18 U. S. COAST AND GEODETIC SURVEY 
Fray [g=3/2 U (e/d)® (1/2—3/2 sin? ¥)(2/3—2 sin? D) —86) 
F 3, /g=3/2 U (c/d)? sin 2Y sin 2D cos t (37) 
Fy. /g=3/2 U (c/d)? cos? Y cos? D cos 2t (38) 
57. Referring to figure 3, the following relations may be obtained 
from the right spherical triangles 1/7’ and MP’M’ and the oblique 
spherical triangle MP’T: 
sin D=sin IJ sin j (39) 
cos D cos t=cos MP’ (40) 
cos MP’=cos X cosj7+sin X sin j cos I (41) 
cos D cos t=cos X cosj7+sin X sin j cos I 
=cos’ 4] cos (X—7)+sin? $7 cos (X¥+7) (42) 
58. Replacing the functions of D and ¢ in formulas (36) to (88) by 
their equivalents derived from equations (39) and (42), there are 
obtained the following: 
Py /g=3/2 U(c/d)?(1/2—3/2 sin’ Y) X 
[2/3—sin? [-+sin? I cos 24] (43) 
Te gs) 2Ui(c/d) simi2¥ x 
[sin I cos? 4 cos (X¥+90°—27) 
+1/2 sin 27 cos (X—90°) 
+sin J sin? 4] cos (X¥—90°+27)] (44) 
ei gG—ol2) Ueld)*eos? YX 
[cos* 41 cos (2X—27) 
+1/2 sin? I cos 2X 
+sin* 4J cos (2X+2))] (45) 
The above formulas involve the moon’s actual distance d and its 
true longitude 7 as measured in its orbit from the intersection. While 
these are functions of time, they do not vary uniformly because of 
certain inequalities in the motion of the moon, and it is now desired 
to replace these quantities by elements that do change uniformly. 
59. Referring to paragraphs 23-24 and to figure 1, it will be noted 
that longitude measured from intersection A in the moon’s orbit 
equals the longitude measured from the referred equinox 7’ less are 
£, and longitude measured from intersection A in the celestial equator 
equals the longitude measured from the equinox 7 less are v. 
Now let 
s’=true longitude of moon in orbit referred to equinox 
s =mean longitude of moon referred to equinox 
k =difference (s’—s) 
Then 
j=s'—t=s—E+k (46) 
60. In figure 4 let S’ and P’ be the points where the hour circles of 
the mean sun and place of observation intersect the celestial equator, 
Y the vernal equinox, and J the lunar intersection. Then X will 
equal the arc P’I and »y the arc IY. Now let 
h=mean longitude of sun 
T=hour angle of mean sun 
X=T+h—p (47) 
Then 
