THE LAW OF THE FALL OF RIVERS AND THE VALUE 
OF THE DEDUCED CURVES IN RIVER 
IMPROVEMENTS. 
F. Oppikofer, C. E. 
The idea that the formation of a river bed is largely influenced by 
the inclination of the river to follow a certain curve is not new. 
The cycloid, the curve of quickest descent, appears to be the lawful 
basis of all river formations. In other words, any river will in the 
course of time — geological time, in some instances — from the source 
to the mouth, if undisturbed, form a cycloid. True, no river profile 
investigated by me had formed this curve to perfection as, in some 
cases, sufficient time had not passed for the waters to wear off rock 
formations or the stream emptied into the main river or sea before 
coming to the zero point of its curve. 
Small falls and shoals, as occurring in the Trinity and Brazos, con- 
stitute in the total length but insignificant deviations; in fact, the sur- 
face of any high water in these streams forms the cycloid to perfection. 
While probably few rivers present an unbroken or unfinished curve, the 
fact that time and nature will accomplish this, should be the guide in 
all permanent river improvements. 
To demonstrate the practical value of this discovery, the Mississippi 
may be used as an example. By computing the cycloid of this river 
from source to gulf and placing this curve over the plotted existing 
profile, any deviation would show at a glance where nature intends to 
fill or scour; also how near this river has arrived at the zero point of its 
curve, or how much farther the delta will form gulfwards to reach the 
zero of the cycloid. 
The effect of cut-offs could be studied. A cut-off in the middle would, 
while deepening the upper reaches, develop a shoaling in the lower 
parts. The data for researches has been meager, but studies of profiles 
of some rivers in Switzerland and this country tend to prove the 
existence of this law. 
The computed curve of surface of high water of the Mississippi River 
from Cairo, Illinois, to Fort Jackson, a distance of 1050 miles, with a 
fall of 315 feet, varies only a few feet from the recorded high water by 
the United States engineers, and the zero of the curve comes about 250 
miles gulfwards, the ultimate mouth of the river. 
