60 nREPORT—1883. 
terms which are merely a reproduction of those already obtained, and 
neglecting terms in q? and q*, we have 
X?— Y?=19) mep* cos (2x —20,—s+2h—/) 
—13mep‘ cos (2y—20,+s—2h+p) 
XZ= — 19) me pq sin (x—20,—s+2h—p) a 
+13 mep*q sin (y—20,+s—2h+p) 2) 
+43mepq(p?—q*)[sin(x+s—2h+p) +sin(x—s+2h—p)| 
1(X?4 Y?—27Z7)=2 (pt—4p?qt+¢') t8me cos (s—2h+p) 
It must be noticed that 49° me arises by the addition of the coefficient 
of the Evection in longitude to three halves of that in the reciprocal of 
the radius vector; that 1% me is the difference of the same two quantities ; 
and that +3 me is three times the coefficient in the reciprocal of radius 
vector. When the development of the lunar theory is carried to higher 
orders these coefficients differ considerably from the amounts computed 
from the first term, which alone occurs in the above analysis. Hence, 
when these coefficients are computed, the full values of the coefficients in 
longitude and reciprocal of radius vector must be introduced. According 
to Professor Adams, the full values of the coefficients are, in longitude 
022233, and in c/r °010022. 
The ratio of the mean motions m is about +5, and is therefore a little 
greater than e, hence me is somewhat greater than e?. Thus we may 
abridge (25), and write the expressions thus :— 
X?—Y?= 123 mep* cos (2x—20,—s+2h—p) 
—1imep' cos (2y—20,+s—2h+p) (26) 
XZ=—195 mepg sin (x—20,—s+2h—p) : 
A(X24¥2-272)= 2 (p*—4p%q?+q')43me cos (s—2h+p) { 
The equations (26) contain the terms to be added to (20) on account 
of the Evection. 
The Variation. 
Treating this inequality in the same way as the Evection, we have 
l=o,+ Jim? sin 2(s—h) 
a 
LAE Sead +m? cos 2(s—h) 
2 
R=1+3m? cos 2 (s—h) 
W(a)=cos a+ 3m? [cos (2(s—h) +a) +cos (2(s—h)—a) } 
® (a)=cos (26,+a)+?3m? cos (20,4 2s—2h+a) 
+1? cos (20,—2s4+2h+a) 
Whence we have to a sufficient degree of approximation, 
X?— Y2=*3 m7p* cos (2x —20,—28+2h), XZ=—0 
2 (X?+ Y?-22?)=3 (p*—4p?¢? + ¢*) 38m? cos (2s—2h) (27) 
