86 
A NEW METHOD OF ESTIMATING STREAM-FLOW 
period of observation, 1909 to 1913, was computed from the relation r = -7=, in 
V n 
which r is the probable error of the mean v, r the probable error of a single observa- 
tion of Solution V \, viz, ±14.4, and n is the number of residuals involved. 
The tabulation indicates that there is a strong systematic distribution of the 
residuals with reference to the season, positive residuals occurring from May to 
July and negative ones from August to September. Moreover, the probable errors 
of the mean monthly residuals are small in comparison with the mean v's them- 
selves for the first two and last two months, especially. This is a very strong indi- 
cation — proof — of a systematic error, and corresponds with the evidence shown in 
the accumulated sums in Table 23 and as plotted on Plate 5. That is, the run-off is 
probably greater than 0.006 foot per day in May and June, and less than that in 
September and October. 
While the evidence of systematic error is indisputable in as far as the season is 
concerned, note that there is little or no evidence of systematic error for the solution 
as a whole. For the whole period 1909 to 1913 the mean residual is — 2=±=0.53. The 
Table 27 — Mean residuals from Solution Vt, Lake Michigan-Huron 
Month 
1909 
1910 
1911 
1912 
1913 
Mean 
monthly 
V 
No. 
of 
v'a 
May 
+3 
+5 

-5 
-9 
-8 
-2±1 
159 
+5 
+4 

-7 
-9 
-7 
-2±1 
170 
+5 
+7 
-1 
-5 
-4 
-3 

167 
+ 5 
+ 7 
+ 5 
- 1 
-10 
-10 

157 
+4±1 
+6=1 
+1*1 
-4±1 
-9±1 
-8±1 
99 
116 
150 
145 
126 
119 
June 
July 
+ 1 
- 3 
— 11 
-10 
-5±1 
102 
August 
September 
Mean annual v 
No. of v's 
Mean residual, v, for whole period 1909-1913= —2*0.53. 
Mean residual, v, for 1910-1913 inclusive= —1*0.6. 
probable error is nearly \ the value of the residual in this case. But note that the 
omission of the + residuals, which would have occurred in May and June 1909 if 
observations for those months had been used, would have reduced this mean v. If 
one consider only the years 1910 to 1913, thereby eliminating the effect of omitting 
May and June 1909, the mean v is — 1=±=0.6. 
The effect of omitting the observations for May and June 1909 may also be 
seen by examining the mean annual residuals. For 1909 it is — 5=*= 1, strong indica- 
tion of systematic error. For the other four years there is no indication of syste- 
matic error. 
The comparison between the actual and theoretical distribution of the residuals 
is shown in Table 29. If the evaporation from the lake surface is fully expressed by 
the form of equation developed and the evaluated constants in it, on the one hand, 
and if all of the influences except evaporation, which affect the mean elevation of 
the whole lake surface have been properly evaluated, leaving nothing in such 
evaluations but accidental errors due to seiche effects, on the other hand, then the 
actual distribution of residuals will be the same as the theoretical distribution of 
accidental errors, neglecting the small difference which will result from using a 
finite number of observations. 
