74 MANUAL OF CURRENT OBSERVATIONS 
180. For obtaining the times or values of 7 corresponding to the maximum and 
minimum velocities of the current constituent formulas may be derived as follows. 
Referring to formulas (1) and (2) for the component velocities, the square of the re- 
sultant velocity V for any 7 may be expressed by the formula 
V?=V?2+ V2=E? cos? (T—K,) +H? cos? (T—K,) 
Considering the resultant velocity as positive regardless of direction, its maximum 
and minimum values will be attained under the same conditions that will render V2 
a maximum.or minimum. For the first derivative of V? with respect to 7’ we have 
d(V?) /dT=—2E? cos (T—K,) sin (T—K,) 
—2H? cos (T—K,) sin (T—K,) 
=— H? sin 2(T—K,) —H? sin 2(T—K,) 
=— (FH? cos 2K,+H? cos 2K,) sin 2T 
+ (H? sin 2K,-+ H? sin 2K,) cos 27_________________- (4) 
Equating the above derivative to zero for maximum and minimum velocities, we have 
FH? sin 2K,4 H? sin 2K, 
TEZCOS 2 KOA ETN CORRS Kas Se et (5) 
Tan 2T= 
From tan 2T determined by the above formula, four values for T differing from each 
other by 90° are possible, two of these values being for maximum velocities and the 
other two for minimum yelocities. Two of the values, one for a maximum and the 
other for a minimum velocity, are obtained when 27 is taken less than 360° and the 
other two when 27 is taken between 360° and 720°. 
181. The distinction between the values of 7 corresponding to the maximum and 
minimum velocities may be later determined when the corresponding velocities have 
been actually computed, or may be made immediately by reference to the second 
derivative of V? with respect to T. From the first derivative (4) we obtain 
d?(V?) /(dT)?=—2(EH? cos 2K,,+ H? cos 2K.) cos 2T 
—2(H? sin 2K,,-+H? sin 2K,) sin 27 (6) 
Values of 7 which render the above derivative negative will correspond to the maximum 
velocities and those which render it positive to the minimum velocities. From (6) 
it is obvious that if the coefficients of cos 27 and sin 27 each have the same sign as the 
function itself, the second derivative will be negative, but if each coefficient has the 
opposite sign the derivative will be positive. As these coefficients are the same as 
the terms in the fractional expression for tan 27 in formula (5), it follows that for a 
maximum velocity the signs of the sine and cosine of the angle will be the same respec- 
tively as the signs of the numerator and denominator of the fraction represented 
and for a minimum velocity both signs will be reversed. 
182. Formula (5) may be solved graphically as follows: From any poimt A (fig. 32) 
draw line AB to represent in length and direction H; and 2K, respectively; from point 
B draw BC to represent in length and direction H? and 2K, respectively. The con- 
necting line from A to C will indicate by its direction the value of 27 corresponding to 
a maximum velocity. The reverse direction from C to A will indicate the value of 27 
corresponding to a minimum velocity. 
183. A formula for the computation of the values of YT corresponding to maxi- 
mum and minimum velocities which is expressed in terms of the ratio of the amplitudes 
and the difference between the epochs of the north and east components of the current 
