80 MANUAL OF CURRENT OBSERVATIONS 
difference in head by the subscript d. Since the usual constituent epochs known as 
“Ikappas”’ refer to the local meridian, it will be necessary for the purpose of comparison 
between places on different meridians to use the Greenwich epochs, which will be desig-— 
nated by the capital letter ““G’’. These should be distinguished from the epochs desig- 
nated by the British Admiralty by the small letter ‘‘g’’, which involve the time meridian 
and correspond to the modified kappas («’) of this office. The relation existing between 
these epochs may be expressed by the following formula: 
Cie UNS) NT) Ga cay) Ope be aes ie Oat aNey ere engl ee Ie (14) 
in which LZ and § are, respectively, local longitude and the time meridian of the place 
(positive for west and negative for east), and a and p represent, respectively, the speed 
and subscript of the constituent. 
193. For any one constituent let 7 represent time as expressed in degrees of the 
constituent reckoned from the phase zero of its Greenwich equilibrium argument. 
Also let Y, and Y, represent the height of the constituent tide for any time TJ as referred 
to the mean level at locations A and B, respectively; and let Y, equal the difference 
(Y,.—Y>). Formulas for heights and difference may now be written 
We =Jeln COS (I—E,) itor llooauiom OAM ee cece (15) 
WyeHle Cos (I—E)) stor Jlooatiom reece ects (16) 
Y,=H, cos (41 —G,) —H, cos (T—G) 
=(H, cos G,—H, cos G,) cos T+ (AH, sin G,—H, sin G,) sin T 
Se Pes COSI (Lal) hes ye SE Ae (17) 
in which 
[Eh =s Risk el Jal, Gos (Ch—(Cesooc ase oocasee mele eecnocaeesoe (18) 
EA SUN Cag lp SI 
EEN C0S)( GE EL ACOSUG Rae Sa oh ae ory ea a 
G,=tan 
The proper quadrant for G, 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. Formulas (18) and (19) may be solved graphically (fig. 34) by drawing 
from any point C’a line CD to represent in length and direction H, and G, respectively; 
from the point D a line DE to represent in length and direction H, and (G,+180°), 
respectively. The connecting line from C to EH will represent by its length the amplitude 
Hf, and by its direction the epoch Gy. 
194. By the formulas given above separate computations are made for each of 
the tidal constituents. The values obtained for H; and G, are the corresponding ampli- 
tudes and Greenwich epochs in an harmonic expression for the continually changing 
difference in elevation of the water surface at the two entrances to the strait. When 
only a single time zone is involved, the small g’s or modified kappas («’) pertaiming to 
that zone may be substituted for the Greenwich epochs (@) in the formulas. For the 
- prediction of the current further modifications are necessary in the amplitudes to reduce 
to velocity units and in the epochs to allow for the lag in the response of the current to 
the changing difference in water level at the two entrances to the strait. 
195. Since the velocity of an hydraulic current is theoretically proportional to 
the square root of the difference in head, we may write 
(Velocity)?=constant (C) Xheight difference_______________-_______._ (20) 
