evidence from the least-square solutions that the stream-flow 
follows a different law during heavy rains or rapid melting is very strong, but if one 
is inclined to be skeptical about it, the following sample of external evidence, not 
connected with the least-square solutions, is given: 
At the Wagon Wheel Gap streams the maximum R' , expressed as a percentage 
of the total change in storage which reaches the stream by percolation on any day, 
is less than 1 per cent. The maximum flow is about 6 c.f .s. per square mile of drain- 
age area. At the Miami Conservancy District, according to figure 4, page 30, of 
Hydraulics of the Miami Flood Control Project by Sherman M. Woodward, the 
maximum R' expressed as a percentage, during the March 1913 flood, was probably 
