A NEW METHOD OF ESTIMATING STREAM-FLOW 147 
stream-flow were obtained on both Streams A and B. These observation equations 
were confined to the summer months only, and no r was used which reached back 
in its computation into the spring or winter freezing-melting period. The least- 
square solutions of this kind contained from 30 to 60 observation equations each, 
one equation for each day, normally in August and September of 1912 and 1913. 
Into each observation equation were written the known, or assumed, quantities, 
the r's and the D"s. By the method of least-squares, the unknowns S c to R'„ were 
evaluated. The result was an approximate, numerical expression for the flow of the 
stream when the addition to storage in the drainage area was all from rain (except 
some of it — an uncertain part — which came from springs). 
(2) The residuals from the least-square solutions mentioned in step (1) which 
were negative and greater than 3.5 times the probable error of a single observation 
of the solution, became approximate F"s for use in equation (57) and made possible 
a determination of approximate flood-flow expressions for the two streams. It is 
important to note here also that the approximate numerical expression so derived 
applied to the flood-discharges in times of heavy rain. 
(3) The approximate expressions of normal flow and flood-flow obtained in 
steps (1) and (2) , that is, the approximate S c 's, R"s and R'/s for both streams, were 
then used in equations (42) to (55), and approximate values of C, F and M com- 
puted, with a T' at first assumed at 30° F. This procedure of using the approxi- 
mate expressions of the stream-flow derived from observations in the summer 
months to evaluating that part of the stream flow in the winter months which is 
not accounted for in terms of C, F and M, is justifiable on the basis of assumption 
No. 10, page 140. 
(4) With approximate values of C, F and M derived as stated in step (3), it 
F 
now became possible to (a) derive an approximate value of — - from equation 
E a 
(35a) ; and (6) to revise the first expression for the normal flow derived in step (1) 
above, and to derive additional R"s in equation (56). This process gave a more 
exact expression of the normal stream-flow than was obtained in step (1) above. 
(5) Using the results of step (4) it now became possible to revise the results of 
step (2), and so secure a better expression for the flood-flow than that obtained 
in step (2). 
(6) Using the results of steps (4) and (5), it now became possible to repeat step 
(3), and so secure better values of C, F, M, and T", than those derived in step (3). 
The method of procedure is thus seen to be a cyclical one, which gradually 
converges toward the truth, or toward as much of the truth as is obtainable with 
the adopted assumptions and forms of expression. Steps (1), (2) and (3) constitute 
one cycle; steps (4), (5) and (6) another. The convergence toward the truth is 
indicated by a decrease in the probable errors of the constants themselves, by a 
general decrease in size of the residuals, and by a tendency for the residuals to 
become accidental in their time distribution. 
In the study on Streams A and B, in such steps as indicated in (1) to (6) above, 
a total of 69 least-square solutions was made, each containing from one to twelve 
unknowns, and each containing from nineteen to 280 observation equations. The 
final numerical results on the two streams, as hereinafter presented, were based 
upon only 8 of the above 69 solutions. The other solutions were necessary from the 
nature of the problem involed. They served to build up gradually the method to 
its present status as already partially presented in general terms. 
