AN OBSTACLE ON A TEAIN OF ELECTRIC WAVES. 
329 
oscillator, and therefore at these points the principal part of the magnetic force 
vanishes. When a—p^ is small, the principal value of the magnetic force due to the 
assumed distribution is 
K- r) fWu 
— -—=— sin 26 tan 3- 
V dz Ei + K ^ 
where 
Uc = 
2/1 
_X \Ri R/_ 
i 
(a—pd cos 3-, 
pi being the distance from the axis of the point where the line OP cuts the plane of 
the disc; whence, if the point P {x, y, z) is on the upper side of the disc, the above 
has the value 
j IK (V^— Ri““ R) 
■ sin 3^ sin 2(f) 2 du, 
V _tuj "1” _tv J — 00 
and therefore the resultant magnetic force at the point P is 
2 -V 2 gV ..-0 du. 
Ug 
In the immediately preceding investigation it has been assumed that p is not 
small; when p is small and 2 > Zq, 
and 
r = z-z, + ^piy{z-z^)- {ppi cos {4)i-(f)))/{z-zd)..., 
n = Zn + lpdjzn, 
13 cos (f) — a sin (f) 
LK 
,ii p) ra ,'2Tr 
^ sin ((^1 + IiM,+ i/o-*’o)} dp^ d6i, 
vZ JO Jo 
27rV 
that i 
IS 
/3 cos (f) — a sin (f) 
.2 0 p 
LK 
2ttV dz 
-j 27rt sin 2</>e‘d’^^* ‘’-ApiUi/^o+j/O—'o)]} ^^2 1 ^^ '^3i{Kppil{z—zJj] dpi, 
or 
^coS(/) —asin (/j = "'> sm 2(f>. z^ '^{z—zj) M e Ji {/<:pp]/(2 — 2 o) I pd dpi. 
V 
This quantity tends to zero with p, but increases rapidly as p increases from zero, 
thus the resultant magnetic and electric forces at a point on the axis have the same 
principal values as if there were no disc, and diminish rapidly in the neighbourhood of 
the axis. Similar results hold for any surface of revolution or for a cylindrical 
obstacle with a line source parallel to its axis. 
VOL. ccxii.—A. 2 u 
