/: 



Jo 



1897.] Mr Pocklington, Electrical Oscillations in Wires. 327 

 Hence (2) becomes 



V COS vv f n 



= — 2rA dd> II cos rd> 



ti Jo 



+ a 2 A a cos rO I d<f> U {cos (r + 1 ) cf> + cos (r — 1 ) <f>} &&, 



or I d(f) Il cos r<£ (r 2 - a-a" cos </>) = (3), 



Jo 



the disappearance of justifying the assumption made as to the 



form of X. In this equation for finding a we may give to •sr and 



z in II any values which correspond to a point on the surface of 



the wire. The simplest values are ot = a + e, z = 0, and these 



should therefore be chosen. 



6. The special case r—\, which corresponds to the funda- 

 mental node of the wire, is that of the most interest and will be 

 investigated in detail. 



In this case, (3) becomes on substituting for II its value 

 (when z = 0), 



piasja? - 2am cos (j) + sr 2 



dcf> = = {cos cf> - \o?a? (1 + cos 2(f))} = 0, 



o va 2 — 2a-n7 cos <f> + ot 2 



or, putting ot = a + e, 



' \v?a? — cos </> + 2 ft 2 ^ 2 cos 2<£ 



o J2a {a + e) (1 - cos </>) + e 2 



fn 2iaasin£0 _ i 



+ Jo^ 2a sin 10 ' (K* 8 - cos <£ + ^ a ' Ja2 cos 2< />) = 0...(4), 



where in the second integral e has been put = 0, since we are 

 neglecting small quantities. 



The first integral in (4) is, calling 2aa = x, log — = L, and 



6 



neglecting small quantities, 



The second integral is, putting </>/2 = -ty, 



■n 



2 qix sin i^ \ (x 2, 1 



ft\|r — : — - — ■! — (1 -j- cos 4-vM cos 2-vir 



o r sm ^ (8a T a r 



P T , sin(#simlr) f^ 2 N 1 



Jo smi/r { 8a a T 



h ^ t oo S _(^Wg (1+cos4t) _l cos2f j... (5) . 



/, 



