8 MEMOIRS OF THE NATIOI^AL A(3ADEMY OF SCIENCES. 



coutrols the distributioti of velocities of curreut within its area, aud that through the interactions 

 of these velocities its parts are interdependent. Each element of its curve is so adjusted to the 

 adjacent curreut and to the detrital load of the stream that it can neither be eroded nor receive a 

 deposit, aud the stability of the profile depends on the fact that an element not adjusted to the 

 contiguous current and load becomes subject either to erosion or to deposition until an adjust- 

 ment is reached. The distribution of velocities within the cross-section is symmetric, the swiftest 

 threads of the curreut being in the center and the slowest adjacent to the banks. 



If, now, curvature be introduced in the course of the channel, centrifugal force is developed. 

 This centrifugal force is measui'ed by the square of the velocity, and is therefore much greater for 

 swift central threads of the current than for slow lateral threads. As pointed out by Thomson* 

 and others, the central threads, tending more strongly toward the outer bank, displace the slower 

 threads of that bank, and the symmetry of distribution of velocities is thus destroj'ed. In other 

 words, the centrifugal force develojied by curvature exercises a selective influence on velocities, 

 and transfers the locus of maximum velocity from the center of the channel toward the outer bank. 

 The conditions of symmetry in the profile of the cross-section are thus destroyed: the outer bank 

 is eroded; a deposit is accumulated on the inner bank. Moreover, there is no compensating tend- 

 ency to restore an equilibrium, for the erosion of the outer bank increases the sinuosity of the 

 channel instead of rectifying it. , 



Curvature of course thus causes a stream to shift its channel laterally, and in this manner 

 enlai'ge its valley. It is the most important condition of lateral corrasion. 



As shown by Ferrel, the deflective force due to terrestrial rotation varies directly with the 

 velocity of the stream. Therefore, it likewise has a selective influence on the velocities within the 

 cross-section of the channel ; and it likewise tends to produce erosion at one side and deposition at 

 the other.t For given amounts of deflective force its selective power is not the same as that 

 of the centrifugal force developed by curvature of course, for centrifugal force varies with the sec- 

 ond power of the velocity, while the rotational deflective force varies only with the first power; 

 but its selective power is of the same kind, and may be quantitatively compared. For the purpose 

 of this comparison I will develop an equation: 



Let F = deflective force, per unit of mass, due to rotfition. 

 n = angular velocity of the earth's rotation. 

 V = velocity of stream. 

 A = latitude of the locality. 

 P = radius of curvature of the stream's course. 

 /= the centrifugal force, per unit of mass, developed by such curvature. 



Then 



/="^ (1) 



and, from Ferrel, | 



F = 2to sin A (2) 



Let V, = velocity of a rapid-flowing thread of the current, and 

 0, = velocity of a slow-flowing thread of the current. 

 Eepresent by F,, i^„/, and/", the corresponding deflective forces due to rotation and curvature, 

 then 



F,. -F, ^ (*', - i\) X 2 n sin A - . . (3) 



and 



f:-f,='-l^^l w 



* Trans. Brit. Ass., 1876, Sections, p. 31. 



t This proposition, which it is the prime object of the present paper to set forth aud develop, was believed, at 

 the time it was read, to be novel, but proves to have been anticipated by more tlian six years. In October, 1877, Mr. 

 A. C. Baines read before the Pliilosophical lustitiite of Canterbury, New Zealand, a paper "On the influence of the 

 earth's rotation on rivers," in which he arrived, by a very ditt'erent route, at essentially the same conclusion. See 

 Trans. N. Zeal. Inst., X, pp. 9-i-iM. 



t Ferrel's equation is given on page 2'.), volume 31 (second series), Am. .Jour. Sci. Instead of the sine of the lati- 

 tude, here substituted, it includes the cosiue oC the polar distauce, which is, of course, equivalent. 



