HARMONIC ANALYSIS AND PREDICTION OF TIDES 103 
296. Substituting in (432), we have 
aM, sin (2n7—60)+2aM, sin (4n7—2a+ B—26) = 
—aM, sin 6—2aM, sin (26+2a— )=0 (435) 
But 
2a— p=—2M.°+Me (436) 
From (427) 
= 2M MM — =: 00" 
180° 
according to whether the duration of rise is greater or less than ’ 
or whether 6 is negative or positive. 
Then 
2a— p= +90° (437) 
according to whether 6 is positive or negative. 
Substituting this in (435) 
—aM, sin 64+2aM, cos 26=0 (438) 
and 
M, | 1 sin 6 
M, = 205 20 (439) 
the upper or lower sign being used according to whether 6 is positive 
or negative. As under the assumed conditions 6 must come within 
the limits +45°, the ratio of “¢ as derived from (439) will always be 
2 
positive. 
ao The duration of rise of tide due solely to the constituent Mg is 
18 
a 
The duration of rise as modified by the presence of the assumed 14, 
is 
pA 28 (440) 
a a 
Therefore 
6=4(180°—aDR) (441) 
Substituting the above in (439) we have 
M,__ , :sin (90°—3aDR) __ — ,cos 3aDR (442) 
~M, ~?cos (180°—aDR) — ? cosaDR ¢ 
and ue 
__ 00s 3aDE 
Mi T3 cos aDR M: >) 
M, must be positive, and the sign of the above coefficient will depend 
upon whether aDA is less or greater than 180°. 
298. The approximate constants for S:, Ne, Ky, and O, may be 
obtained from the observed high and low waters as follows: Add to 
each low-water height the mean range of tide. Copy the high and 
modified low water heights into the form for hourly heights (form 362), 
always putting the values upon the nearest solar hour. Sum for the 
desired constituents, using the same stencils as are used for the regular 
