PARABOLOID FORM OF SALTATORY PATHS 205 



tend to define curves of parabolic order, and that those thrown toward the 

 thread of the current by intercollision must move fastest and farthest. 

 lUustrations of this tendency apj^ear primarily in the paraboloid profiles 

 of mature streams (that is, in hydraulic grades), and secondarily in the 

 prevailing land forms sculptured by running water; the demonstration 

 appears in Humphreys and Abbott's classic determination — the most 

 notable contribution of American engineering to hydrophysics — that the 

 curves of differential stream-flow are parabolic. It follows that stream 

 water may not justly be considered to move in conformable films or even 

 as contiguous filaments, but must be conceived to move as an assemblage 

 of essentially incompressible and noncoherent and both nonattractive and 

 nonrepulsive yet virtually frictionless particles, each responding individ- 

 ually to the forces of gravity and coUisional resistance with the con- 

 stantly varying momenta of inertia arising therein, and all responding 

 interdependently to the sum of these conditions and forces and also to the 

 perturbations introduced by the extraneous particles forming suspended 

 and saltatory load. 



The veridity of this concept of the particular mechanics of running 

 water is corroborated by the inability of pure water to corrade, since the 

 particles are themselves virtually frictionless; by the development or 

 intensification of the undulatory tendency soon as the liquid comes in 

 contact with extraneous materials producing friction; by the invariable 

 development of current ripples themselves assuming paraboloid patterns ; 

 by the deadening of breakers through load of any material (sand, mud, 

 etcetera) introducing perturbation among the particles; by the geo- 

 metrically increasing capacity accompanying increase either of velocity 

 or of any load tending to perturb the particles through surface friction, 

 and indeed generally by the natural behavior of moving water. The 

 concept has the merit, too, of explaining physically the paraboloic flow- 

 curves worked out empirically by Humphreys and Abbott, without re- 

 course to arbitrary and probably misleading assumptions as to friction. 

 The concept also clarifies Powell's practical induction as to the increase 

 of efficiency with load; for it is evident that the internal work (neglecting 

 tliat directly due to wetted perimeter) varies with friction among the 

 extraneous particles — that is, approximately with the aggregate super- 

 ficies of these particles, capacity and competence varying only with their 

 mass (both with the same velocity function) — while the practical case is 

 that in which the volume is variable from freshet to low-water stage in 

 such degree as measuraljly to balance velocity and thus eliminate the 

 retardation of flow due to the increased viscosity occasioned by the load. 

 At the same time the concept elucidates the habit of running water to 



