“266 Scientific Proceedings, Royal Dublin Society. 
If we assume that the mean velocity of the discharge at any 
point is proportional to the depth at that point, we may calculate 
the total water-discharge of the river as follows :— 
Let y denote the depth of the river, and « the corresponding 
distance from the bank of the river; then we shall have, to ex- 
press the curve of the river-bed, 
. y=$(x) 5 
also, we have 
vky ; 
where i is the coefficient expressed in the last column of the 
preceding Tables: 
The cross section of any elementary slice of the river is ydz, 
and the corresponding discharge is 
vx ydx, 
or 
ky*dex. 
Hence, 
! 
Q=k f yx ydx (1) 
12 
where Q is the total water-discharge ; or, finally 
3 Q=2kxy, XA; (2) 
where y, is the depth of the centre of gravity of the cross section 
of the river, and A is the area of the cross section. Both these 
quantities may be found, without calculation, by experiments 
made upon a zinc templet, drawn to scale, representing the cross: 
section of the river. 
It is well known that 4d x y, is the volume of water, whose 
weight denotes the hydrostatic pressure upon the river section, 
regarded as a boundary wall; hence we have the proposition— 
The quantity of water discharged by a river in a given time is 
proportional to the hydrostatic pressure on the river section, 
multiplied by a coefficient which varies with the river basin. 
We may now apply the foregoing to calculate the discharge ce 
_the Parana at Rosario, on the o5th January, 1871. 
The cross section of the river may be divided into six parts, 
according to the varying slope of the bottom, as follows :— 
