140 A NEW METHOD OF ESTIMATING STREAM-FLOW 
THE FREEZING-MELTING THEORY 
If the flow of a stream is not affected by current or earlier freezing and melting 
on its watershed, equation (33) becomes simplified by the fact that the net melting 
as expressed by equation (36) is zero. This would be true of any stream in which 
little or none of the precipitation on its watershed is in the frozen form, as for 
instance, roughly speaking, streams below latitude 35° to 37° N in the eastern two- 
thirds of the United States. For streams affected by freezing and melting of the 
snow and ice on their watersheds, equation (33), involving equation (36), becomes 
more complicated by virtue of the lag between precipitation in the frozen form and 
its subsequent run-off as water, and because of the reverse process — a subtraction 
from the storage in the drainage area by freezing. 
Equation (33), involving equation (36), when applied to normal stream-flow in 
cold weather, is based upon the following assumptions: 
Assumption No. (9) — There is a mean air temperature on the watershed for a 
day which will be called T' for which the melting during the 24-hour period is just 
balanced by the freezing during that period, and there is therefore no net addition 
to or subtraction from the stored water in the drainage area by melting and freezing. 
Assumption No. (10) — The stored water in the drainage area moves according 
to the same laws, to the degree of accuracy which can be detected by observations 
of stream-flow, in cold weather as in warm weather. The stored ice and snow does 
not move. 
Assumption No. (11)— Whenever the mean air temperature for a day in the 
drainage area is below T' the amount of water which is changed into ice on that day, 
and thereby withheld from storage as water, is (T' — t)F, in which t is the observed 
mean air temperature for the day and F is a constant to be determined from the 
observations. 
Assumption No. (12) — Whenever the mean air temperature for a day in the 
drainage area is above T' , and there is an abundant amount of ice or snow on and 
in the ground available for melting, the amount of water which is added to the 
storage in the drainage area on that day is (t — T')M; in which M is a constant to 
be derived from the observations. 
On the above four assumptions r,, the addition to the water stored in the 
drainage area on the current day, is 
ri = (Observed rainfall) — (Estimated evaporation from land — (Observed discharge) 
,-(T'-t)F } 
+ or (38) 
l+(t-T')M) 
Note that equation (38) takes no account of precipitation in any other form 
than rain. It is written as if T' were known. It can not be known from the start 
and must be determined from the observations. Let T" be an assumed value of 
T' , and let it be assumed that when the mean air temperature for the day is T", 
there will be an amount of melting C, that the melting at higher temperatures will 
be C-\-(t—T")M, and that the amount of water held back from storage by freezing 
at lower temperatures will be -\-C—(T" — t)F. Then from a given set of observa- 
tion equations, as indicated later, a value of C may be derived. If this derived 
value of C is positive, the necessary correction to the assumed temperature T" to 
