218 
Proceedings of the Royal Society of Edinburgh. [Sess. 
On multiplication by ( 1 — — cos 0 ) the right-hand side becomes 
^ 1 a w y^(-) p cos w 2p 0 | (2(n - p) + 1) ! (2(n - p) - 1) ! ) 
o2 n r n o pi 1 (n - p) ! (n - 2p)i (n -p - 1)! {n - 2p - 1)! f ’ 
the upper limit for p being (n— l)/2 when n is odd, while, when n is even,, 
it is n/2 in the first term of the bracket, and (n — 2)/2 in the second term. 
From this the similar sum, with — cos d replacing cos 6, has to be sub- 
tracted. Thus n must be odd, and the final sum becomes double of the 
above. Hence 
F cos c/> 
1 
, \ n—i - 
') ? 
(-Y(2(n-p)-l)\ 
'pi ( n —p — 1) ! {n — 2p - 1) ! 
n + 1 
n - '2p 
cos' 
- ip e , 
where M = ma is the magnetic moment of the magnet. 
The corresponding expression for the other component is 
F sin <4 = Msin6 y. 1 a "y, (-) y ( 2 ( ra -/ ) ) + !) ! 
r 3 ^ 2 n r n q p\ (n - p ) ! {n 
2 p)\ 
cos” 2p 0 , 
where n is even. So, writing n for n—1 in the expression for F cos (f> r 
we obtain, with n always even, 
17 , , . \ M xS 1 CL' 
1 COS (0 + <t>) — — ^ 
n/2 
( -)>'(2(«-J))+l)! cos n-^ e 
2” r n o P ! (n — p)i ( n - 2p) ! 
% -g>+ 3 coatt-l 
_ n — 2p + 1 
F rin (» + *)-“ 2"^ ^2 
VI — -- — — — — — y-Z L — cos” -^0 cos 0 sin 0 . 
^pi (n - p)\(n - 2p) ! n - 2p + 1 
These are respectively expressions for the components of force, at (r, 6)> 
taken parallel to, and transverse to, the direction of the magnet, and can 
be identified with 
F cos (<? + *) = ) V + 1 ) P ^+« , 
F sin (6 + <f>) = sin 6 P V+D > 
where n = 2ja, and P is the zonal harmonic in cos 0 of order 2(/x-fl). 
