HARMONIC ANALYSIS AND PREDICTION OF TIDES 41 
THE M: TIDE 
121. The separation of constituents from each other by the process 
of the analysis depends upon the differences in their speeds. Constit- 
uents with nearly equal speeds are not readily separated unless the 
analysis covers a very long series of observations but they tend to 
merge and form a single composite constituent. In formula (63), 
terms A;,and A;; have nearly equal speeds, one being a little less and the 
other a little greater than one-half the speed of the principal lunar 
constituent M,. These two terms are usually considered as a single 
constituent and represented by the symbol M;. Neglecting for the 
present the general coefficient and common latitude factor, the two 
terms may be written as follows: 
term A;;=1/2 e sin I cos’ 3J cos (T—s+h—p—90°-+ 2—») (188) 
term A.3;=3/2 e sin I cos I cos (T—s+h+p—90°—y) (189) 
The latter term, having a coefficient nearly three times as great as 
that of the first term, will predominate and determine the speed and 
period of the composite tide while the first term introduces certain 
inequalities in the coefficient and argument. 
122. For brevity, let A and B represent the respective coefficients 
of terms Aj, and A; and let 
6=T—s+h+p—90°—»p (190) 
Also let P equal the mean longitude of the lunar perigee reckoned from 
the lunar intersection. Then 
Bee (191) 
We then have 
term A,,=A cos (@—2P) 
=A cos 2P cos 6+A sin 2P sin @ (192) 
term A.3;=B cos 6 (193) 
M,=A,,+ A.3,= (A cos 2P+B) cos 0+ A sin 2P sin 6 
A sin 2P 
_( A2 2)4 Reo) Aen ee i 
Geis a as 2P+B’)? cos E tan ne SDE==B 
a Ee Ecos (T—s+h+p—90°—v—Q,) (194) 
in which P 
cos cos’ i 
1/Q.=| 1/4+3/2 £25 00s 2P+9/4 (195) 
Q,=tan! ues (196) 
3 cos I/cos? 4/+cos 2P 
If J is given its mean value corresponding to w, formula (195) may be 
reduced to the form 
1/Q.= (2.310-+1.435 cos 2P)} (197) 
Values of log Q, for each degree of P based upon formula (197) are 
given in table 9. 
