42 U. S. COAST AND GEODETIC SURVEY 
123. The period of the composite constituent M, is very nearly an 
exact multiple of the period of the principal lunar constituent Mz, 
and for this reason the summations which are necessary for the 
analysis of the latter may be conveniently adapted to the analysis of 
the former. With other symbols as before, let 
Qe Ts hone tee (198) 
Terms A,, and A.; may then be combined as follows: 
term A,,—A cos (6—P) 
=A cos P cos 6+A sin P sin 6 (199). 
term A .,;—B cos (6+ P) 
=B cos P cos 6—B sin P sin 0 (200) 
M.=Ay.+:A.3= (A+B) cos P cos 6+ (A—B) sin P sin 6 
= (A?+2AB cos 2P+B’)* cos | o—tan-"( 5 tan P)| 
: 2 
a2 Sn OSE cos(T—s-+h—00° +89 + Q) (201) 
in which 
—tan-( 2 cos L—1 
Q=tan 7 cos Tani *20 P) (202) 
If J is given its mean value corresponding to w, formula (202) may 
be reduced to the following form which was used for computing the 
values of Q in table 10. 
tan Q=0.483 tan P (203) 
124. Formulas (194) and (201) are the same except in the method 
of representing the argument. The elements +p —Q, in the first 
formula are replaced by +£+Q in the latter, but it may be shown 
from (196) and (202) that 
Q,+Q=P=p-—t (204) 
p—Q=E+Q (205) 
The complete arguments are therefore equal but in formula (201) 
the uniformly varying element p has been transferred from the V 
of the argument and included in the value of Q where it is treated 
as a constant for a series of observations being analyzed. The speed 
of the argument as determined by the remaining part of the V is 
then exactly one-half that of the principal constituent M, and with 
this assumption the summations for the latter may be adapted to the 
analysis of the former. It is to be noted, however, that the wu in 
this case has a progressive forward change of nearly 41° each year. 
The true average speed of this constituent is determined by the V 
of formula (194) which includes the element p. 
125. The obliquity factor for the composite M, constituent may be 
expressed by the formula sin J cos*4JX<1/Q,. According to the work of 
Darwin (Scientific Papers by Sir George H. Darwin, vol. 1, p. 39) the 
