: 
2 
: 
‘ 
: 
j 
_ y serves to reduce the observed depths of axis to the position of 
2 
_ Report of Messrs. Humphreys and Abbot. 31 
those when it either did not blow at all or blew directly across 
the stream. The wind-forces were estimated according to the 
usual scale of notation, 0 being a calm and 10 a hurricane. For 
the first two classes of observations; the sum of the products of 
the numbers of observations at each point by the force of the 
wind was made out for each, and the difference between the two 
sums, divided by the total number of observations at all the 
oints, was presumed to give the effective force of the wind: 
Five sets of determinations were thus obtained, in each of which 
the number of observations, the depth of the axis, the mean 
velocity of the river, and the resultant force and direction of the 
wind were given. 
If « be the unknown depth of the axis due to a calm, and 
d, d,, d,,, &c., the observed depths, and if y denote the amount of 
movement of elevation or depression of the axis which would 
among the data be denoted by 7 7, 7,, &c., then the movements 
‘A the axis will be /y, fy, 7, y, &e. Accordingly, we shall 
ave 
ttfy=d, t+f,y=d,, c4+-f,y=d,, &e., 
the negative sign being used when the wind is down-stream, and 
the positive, when it is up. Multiplying both members of each 
of these equations by the number of observations from which 
its constants were deduced, and adding the whole, member for 
member, we obtain one equation containing both a and y. 
- But if we consider that the sum of all the movements of the 
axis, taken without regard to sign, must be equal to the sum 
of all the differences between x and d, w and d, &c., taken 
positively—that is, taking 2-—-d when x is the greater, and d—z 
when dis the greater, we shall, by multiplying once more all 
these differences and the corresponding movements (fy, fy, &e.) 
by the numbers of observations to which they respectively be- 
long, and adding the products as before, obtain a second equa- 
tion containing both x and y. From these equations combined, 
both « and y are determined. The deduced value of « is 817 
which is the depth of the axis when the wind force is zero, in 
decimals of the total depth of the river. The deduced value of 
calm. A comparison of the values of x so obtained (which should 
_ all agree with each other, and with the value of « given by the 
