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XXXIV. Stability of Motion {continued from the May, June, 

 and August Numbers). — Broad River flowing down an 

 Inclined Plane Bed. By Sir William Thomson, F.R.S.* 



41. /CONSIDER now the second of the two cases referred 

 V7 to in § 27 — that is to say, the case of water on an 

 inclined plane bottom, under a fixed parallel plane cover 

 (ice, for example), both planes infinite in all directions and 

 gravity everywhere uniform. We shall include, as a sub- 

 case, the icy cover moving with the water in contact with 

 it, which is particularly interesting, because, as it annuls 

 tangential force at the upper surface, it is, for the steady 

 motion, the same case as that of a broad open river flowing 

 uniformly over a perfectly smooth inclined plane bed. It is 

 not the same, except when the motion is steadily laminar, 

 the difference being that the surface is kept rigorously plane, 

 but not free from tangential force, by a rigid cover, while the 

 open surface is kept almost but not quite rigorously plane by 

 gravity, and rigorously free from tangential force. But, pro- 

 vided the bottom is smooth, the smallness of the dimples and 

 little round hollows which we see on the surface, produced 

 by turbulence (when the motion is turbulent), seems to prove 

 that the motion must be very nearly the same as it would be 

 if the upper surface were kept rigorously plane, and free from 

 tangential force. 



42. The suh-case described in § 31 having been disposed of 

 in §§ 32-40, we now take the including case, described in the 

 first half- sentence of § 31 ; for which we have, as steady 

 solution, according to (5), 



U = /3y-i C / (57), 



if we reckon y from the bottom upwards. Thus (7), (8), (9), 

 (11), (12) become 



J + ^-io^g + ^-^^^.-J . (58), 



%+&-**% =^-| • m, 



§^-W) d £ =^-g • (60), 



2(/3-«/)£= -VV • (61), 



-ar+^s+^^^ar"* 7 ^ ■ ■ ■ (62) - 



* Communicated by the Author. 



