DIURNAL VARIATION OF TERRESTRIAL MAGNETISM. 
189 
This is in agreement with Rodriguez’s theorem, if the definition of Q/ depending 
on the operation 
(1 -/x 2 )! d n+ ° (/x 2 -l) 72 ”n! dp n+,T 
is extended to negative values of cr, for in that case 
o -o- _ 1 a 
2 n n ! 1 ^ ’ dp n ~ a 
, )<r 1 (ft-o-)lcfr+'fc a -l)" 
y ’ 2 n n\(n + ar)\ dp H+ ' r 
-( (n + cr)! • • • • 
(23). 
It follows that the operation (17), in the case where Q n IJ cos a replaces 
Q/ cos (crX —a), reduces to 
sin 0 
9 c 
2 n + 
— {(r 1) {n +1) Q „_i n (n + 2) Q H+1 1 [cos (X — a) + cos (X + a)]. 
This result may also easily be obtained independently, but in view of the ultimate 
application of (22) it is important to include the special case in the general one. 
It will be appropriate here to obtain another formula which will be used sub¬ 
sequently. Let it be required to find 
, a d Q n , d . a a d 0/ 
COt - + ~Tn Sln V C0S # —T7T- • 
dX dd dd 
From the fundamental equation we find this to be equal to 
(24). 
—n . ti + 1. sin 6 cos 0Q/—sin 2 d 
and as 
and 
dQn 
dd ’ 
71.71+1 
-71.71 +1. cos d Q/ = - 2 - n+1 {(n + o-) Q\ +1 + (ti-o-+ 1) QVi} 
m @ dd ~~ 2 n + 1 ' ' ( nJrCr ) QTn-i—n (n — o-+ 1) Q\+i|, 
(24) becomes equal to 
sin 6 
2 ^-j-^{(w + 2)R(n-a-+l)QV 1 + («-l)(n+l)(7i + o-)QVi} • . (25). 
We are now in a position to attack our main problem. The equations to be solved 
are 
y _ dS ^ 
A ~ P 7m + 
dU 
dO sin 6 dX 
P Y = P 
c/S c/It 
sin 6 dX dd 
(14). 
