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Report of Messrs. Humphreys and Abbot. 27 
city. The next object was to determine the law of change. The 
— which presented themselves were the high water mean, the 
water mean, and the grand mean curves; to which another 
purpose. e idea was then conceived of sont the same 
investigation in regard to the velocities in horizontal planes; or 
at the surface of the river. Data for this i inquiry had been am- 
ns furnished by the observations for daily discharge. These 
observations, which were made at a depth of five feet below the 
surface, were grouped according to the even feet of approximate 
mean velocity of the river; and thus were obtained material 
for eight mean curves, corresponding to as many different mean 
velocities. From these was deduced a grand mean curye, as 
in the case of subsurface velocities. The result was a very clear 
disclosure of the parabolic law 
In proceeding to the study of the law of variation of curva- 
ture, equations were deduced for each of the eight mean curves, 
The reciprocals of the parameters of these parabolas were plotted 
as ordinates, the corresponding mean velocities of the river being 
the abscissie—the reciprocal of the parameter of the limiting 
parabola, or straight line, which is zero, indicating that the curve 
intersects the axis of abscissx at the origin of codrdinates. A 
eurve resulted which cited closely to the sect om and 
thus furnished a general expression for the reciprocal of the 
parameter of any parabola of surface velocities corresponding to 
any given mean velocity of the river. This r 
by applying it to the formation of a general quail for the 
curve of velocities five feet below the surface; and empl 
is equation to SS the velocities potas wd the 
— mean curves which as compone 
the grand mean. The differences were all sana, seh 
pecs than those in the grand mean subsurface curve; a conse- 
Fee probably of the fact that each of these eight curves was 
upon a much more limited series of observations than 
that. A law of parameters having been thus deduced for the 
horizontal curves, the —— naturally suggested itself that 
a similar law governs those of the vertical curves also, Mate- 
Tials which are insufficient to reveal the exi istence of an unknown 
mean nena of the pao mean curve were the 
pair best determined, next to those of the straight line, which 
