DIURNAL VARIATION OF TERRESTRIAL MAGNETISM. 
191 
Of this we take separately 
— sin 2 9 cos X ~ 
(IX 
= — H ' -~ ^ [cos {( cr + 1 ) X — a} + COS {(cr— 1) X — a}] x]j n a 
A 
= — cr sin 0^4+—COS {(cr+l) X —a 
2.2 n + 
s 
— — cr sin 
0 (n + o-)(n+o--l)xlj H ^ 1 -(n- a- + 2){n-(T+l)xJj n+ , a 1 r/ x \ x _ a i 
2.2?i + l lV ’ J 
The remaining terms depending on y are 
x cos 9 
2 n + 1 _ 
cr cos 9xjj„ ,r ( cos {(o-+ l) X — a} + COS {(cr—1) X — a}) 
+ sin 6 —(cos {(cr— l) X — a} — cos {(cr + 1) X —a}) 
ad 
or, making use of (Lj) and (L 2 ), 
x cos 9 sin 9 
2 n + 
- i/// +1 cos [(cr+ 1) X —aj + (a, + cr)(n —or+1) i/// 1 cos [(cr— 1) X —a]. 
The terms containing y, leaving out the longitude factors, are therefore 
y sin 9 
2 — 2n+ l — 2<x) 'A»+d +1 + (w + 2cr + 1) i//„_ 1 CT+1 + (n — cr + l)(n — cr + 2)(n + 2cr)i//„ +1 CT 1 
+ (li+ cr) (u + cr — 1) in — 2cr + 1) ifi n -\ 1 J. 
Collecting our results together, we find as the effect of the operation, eliminating II 
on the left-hand side of equations (14), 
7 
sm 
Q (n — cr+ 1) (n— cr + 2) (n + 2cr) xfj n+1 <r l + (n + cr) (n + cr—1) (n — 2cr+ 1) xf/ H - " 1 
— cr sin d 
2. 2n + 1 
^ <r , 9 / ( /l ~ cr + 1 ) <ftVn+ (^ + cr) 1 
7 2n +1 
<r+1 
COS (crX —a) 
• /] (il — 2cr) il/„ +1 <r+1 + (n + 2cr + 1) i \i n -\ <- / . i \ \ , 
+ y Sin o ^ ; y " +1 -4- Tn — cos • (cr + 1) X — a f 
' 2.2n+l IV 7 J 
COS {(cr —1) X — a} 
(26). 
The expression reduces to about half its terms when n = cr, and for the two special 
cases which form the main subject of the present inquiry we have then 
Case I. n = 1, cr = 1. 
sin~ + < llx ) = 7 ^ 2 ° C0S W + t/W cos ( X_a ) cos (2X—a). 
