Report of Messrs. Humphreys and Abbot. 209 
n the endeavor to ascertain the law of change, the slopes 
ome first plotted as abscissz, to the a ee as ordinates ; 
and straight lines were drawn connecting the points representing 
the top and bottom of each rise. These lines were not parallel, 
showing that the rate of increase of slope varies for different 
rises. In the further study of their relations, it was discovered 
that the difference of slope divided by the rise is the abscissa of 
a curve sensibly parabolic, in which the gauge-reading at the 
top of the rise, measured from low water mark, is the corres- 
ponding ordinate. Or, if x denote the rise, ¢ the primitive gauge- 
reading, and e+a the gauge-reading at flood; also, if s, and s,, 
represent the primitive slope and the slope at flood, then the 
following equation will be true :— 
Sy 
ae 1 
f0— 8) (e-pe)?, or 8 — §= 5P(e-+-a)?2. 
x 
The value of = is to be determined by dividing s,,—s, (of 
both which aif ‘the values are deduced, as just stated, by the 
formula, after the observations have determined We cross-section, 
discharge, perimeter, and rise of the river) by (e+2)?x. For the 
same locality it is found to be constant; but it is different at 
different points in the length of the riv ver, 
If now we put a, Q, p, W,v, for the cross-section, discharge, 
perimeter, width, and mean velocity of the river in the primitive 
stage, and a,, QP, W ,,and v,, for the same quantities after the 
tise; and if, in estimating the increased perimeter of the river 
occasioned by the rise, we neglect, as we may safely do for a 
large stream, the inclination of the banks, the new perimeter 
aga equal to the primitive perimeter increased by 2x, and we 
ay a4+W 
PutWy aoe W +22 
Also, as — denominators are equal and numerators also, 
we shall hay 
<i awiss or 4@,,v,=Q,,=4,v,+W,»,,2 
Qu ore 
wis a W, 2, 
Now, if the quantities a, Q,, py» W, % oe z, which is ‘sd 
function) be given, and it be required to know how muc 
river will rise if Q, be made Q. "he oh fei may be solved, 
and h ne, pao 5Y eee by an easy process of trial and 
error. ber equato ted from the formula 
as. eee 2\7 2 
Am. Jour. Sc1.—Szconp SERIEs, ate ane No. 107.—Szpt., 1863, 
27 
