58 REPORT—1883. 
X2—Y2=(1— Je?) p! cos 2 (x—2,) + (1—3e?) 2G? cos 2x 
+iep! cos (2x—3¢,+2,) 
—Llep? [ p? cos (2x—0,—2,) — 6q? cos (2x — o,+a,) | 
+17 ¢%p4 cos (2y—40,+22,) 
XZ=—(1— 2) [p%y sin (x—20,) —pg? sin (x+2,)] 
+ (1—3¢) pq (p?—9’) sin x— Zep*g sin (x—3¢,+- @,) 
+-Lepq [p? sin (x—2,—@,) +3 (p?—g?) sin (x0, + @)] 
+ Sepq (p?—@) sin (x + 9,— @)) — Ye*p?q sin (x—40, + 207,) 
4 (X24 ¥2 222) =2 (p4—4p%q? + q*)[(1— 32) + 3¢ cos (¢,—=,)] 
+ 2p?q?[ (1— 12?) cos 20,+e cos (30,—2,) | 
(20) 
The terms which have been retained in violation of the rule of 
approximation are that in X?— Y? with argument 2y—¢,+,, and that 
in 1(X?4 Y?—22Z?) with argument 30,—7a,. 
The only other term which could have any importance is 
Je 2p%q? cos (2x-+0,—@,) in X?—¥?. 
Before proceeding to consider the tides due to lunar inequalities it 
will be well to consider two pairs of terms in the expressions (20). 
First, in X?— Y? we have the terms 
—lep? [p? cos (2x—0,—az,) —6 9? cos (2x—,+2,)] 
The expression within [ ] may be written 
(p?—69? cos 2@,) cos (2x—¢,—z,) + 6q? sin 2a, sin (2x—0,—2,) 
=p V p?—12¢? cos 2a, cos (2x—0,—2,—R) approximately ; 
where 
sin 2a ; 
tan R= Se 
ae ecot? $ I—cos 2a, Be 
Thus this pair of terms may be written 
—hep J {1—12 tan? I cos 2@,} cos (2x—0,—a7,—R) . (20%) 
Secondly, in XZ we have the terms 
+ epq [p? sin (x—9,- 2) +8 (p? 9") sin (x—2, +7) ] 
This is approximately equal to 
+ hep3q [4 cos a, sin (x—2,) +2 sin a, cos (x—2,)] 
=ep*q / {3+3 cos 2a,} sin (x-—9o,+ Q) ; ‘ (2038) q 
where tan Q=) tag, . ny + 
The object of the transformations (20%), (20%), which may seem 
theoretically undesirable, is as follows :— 
The numerical harmonic analysis of the tides is made to extend over 
