VARIATION OF TERRESTRIAL MAGNETISM. 
501 
where 
P 
^ , and — i) = (cos J 
. . TT 
I Sin — 
■i 
Generally in the above expansion we may put therefore 
^ ? — 3 !?/3 (qqq j . 
so that the ratio of the vertical forces will be 
where 
X = 1 - 
X + 
X' + iY' ’ 
n + l.?i +2 TT (v_i , n.n + l.n + 2.n + 3 27r 
-z-COS 7 § “ + -_ XI 
2.4 
cos — 8' 
4 
{n — 1) . . . {n + A) ‘6 'it 
-2X6- ^ ■’ 
,, 11 + \.n + 2 . TT cv i + 2.71 + 3 . 27r 
Y =--- sin - b -—7-sin —- d ^ 
2 4 2.4 4 
(9. - 1) . . ■ {11 + 4) ^ 
^ 2.4.6 4 ’ 
xr/ , 71 — l.■?^ TT cv 1 . a — 2.71 — 1.71.91 + 1 27r ~ , 
X=l-- 2 -cos- 8-^+- — - cos- 8 1-..., 
-ry, 91 — 1.91 .77 p, j 91 — 2.91 — 1.91.91 + 1 . 277 . , 
Y = 2 sin - 8-’- ^ - sin ^ 8 1 + . . . 
The ratio of the amplitudes, if 8 is large, becomes 
x/(X^ + Y^) 
^(X'3 + Y'2) 
= 1 -(291+ 1)(28)-+ 
and the angle between the two vertical forces 
tan“i ^ — tan“i^, = (291 + 1) (28 )““. 
The resultant of these two forces, resolved in a direction parallel to either of them, 
is therefore equal to the component which is at right angles, and the resultant will 
consequently be at an inclination of 45°. 
