BELATION OF CAPACITY TO SLOPE. 



113 



measurement of slope. (See p. 99.) In the 

 term log (1 a), which enters into the value of 

 j, (equation 42), 1 is the unit of slope; and the 

 relative magnitude of 1 and a changes as the 

 unit changes. Other terms of the formula are 

 also (implicitly) functions of the unit, but the 

 various influences are not compensatory, and 

 the resultant is of such nature that the values 

 of 7, vary inversely with the magnitude of the 

 unit. It will be recalled that in the notation of 

 this paper the symbol S pertains to the unit 



distance 



' i ' 100) 



jjW -- The curve marked j t in figure 35 rep- 



resents values of the exponent computed with 

 use of the smaller unit, and the curve marked 

 Tc represents a coordinate system of values corn- 



Slope 



FIGURE 35. Variations of exponents d, ;'i, and k, in relation to slope. 

 The scale of slope is in per cent. 



puted with use of the larger unit. The latter 

 curve, produced, would intersect the curve of ^ 

 at a point corresponding to s= 1 or 5= 100. 



It would be possible, by employing the larger 

 unit in the notation for slope, to construct a 

 table equivalent to Table 16 in which the values 

 of 7\ would all be smaller and would have in each 

 series less range. For many of the series they 

 would approach closely the associated values of 

 n. This reduction of exponents would be ac- 

 companied by an enormous increase in the 

 values of coefficients, each value of c t in the 

 table being magnified by the factor 100". The 

 increase would result from the fact that c t is the 

 capacity corresponding to unit slope, while the 

 unit slope in that case would be 45. Con- 

 sidered as capacities, the values of c t would be 

 in a sense fictitious, because the laws of trac- 

 tion wifh which we are dealing do not apply 

 (see p. 63) to so high a slope as 45. 



20921 No. 8614 8 



To recur for a moment to the general account 

 of the index of relative variation, it will be 

 recalled that the index was shown to be inde- 

 pendent of the units of the observational quan- 



tities. In this particular instance the value of 

 o 



the index, n^ - , has two factors, of which the 







first is an abstract number and the second is a 

 ratio between slopes and is independent of the 

 unit of slope. The ordinates of the curve 

 marked \ in figure 35 are therefore independent 

 of the slope unit, and the curve is a fact of ob- 

 servation, plus the assumptions of the formula 

 of adjustment. 



PRECISION. 



Because the values of \ were computed from 

 values of n and a, their precisions are involved 

 with those of n and a, but the relation is not 

 simple. The precision of n depends partly on 

 the harmony of the observations of load and 

 slope and partly on the precision of a. It is 

 not feasible to measure the precision of a for 

 individual series of observations, and the pre- 

 cisions of individual values of n and i t are 

 therefore indeterminate. All that has been at- 

 tempted is to derive a rough estimate of average 

 precision from average values of the quantities 

 involved. The estimated average probable 

 error of values of n is 3.9 per cent, and the cor- 

 responding estimate for values of i t is 4.6 per 

 cent. 



The precision of i t varies with slope within 

 each series represented by a column in Table 

 15, being relatively high for the steeper slopes. 

 The probable error, if measured in the same 

 unit with i,, is much larger for gentle slopes 

 than for steep ; if measured in percentage, it is 

 somewhat larger for the gentler slopes. Meas- 

 ured in percentage, its value for the steeper 

 slopes approximates the corresponding prob- 

 able error of n. 



EVIDENCE FROM EXPERIMENTS WITH MIXED 

 DEBRIS. 



The observations on capacity and slope when 

 the d6bris transported consisted of a mixture 

 of two or more grades were reduced in the same 

 general manner as those for single grades. It 

 was not thought advisable to make any adjust- 

 ment of the values of a, but each logarithmic 

 plot was treated independently. For about 

 one-third of the mixtures the best value 



