78 MANUAL OF CURRENT OBSERVATIONS 
directly for values of R which do not exceed unity. If H, is greater than H7,, substitute 
the reciprocal of R for the upper argument of the table and interchange the values of 
k, and K,, in obtaining the difference for argument D. The tabular values for time will 
then be expressed in terms of the east component phase (7—K,). The corresponding 
tabular values for direction are reckoned counter-clockwise from the east and must be 
subtracted from 90° in order to obtain the correct azimuth as reckoned from the north. 
In Table 9 the reciprocal of & may be substituted for R when H, is greater than H,. 
All tabular values in this table are positive and express the ratios of the maximum and 
minimum resultant velocities to the larger one of the two component amplitudes. 
187. Reference of harmonic constants to any axis.—In a locality where the current, 
although rotary, has a predominating movement along a particular axis, it may be 
desirable to refer the harmonic constants to that axis im order that they may be used 
for prediction purposes. Such an axis would in general be the major axis of the current 
ellipse for the M, constituent, the direction of which may be determined by the method 
described in the preceding paragraphs. Assuming that the harmonic constants for the 
north and east components of the several constituents are available they may be re- 
ferred to any axis desired by means of the formulas derived in accordance with the 
following explanation. 
188. Use the same symbols as before to represent the north and east component 
velocities of any constituent, and let H,, K,, and V, represent respectively the ampli- 
tude, epoch, and velocity of the constituent along an axis with azimuth A. Then 
resolving the north and east components, as represented by formulas (1) and (2), alone 
this axis and taking the sum, we have 
V.=H, cos (T—K,) cos A+H, cos (T—K,) sin A 
= [, cos A cos Kk, +H, sin A cos K,) cos T 
+ (1H, cos A sin K,+H, sin A sin K,) sin T 
ETS LCOS (IRE a VEG) Ra OAN se met ye 1 PR iP asp We We ey a ae (11) 
in which 
H,=|H,? cos? A+H? sin? A+2H,H, cos A sin A cos (K,—K,,)|*_-----__- (12) 
H, cos A sin K,-+H, sin A sin K, 
an lten n @ e ‘ 
EG /&L, GOS Al COS IK AMIEI, Sit, Al COS I (is) 
The quadrant for K, is determined by the signs of the numerator and denominator of 
the above fraction, these being the same respectively as for the sine and cosine of the 
angle. 
189. By means of formulas (12) and (13) the amplitude and epoch for each con- 
stituent as referred to the new axis may be computed. The formulas may be solved 
graphically (fig. 33) by drawing from a point C a line CD to represent in length and 
direction (H, cos A) and (K,), respectively; from the point D a line DE to represent 
in length and direction (H, sin A) and (K,), respectively. Then the connecting line 
from C to # will represent by its length and direction the amplitude (H/,) and the epoch 
(K,), respectively. 
190.. Hydraulic Current.—The term “hydraulic current” is applied to a current in 
a strait that is caused by a difference in the head of water at the two entrances. When 
this difference in head results from tidal action that causes the water at one end to be 
alternately higher and lower than at the other end, the movement is periodic and may 
be treated as a reversing type of tidal current. The currents through the East River 
which connects New York Harbor with Long Island Sound afford an example of hy- 
draulic currents. When there is tidal action at each entrance to a strait, difference in 
