HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 57 
Now substituting from (16) in (12), giving to a its appropriate value, 
we have 
X?—Y2=(1—Ye?)[p4 cos 2 (y—o,) +g! cos 2 (x+9,)] 
+ (1— $e?) 2p?q? cos 2x 
+ ge |p‘ cos (2x—3¢,+a,)+ 9! cos (2x+30,—a,) ] 
7 Ze [y* cos (2x—¢,—7,) 2 i cos (2x+¢, +a,) | (17) 
“F 3e 2p?y? [cos (2x +¢,—2,) + COS (2x—¢,+2,) ] 
+ 1 e?[p4 cos (2x —40,+20,) +91 cos (2x +49,—22,)] 
+ $6?2p?¢? [cos (2x + 27,—2a,) + cos (2x —20, +2a,) ] 
Clearly —2XY is the same as (17) with sines in place of cosines. 
Also since YZ is the same as X?— Y? when y replaces 2x, —p*q replaces 
p*, pq (p?—@") replaces 2p?q?, and pq’ replaces g‘, and since XZ is the 
same as YZ with sines in place of cosines, we have from (i7) 
XZ=—(1—'Ye)[ pry sin (y—2e,) —pq? sin (x +2.) ] 
+ —2e*) pa (v?—9") sin x 
— ge[p*q sin (x—30,+7,) — pq? sin (x +30,—2,)] 
+ 3e[p*q sin (x—2,—a,) — pg’ sin (x+0,+2,)] (18) 
+ 2epq (p?—9°)[sin (x-+ ¢,--2,) +8in (x-0,+2,) ] 
— el [piy sin (x—4e,-+20,) —pq? sin (x +40,—20,)] 
+ 3° pq(p?—4q’) [sin (x+20,—20,) +sin (y—20,4+2z,)] 
Lastly, 
§ (X24 ¥2_ 272) =} (p'—4p%¢?-+-9°)[ (1 —fe2) + 8¢ cos («,— =) 
+ $e? cos 2 («,—a,) ] 
+2p?q?[(1— 4?) cos 20, + Ze cos (30,—a,) —he cos («, +2,) 
+ We? cos (40,—2a,)] . (19) 
Hitherto no approximation has been admitted with regard to J, the 
obliquity of the lunar orbit to the equator. 
The obliquity of the ecliptic is 23° 27'-5, and I oscillates between 
5° 88 greater and 5° 8'-8 less than that value. The value of g or sin 4, 
when I is 23° 273, is -203, and its square is ‘041, and its cube ‘0084. 
rv eccentricity of the lunar orbit e='0549; hence q? is a little smaller 
than e. 
The preceding developments have been carried as far as ¢?, principally 
on account of the terms involving 47e2, which, as e is about zs, have 
nearly the same magnitude as if the coefficient had been Le. 
It is proposed, then, to regard y? and q® as of the same order as e, and 
to drop all terms of the order e?, except in the case where the numerical 
factor is large. This rule will be neglected with regard to one term for 
a special reason, which appears below; and for another, because the 
numerical coefficient is just sufficiently large to make it worth retaining. 
| Adopting this approximation, we may write (17), (18), (19), thus,— 
