144 A NEW METHOD OF ESTIMATING STREAM-FLOW 
FORM OF OBSERVATION EQUATION FOR DETERMINING NORMAL STREAM-FLOW 
In what has preceded, culminating in equation (55), there have been presented: 
(a) the form of equation which expresses normal stream-flow, viz, equation (33) 
involving equation (34), (35a) and (36); (6), the form of equation which expresses 
flood flow, viz, equation (37); and (c), the form of observation equation for deter- 
mining C, F and M, viz, equation (46). 
It should be obvious that the constants, S c , R\, R\, . . . R\» in equation (33) 
could not be determined at the outset in this research from observations in the 
winter months without knowing how to compute the net melting, equation (36), 
F 
or the — - of equation (35a). Likewise it was impossible to evaluate all of the R"s 
in equation (33) from simultaneous observations of stream-flow and the meteor- 
ological elements in the summer months only, when net melting was not involved, 
because on the Wagon Wheel Gap watersheds the air temperature is above freezing 
for only about seven months out of the year, and to compute r t one must go back 
8.4 months from the date for which r, is wanted. (See definition of r l0 after 
equation (33).) 
The method of procedure was as follows: It was assumed at the outset that 
F 
— was the same as for the Great Lakes' region, viz, 0.62; that is, it was assumed 
that the rate of evaporation from the land surface of the Wagon Wheel Gap area 
was 62 per cent as large as that from a water surface, as determined for the Great 
Lakes' drainage area. Further, the observations were confined to the summer 
months only, which eliminated net melting from equation (33). The observation 
equation for the first least-square solution then had the following form: 
S c +r l R\+r,R\+r z R' z +r i R\+ . . . + r n R' n -D' = v (56) 
In this equation r» = (observed rainfall)— E w (0.62) — (observed run-off) on 
the current day, r 2 has the same definition but relates to the preceding day, r, to 
the day before the preceding day, and so on, similarly, to the definitions already 
given in connection with equation (33). 
D' is the observed stream-flow on the current day, or day to which the equation 
pertains, expressed in 0.001 c.f.s. 
S c is the constant part of the stream-flow in the same units as D', and R\, 
R' 2 , . . . R' „ express the effects of current and earlier changes in storage in the 
drainage area, on the normal flow of the stream on the current day. 
The right-hand member of equation (56) v is a residual and represents the dis- 
crepancy between the stream-flow as computed from theory — the S c and rR"s— 
and the observed stream flow, D'. 
One observation equation in the form of equation (56) was written for each 
day. From these the least-square solution served to determine the most probable 
values of the unknowns, and from the residuals, v, the accuracy of the unknowns 
(their probable errors) was computed. 
Because of the relatively short period of the summer months on the Wagon 
Wheel Gap watersheds; because, to compute r iQ for any day, one must use data 
which goes back a period of 8.4 months; and because only three years of data were 
available for use in deriving the laws of flow (viz, 1911-1913) — as it was desirable 
to apply the formulas to years (1914-1915) not used in their derivation — equation 
