494 
THE ASTRONOMER ROYAL’S EXPERIMENTS 
where k depends only on the intensity of the current, and where P,, P 3 , P 5 are defined 
by the equation 
l 
'Pi — c 2x cos 0 + x 2 - 
1 — |— P* — j - P — I - P* 3^ 3 ~ P • • • * 
If, therefore, X represents the resolved part, perpendicular to the plane of the circle 
and towards it, of the force exerted by the current on a unit of magnetism placed at Q, 
and if Y represent the resolved part of that force parallel to the plane of the circle and 
directed from its centre outwards, then 
X: 
d\J . . d\J . 
sm 0 — — cos 0 , 
> . dQ 
dr 
dV d U 
1 = — scos 0 -j- — sm 
> . dd 
dr 
To calculate these quantities, we know that 
P, = cos 0, 
P 2 =f (cos 3 0 — f cos 4), 
P 3 =- 8 - (cos 5 0 — ^ COS 3 0+^-| cos 0). 
We shall only consider the case of those points for which r is greater than a. Sub- 
stituting these values in the expression which in such instances holds for U, we have 
U = 2tA: j — \ cos0+y|. - 4 (cos 3 4 — f cos 4) 
“ i'rf (cos 5 4 — ^ cos 3 44-^jf cos 4) 
From which, after some reduction, we obtain 
-^= — i ( — 1 +-3 cos 2 4) T -j - yq . (9 — 90 cos 2 4 4- 105 cos 4 4) ~ 
-ris(- 75 4- 1575 cos 2 4- 47 25 cos 4 04- 3465 cos 6 0)^' 
' r 1 
+ (a 
X = si n a. ( -I- ! cos . ^ 3 — Jg- ( — 27 cos 0+105 cos’ 
+T 2-8 (525 cos 0 — 3150 cos 3 44-3465 cos 5 4) % 
/ rpl 
~ } (2; 
Each of these expressions consists of a series of terms in ascending powers of -, which 
V 
will be converging. 
