46 REPORT—1885. 
Thus we may put 
cost‘}I __cos?A sinIcos*}I.. _ sin2A 
= a ’ . == - 
cos‘ 4w cost}: cos” A, sinwcos?4w costsi sin2A, (9) 
sin? I __sin?A a 
sat =, tan? 47=1tan?A 
sin?(1—3sin?i) sin?A/ ere 
An approximate formula for A and the value of A, are 
A=16°51+43°-44cos N—0°'19 cos 2N, A,=16°86 . . (10) 
The introduction of A and A, in place of I and w entails a loss of 
accuracy, and it is only here made because former writers have followed 
that plan. It may easily be dispensed with. 
Now iet us write 
D=cos2(s—8&), D'=sin2(s—£) 
|, ne 
II =cos(s—p), II’=sin (s—p) 
From (7) and (8), 
cos? 6 —cos? A sinccosé dé 
= — = ) | i EA a pas 9 
= sin?A : " osin2A dt °° “aS 
Then, if we write for the ratio of the moon’s parallax to her mean paral- 
lax P, we have 
P—1=ccos(s—p), 
and 
1 1 dP 
II =—— P=] Tl’= ee . . . . 
es gaa) a (13) 
Hence D, D’, 11, 11’ are functions of declinations and parallaxes. The 
similar symbo!s with subscript accents are to apply to the sun. 
Now (6) may be written by aid of (9) and (11), 
Qt+h—s—(v.—é)]=20+4+4ell’—Ditan®A . . . (14) 
The left-hand side of (14) is the argument of M, (see Sched. B. i. 
1883), and from (9) the factor of M, is cos’A/cos*A,. Hence, subtracting 
the retardation 24 from (14) we have 
2A 
M,)= _ Moos [ (20+ 4ell’ — D'tan?A) — 2p 
(at,) = 25> Meos [( )— 24] 
expanding approximately, : 
(M,) = © Mf cos2(W—p) 
cos*A, 
cos?A . 
ee ay = 
Pay 4 Me sin2(l—p) 
2. sin2A D! Msin2(wW—,) +. Tlo-8 Saas (15) 
cos*A, 
We shall see later that the two latter terms of (15) are nearly 
annulled by terms arising from other tides, and as in the case of the sun 
the rates of change of parallax and declination are small, we may write 
by symmetry, 
: (8) 8 cos2(d,2) 6°. ae 
