22 
DR. S. CHAPMAN ON THE SOLAR AND LUNAR 
terms in the North, West, and vertical force variations at the surface of the earth 
(r = R) 
(5) 
= {(E n r;l(a) + I n m ( a) ) cos nt+ (E n m(6) + I” m(J) ) sin nt] 
(North), 
(West), 
( 6 ) -R ] - n C ~jt~ = { “ (E” m (a) + I”m (a)) sin nt + ( (b) + l n m (W ) cos nt } -2— Q m " 
v ' R sin 8 d\ sm 8 
( 7 ) = — {( m ^ n m (a) —m +1 I n m (a) ) cos nt + (wE" m (i) —m + 1I" m (w ) sin nt } Q m ” 
These may be written in the form 
( 8 ) (A m n cos nt + B m " sin nt) N m ” (8) 
(9) (B m " cos nt — A m " sin n£) W m n ( 0 ) 
( 10 ) -(A,/ cos nt + B /ra ” sin nt) Q m n ( 0 ) 
the new notation being thus defined :— 
(Radial, outwards). 
(North), 
(West), 
(Radial, outwards), 
(11) A m n — Ei n m (a) +1” „ (a ), — E' ! m ( 4 ) + I" m ( A ), 
(12) A m " = w - (m +1) I ra m w , B,/ = mE" s (w - (m +1) I” m (6) . 
The new symbols N,„” (8) and W m n (0) are defined as follows :— 
dO" 
(13) 
N„« (e) = 
d6 ’ 
w„”(e) = V^Q„\ 
sm 0 
Table B contains a list of the particular values of the three functions Q m n , N m " 
and W m n corresponding to the special values of m and n with which we are 
Table B. 
Qm" (COS 0). 
M n — f) 
iN m 
W n 
v » m — 
ft 
sin 0 
O n 
Qi 0 = cos 0 
Qi 1 = sin 0 
QA = 3 sin 6 cos 0 
Q 3 J = -| sin 0 (5 cos 2 0-1) 
Q 2 2 = 3 sin 2 9 
Q 3 2 = 15 sin 2 9 cos 9 
Q 4 2 = ^ sin 2 9 (7 cos 2 0-1) 
Q 3 3 =15 sin 3 0 
Q l4 3 = 105 sin 3 0 cos 0 
Q 5 3 = 1 -P- sin 3 0 (9 cos 2 0 - 1 ) 
Q 4 4 = 105 sin 4 0 
Q 5 4 = 945 sin 4 0 cos 0 
Qe 4 = -f-sin 4 0(11 cos 2 0 - 1 ) 
Ni° = - sin 0 
Nfi = cos 0 
No 1 = 3 (2 cos 2 0- 1) 
N 3 1 = f cos 0 (15 cos 2 0-11) 
N 2 2 = 6 sin 0 cos 0 
N 3 2 = 15 sin 0 (3 cos 2 0-1) 
N 4 2 = 30 sin 0 cos 0 (7 cos 2 0-4) 
N 3 3 = 45 sin 2 0 cos 0 
N 4 3 = 105 sin 2 0 (4 cos 2 0-1) 
N 5 3 = A|Asin 2 0 cos 0 (15 cos 2 0-7) 
N 4 4 = 420 sin 3 0 cos 0 
N 5 4 = 945 sin 3 0 (5 cos 2 0-1) 
N 6 4 = 945 sin 3 0 cos 0 (33 cos 2 0 - 13) 
wy = 0 
wp = 1 
Wp = 3 cos 0 
W 3 1 = f (5 cos 2 0-1) 
W 2 2 = 6 sin 0 
W 3 2 = 30 sin 0 cos 0 
W 4 2 =15 sin 0 (7 cos 2 0-1) 
W 3 8 = 45 sin 2 0 
W 4 3 = 315 sin 2 0 cos 0 
W 5 3 = sin 2 0(9 cos 2 0 - 1) 
W 4 4 = 420 sin 3 0 
W 5 4 = 3780 sin 8 0 cos 0 
W § 4 = 1890 sin 3 0 (11 cos 2 0 - 1) 
