§ 143. Determination of the Coefficients of the Current Meter.—Suppose a series of observations has been made by towing the meter in still water at different speeds, and that it is required to ascertain from these the constants of the meter. If v is the velocity of the water and n the observed number of rotations per second, let
v = α + βn
(1)
where α and β are constants. Now let the meter be towed over a measured distance L, and let N be the revolutions of the meter and t the time of transit. Then the speed of the meter relatively to the water is L/t = v feet per second, and the number of revolutions per second is N/t = n. Suppose m observations have been made in this way, furnishing corresponding values of v and n, the speed in each trial being as uniform as possible,
| Σn = | n1 + n2 + ... |
| Σv = | v1 + v2 + ... |
| Σnv = | n1v1 + n2v2 + ... |
| Σn2 = | n12 + n22 + ... |
| [Σn]2 = | [n1 + n2 + ...]2 |
Then for the determination of the constants α and β in (1), by the method of least squares—
| α = | Σn2Σv − ΣnΣnv | , |
| mΣn2 − [Σn]2 |
| β = | mΣnv − ΣvΣn | . |
| mΣn2 − [Σn]2 |
| Fig. 144. |