THE OBSERVATIONS. 



37 



from the feeding station to the collecting sta- 

 tion being at least 16 feet longer than the 

 space covered by the observations used in the 

 computation. 



The ninth column shows the depth at a 

 single point as measured by the gage (see 

 p. 21); the tenth gives the mean depth as 

 estimated from the records of water profile and 

 bed profile (see p. 25). 



In the eleventh column, headed "Character 

 of bed," a generalization is given from notes 

 on the mode of transportation, the condition 

 of the water surface during the run, and the 

 condition of the bed of debris after the with- 

 drawal of the water. The original notes are 

 somewhat varied in scope and nomenclature, 

 and it seemed best to make the tabulated 

 record more simple. The words dunes, smooth, 

 and antidunts denote the three modes of trac- 

 tion described on pages 30-33. The word 

 transition is used where the mode of traction 

 was intermediate between one of the rhythmic 

 modes and the smooth mode, and also where 

 different modes of traction obtained in differ- 

 ent parts of the trough. 



In the final column the word free indicates 

 that the contractor (p. 25) was not used, but 

 the experiment trough retained its full width 

 to the end. 



The reduction of the slope observations was 

 preceded by a careful study of them with refer- 

 ence to their systematic and accidental errors. 

 As a result of that study certain criteria of 

 exclusion were adopted, by means of which, it 

 is thought, the influence of systematic errors 

 was materially reduced. The criteria were 

 applied through an inspection of the plotted 

 profiles of water surface and de'bris bed. The 

 observations excluded were those believed to 

 be much affected either by the peculiar con- 

 ditions near the head of the trough or by the 



peculiar conditions near the outfall end, so 

 that the retained observations constituted a 

 continuous series covering the middle. 



The observations to be reduced were as- 

 cribed equal weights. They consisted of a 

 series of level readings (h a , 7i lt Ti 2 . . . 7i n ^, 

 h n ), each giving the vertical distance of a point 

 of the de'bris surface, or of the water surface, 

 below a horizontal plane of reference, together 

 with the horizontal distances of the same 

 points from an initial point in the axis of 

 the trough. The observed points being at 

 equal intervals along the trough, and the zero 

 of distances being made coincident with the 

 first of the observed points, the horizontal dis- 

 tances may be represented by 0, I, 2l, . . . 

 (nl)l. The following formula 1 was used to 

 compute the slope, the number of terms in 



7? 



numerator or denominator being x when n 

 was an even number, and ^ when n was odd. 



100 (h n -h )n+(h n _-h l )(n-2)+(h n ^-h 1 )(n-4)+ . . . ,, , 

 I ' n 2 +(n-2) 2 +(n-4) 2 + . . . 



The numerical operations were simple. 



In division (J) of the table the arrangement 

 is somewhat different, and a column is added 

 at the left. The division records experiments 

 made with debris prepared by mixing, in defi- 

 nite proportions, two or more of the grades of 

 debris to which the preceding divisions pertain. 

 The notation adopted to designate these mix- 

 tures is analogous to that for chemical com- 

 pounds, subscript figures being used to indicate 

 approximate proportions. This notation, to- 

 gether with the more precise indication of the 

 proportions, is given in the left-hand column. 



i EssentiaUy a least-squares formula, although developed from appa- 

 rently independent considerations. 



