56 REPORT—1883. 
Thus when the functions ®, ¥, R are developed as a series of time- 
harmonics, the further development of the X-Y-Z functions consists in 
substitution in (12). 
It will now be supposed that the moon moves in an elliptic orbit, 
undisturbed by the sun. The tides which arise from the lunar inequali- 
ties of the Evection and Variation will be the subject of separate treatment 
below. 
The descending node of the equator on the lunar orbit will henceforth 
be called ‘ the Intersection.’ 
Let c, be the moon’s mean longitude measured in her orbit from the 
intersection, and a, the longitude of the perigee measured in the same 
way. It has been already defined that / is the moon’s longitude in her 
orbit measured from the intersection. 
The equation cf the ellipse described by the moon is 
Saal 
(0 SOS pea Tay Fe ee) 
iz 
Hence 
R=1+4+ 3e?+3¢e cos (I—a,) + $e? cos 2(J—a,)+ .. 
© (a)=R cos (2/+a) 
= (14362) cos(21+-u) +3e[cos(3l+a—a,) +cos(I+a+e,)] | 14) 
+ 3e?[cos(4l+a—2a,)+cos(a+2z,)]J+ ... 
¥(a)=R cos « 
By the theory of elliptic motion 
I=o,+2e sin (o,—aw,)+4e? sin 2(o,—w7,)+ ... . « (15) 
In order to expand ®, ¥, R in terms of o, (which increases uni- 
formly with the time), we require cos (2/+«a) developed as far as e*; 
cos (381+a—a), and cos (l+a+a), as far as e; and only the first term of 
cos (4l+a—2za,). 
Substituting for / its value (15) in terms of ¢,, it is easy to show that 
cos(2/+ a) = (1 —4e?) cos(20,4+a) —2ecos(o,+a+a,) + 2ecos(37,+a—2,) 
+ 3c? cos (a+2a,)+ Ye? cos (40,ta—2a,)4+ .... 
cos (3/+a—a,) =cos (30,+a—z,) 
—38e cos (20, +a) +3e cos (40,+a—27,)+ . 
cos(Il+a+a,) ==cos (¢,+a+a,)+e cos (20,+a)—ecos(a+2a,)+... 
cos (414-a—2a,)=cos (4¢,+a—20,)+ .. 
Substituting these values in (14) we find, 
® (a)==(1— Je?) cos (24,+4) — $e cos (¢,+a+7,) 
+ de cos (30,+a—a,)+ lie? cos (4¢,t+a—2a,)+.... 
R=(1—e?) +3e cos (¢,—a@,) + $e? cos2 (o,—a,)+ .... (16) 
W (a) =(1—3e?) cos a+ Ze [cos (,+a—z,)+cos (¢,—a—z,) | 
+ 2c? [cos (27,+a—2a,)+cos (206,—a—2a,)]4+ .. 
