EffeBs of Eddies y ^c. on the Motion of Riven. 45)3. 



the feftion M B. The current which is formed near the bottom at B, is turned out of its 

 courfe by the flope B C, and proceeds to rife above the furface at Qj fometimes in the form 

 of a curling wave or vertical v/hirlpool. If the extremities of the hollow place form an 

 abrupt angle, as D E, F G, eddies will be produced even at the bottom, in the vertical direftion 

 at D, and fometimes alfo at G. Thefe phenomena may be obferved in an artificial channel 

 with glafs fides. 



Every eddy de/iroys apart of the moving force of the current of the river. For the water which 

 defcends by a retrograde motion, in the inclined plane C D E, fig. 19. cannot be reftored in 

 the direftion of the current of the river but by a new impulfe. It is as it were a ball, which 

 is forced to rife on an inclined plane, whence it continually falls back again to receive new 

 impulfions. It is the labour of Sifyphus. 



Hence I deduce, as a primary confequence, that in a river, of which the courfe is permanent, and 

 the feclions of its bed unequal, the water continues more elevated than it would have done, if the ivhole 

 river had been equally contrafled to the dimenfwis of its fmallefl feElion. The caufe of this pheno- 

 menon is the fame as that which retards the expenditure through the tube with enlarged parts. 

 (Prop. VII. No. 4.) The water which defcends from the elevation above the contra£ted part 

 N, into the bafon P Q^ fig. 19. lofes nearly the whole of the velocity it acquired by defcending 

 from it ; -becaufe the narrow part has a curved flope towards the lower part of the river, which 

 direfts the velocity of the ftream in an horizontal dire£tion. Guilielminihas well remarked, that 

 a fall does not influence the velocity of the lower ftream, becaufe the eddies of the water in the 

 bafon P Q^deftroy the velocity produced by the fall. This velocity increafes the depth, and 

 enlarges the width of the channel at P Q^ Eddies are formed on each fide, at the bottom, 

 and at the furface ; both in the horizontal and vertical direftions. It would be to no purpofc 

 to atttempt to prevent this hollowing out and enlargement of the channel by fuch a fall, by 

 adopting the means of clofe walls ; for the bafon would then obtain its enlargement, where 

 thefe conftruftions might end. 



If the channel have a number of fuccefllve contracflions and dilations, M N, without 

 cafcade or dam, there will ftill be formed, at each dilatation, eddies which will diminifli the 

 \'elocity more than if the channel had an uniform fettion equal to that in M or N. It will, 

 tiierefore, follow, that the furface of the water, after each dilatation, muft rife, in order to 

 recover the velocity it loft by the eddies. If we call the height to which the water muft rife, 

 above the elevation necelTary to have overcome the retardations of a bed of uniform fc£llon, 

 = a, and that the number of equal and fucceflive alternate dilatations and contraftions bte 

 = VI, the height of the rife in the ftream thus alternately dilated beyond that of the fame 

 river unifonnly contra£led will be rr am. I here fuppofe the bottom of the river to be 

 uniform. If this bottom be of fuch a nature to be attacked by the current, the contrafted 

 parts will be hollowed out, and the matter will be dcpofited in the enlarged parts. 



The fecond confequence which I draw from the principle here eftablifhed, refpe£ling the 

 Ibfs of force, caufed by the eddies, is of confiderable importance in the themy of rivers, and 

 appears to have been neglefted by thofe who have treated on this fubjeft. The friftion of the 

 water along the wet banks, and over the bottom of rivers, is very far from being the only 

 caufe of the retardation of their courfe, which, confequently, requires a continued defcent to 



Vol. II. — Feb. 1799. 3 S maintain 



