DIURNAL VARIATION OF TERRESTRIAL MAGNETISM. 
187 
In order to avoid frequent interruption, I prove in the first instance a few formulae 
of transformation which I have found of great utility in these investigations. I start 
from the following equations denoted in my previous communications by Roman 
letters, which it is convenient to retain : 
(2?h- 1) cos 0Qy = (ft-cr+1) QVi + ^ + oOQVi . 
(2n + 1) sin 0Qy = Q„+y +1 -Q H _y +1 . 
= (ft + cr) (ft + cr— 1) Q n _y _1 — (ft — o- + 2) (ft —cr+ l) Q„ + y _1 
rQ/ 
sin 8 
— (ft + cr) (ft+cr— 1) Qn-i* 1 + Q )! -i' 7 + 1 
2c/Q/ 
dd 
— Q K+ i ff+1 + (ft-cr + 2) (ft —cr +1) Q, i+ y 1 
= (ft+o-)(ft-o-+i)Q l r 1 -Q/ +1 . . 
(A) , 
(B) , 
(C) , 
(B), 
(E), 
(Hi). 
Qy denotes the tesseral function derived from the zonal harmonic P„ by the relation 
d a V 
Gy = sin 0, 6 — 7 -—-, where u = cos 8. 
dfS r 
Multiplying (D) by (ft — cr+I) and (E) by (ft + cr), and adding, we find, with the 
help of (A), 
(ft+cr) (ft-<r+l) cos 0Q/ 1 --r^ Qy = -(ft + o-) Q K+ y +1 -(ft-cr+ l) Q n _i <r+1 . 
sm v 
(2ft +1) 
If in the formula (A) we substitute <x+1 for cr, it becomes 
(271+1) cos 8Q/ +l = (ft-cr) Q n+ y +1 + (ft + cr+l) Q„_y +1 . 
From the last two equations we obtain, by subtraction and substitution of (Hi), 
(2n+1) cos 0 %' - Q„' = + 1) Q,-,' +1 . 
cl 8 sm 8 
Now multiplying (B) by n . (ft +1), and subtracting, we finally obtain 
. rlO a 
sin 8 cos d—jjj — ft. 7i+1 , sin 2 0Qy—o-Qy 
- An 8 ( n ~ 1 ) ■+ 1 ) 1 ~ n (»■+ 2 )Q*+i J+1 / K \ 
2ft+l v 
If on the right-hand side of (K x ) we substitute the values of Q„ + y +1 and Q„_y +1 
from D and E, we obtain a corresponding equation 
sin 8 cos 0—jjj —ft .ft+1 .sin 2 d.Qy + crQy 
= { w (^ + 2 )(ft-0- + 2)(ft —cr+l)Q, i+ y- 1 -(ft-l)(ft+l)(ft+o-)(ft + o— i)Q n -y -1 } 
. . ' • (K 2 ). 
2 b 2 
