RELATION OF CAPACITY TO DISCHABGE. 



143 



Table 12, but also whatever errors affect the 

 method of adjustment in connection with the 

 discussion of the control of capacity by dis- 

 charge. That method includes the assump- 

 tion that capacity varies as a power of (Q /c) 

 and also certain assumptions as to the control 

 of K by various conditions. If the errors in- 

 volved in those assumptions could be elimi- 

 nated or discriminated, the general estimate 

 for the values in Table 12 would be somewhat 

 reduced. 



As the chief result of this discussion, the 

 general precision of the main body of material 

 contributed by the Berkeley experiments is 

 characterized by an average probable error 

 slightly in excess of 3 per cent. 



This estimate applies specifically to the ad- 

 justed capacities of Table 12 as those capacities 

 are related to slope and discharge, and it can 

 not be extended to derivatives of those capaci- 

 ties without qualification. It is believed that 

 the precision of the readjusted capacities of 

 Table 32 is higher, and also that of the values 

 of 7, in Table 23, but that the values of i lt j lt 

 and i 3 , in Tables 15, 16, and 32, rank lower. 



In the adjustment of the observations on 

 capacity in relation to slope, and also in the 

 adjustment of observations on depth and slope, 

 many cases were treated in which the observa- 

 tions were either not sufficiently numerous or 

 not sufficiently harmonious to afford good con- 

 trol of the parameters of the adjusting equa- 

 tions. In such cases the parameters were esti- 

 mated in groups, with orderly sequences of 

 values. A similar method was employed also 

 in readjustments in relation to discharge. 

 While this procedure appeared, and still ap- 

 pears, to be the best practicable, it can hardly 

 fail to introduce a certain amount of error. 

 The terms of the adjusted sequences are in- 

 evitably associated with different form ratios; 

 and the laws connecting form ratio with 

 capacity are so different from the other laws 

 of the system as to determine sequences less 

 simple than those actually used. The ideal 

 adjustment would take simultaneous cogni- 

 zance of the complicated interrelations of 

 capacity, slope, discharge, and form ratio; 

 but such comprehensive treatment can not 

 be attempted with profit until we have a 

 better theoretic knowledge of the physical 

 reactions. 



CONTROL OP RELATIVE VARIATION BY 

 CONDITIONS. 



The sensitiveness of capacity to changes of 

 discharge is indicated by i 3 , the index of rela- 

 tive variation. Inspection of the values of that 

 index recorded in Table 32 shows that they 

 vary inversely with discharge and directly 

 with width, and both these tendencies are also 

 to be inferred from the lines of figure 47. 



The table further indicates that the rate of 

 change in the index in response to change of 

 discharge is greater for small discharges than 

 for large. This feature might also have been 

 inferred from the plotted lines, but its system- 

 atic expression in the table is a product of the 



formula, i 3 = n , by which the values were 

 y K 



computed. 



The relation of the index to width of channel 

 is not similarly dominated by the formula, 

 although somewhat influenced by the assign- 

 ment of values of . The variation of the 

 index with width is in general more pronounced 

 for low discharges than for high, but in the 

 data for high discharges occur two exceptions 

 to the general law. These exceptions are 

 ascribed to irregularities in the data. As the 

 width for any particular discharge varies in- 

 versely with the form ratio, it follows that the 

 index is a decreasing function of form ratio. 



The relation of the index to slope is not shown 

 by the table. It was the subject of a special 

 inquiry, including 32 comparisons. In each 

 comparison a value of the index for a particular 

 slope was contrasted with the value for a slope 

 twice as great, the other conditions being the 

 same. In 25 instances the greater index was 

 associated with the smaller slope; in 7 instances 

 with the larger. The mean of the indexes 

 computed for smaller slopes was 1.51; the mean 

 for larger slopes 1.17. The general law appears 

 to be that the index varies inversely with 

 slope. The seven instances of opposite tenor 

 are all associated with large discharges; and 

 their occurrence is ascribed to a systematic 

 error connected with the assignment of values 

 to K. 



Table 35 compares values of the index with 

 fineness. Despite irregularities, it is evident 

 that the values tend to increase in passing from 

 finer to coarser grades that is, their variation 

 in respect to fineness is inverse. 



