REVIEW OF CONTROLS OF CAPACITY. 



193 



They are also connected with the fact that the 

 algebraic forms imperfectly represent the physi- 

 cal phenomena. It is possible that the physi- 

 cal facts are closely represented by some one of 

 the equations, in which case the incompati- 

 bilities of the other two equations are wholly 

 of the nature of errors; but if this be so, our 

 data do not enable us to indicate the truer 

 equation. 



The exponent in the slope factor is greater 

 for the capacity and duty equations than for 

 the efficiency equation, the difference being a 

 large fraction of unity. The exponent in the 

 discharge factor is greater for the capacity 

 equation than for those of duty and efficiency, 

 the difference being a large fraction of unity. 

 The competence constant in the slope factor 

 is smaller in the capacity and duty equations 

 than in that for efficiency, the difference being 

 a small fraction of the constant. The com- 

 petence constant in the discharge factor is 

 smaller for the capacity equation than for the 

 others, the difference being relatively large. 

 The interrelations of parameters and the rela- 

 tions of parameters to independent variables 

 are qualitatively the same for the equations of 

 duty and efficiency as for the equation of 

 capacity. 



THK FORMULA OP 1ECHALAS. 



THE FORMULA. 



The only earlier serious attempt to formulate 

 the transportation of debris, so far as I am 

 informed, is that of C. Lechalas, who wrote in 

 1871, under the title "Note sur les rivieres a 

 fond de sable." 1 His discussion makes use of 

 Dubuat's experiments on competent velocity 

 (1786), Darcy and Bazin's formulas for veloci- 

 ties in conduits and rivers (1878), and observa- 

 tional data accumulated by the French engi- 

 neering corps in connection with projects for 

 the improvement of navigable rivers. In the 

 following abstract of the more elementary 

 part of his discussion the symbolic notation 

 and the terminology of the present paper are 

 to some extent substituted for those of the 

 original. 



Postulate a stream of fixed width and con- 

 stant discharge, traversing a bed of uniform 

 sand, unlimited in quantity but without 



1 Annales dcs ponts et <*>aiiss&s, M&n. et doc. ,5th ser., vol. t, pp. 381- 

 431, 1871. 



20921 No. 86 14 13 



accessions. So long as the velocity along the 

 bed exceeds a certain value the current trans- 

 ports sand. Below that limit, the sand is 

 undisturbed. 



The discharge, Q, and width, w, being 

 known, the mean depth, d, the slope, s, the 

 mean velocity, V n , and the bed velocity, V b , 

 are given by the following three equations, of 

 which (1) and (3) are from Darcy and Bazin. 

 The constants are in meters. 



(1) 



(2) 

 (3) 



The sand travels (1) by rolling, (2) by sus- 

 pension. A particle of water impinging on 

 the bottom gives motion to a sand grain, the 

 motion having a direction which depends on 

 the impact and on the positions of adjoining 

 particles, solid and liquid. The grain is pro- 

 jected free from the bottom or is rolled along 

 it, the particular result depending on the 

 inclination and force of the impact and on 

 various conditions which affect the resistance. 

 Suspension corresponds especially to impacts 

 associated with high velocities. Suspension is 

 rare below a certain critical velocity for each 

 density and size of sand grain. Transporta- 

 tion is slow at low stages of a variable stream, 

 rapid and by suspension at high stages. The 

 grains describe trajectories analogous to those 

 of the water particles, but shorter; and there 

 are frequent returns to the bottom, as well as 

 restings between excursions. Larger grains 

 are lifted less high, or are rolled only, or remain 

 at rest. Small grains afford a better hold 

 (prise) in relation to their weight. The 

 smallest of all are carried in the body of the 

 current. 



The amount by which the pressure on the 

 upstream face of a grain immersed in a current 

 exceeds the pressure on the downstream face 

 is proportional to the square of the velocity. 

 Represent it by a V b 2 , the coefficient a depend- 

 ing on size, form, and position. For the sand 

 of the Loire, the resistance developed equals 

 a 0.25 2 , as that sand is immobile when V b < 0.25. 

 "The difference is equal to the product of the 

 mass of the grains by then- velocity, projected 

 on the same axis as F 6 that is to say, on the 

 axis of the stream. This product, being pro- 



