1912]
ELECTRIC CURRENTS IN A THIN ANCHOR-RING
103
respectively reverses cos fa and cos fa, while it leaves r unaltered. The definite integrals required for the other terms are*
•+TT
log (pj2 + p22 — 2p1p2, cos (/>) d(f> = greater of 47r log p2 and 4nr log plt (5)
cos ?n,<^i> log (pi* + p22 —
= - -- x smaller of f ^ and f ^ , ....... (6)
m V/V \PoJ
m being an integer. Thus
i rr . i /'+rr [+7r
-•— a log r dfa dfa= -—,, I dfa I £#</>logr = greater of log p2 and log PH. (7)
So far as the more important terms in (4) — those which do not involve p as a factor — we have at once
log (8a) — 2 — greater of log pa and log pj ................ (8)
If p2 and pl are equal, this becomes
log(8ft/p)-2 .................................. (9)
We have now to consider the terms of the second order in (2). The contribution which these make to (4) may be divided into two parts. The first, not arising from the terms in log r, is easily found to be
(10)
The difference between Wien's number and mine arises from the integration of the terms in log r, so that it is advisable to set out these somewhat in detail. Taking the terms in order, we have as in (7)
]_ /*+T f+fC
-._ 2 1 I log r dfa dfa = greater of log p2 and log px ........ (11)
TJ7T J — n- J —TT
In like manner
i rr
— - sina fa log r dfa dfa = -| [greater of log p2 and log pj, . . ..(12)
i r r
and -r—; sin2 falogrdfadfa has the same value. Also by (6), with m = 1,
47Ta J J
4-7T2
cos ($2 — fa) log r d fad fa = — £ [smaller of pzjpi and pi/p2]— (13)
Finally j~l sm 0i s"1 $2 log r ^ d<£2
]_ /"+7T r+TT
= j—2 dfa sin 0! (sin ^ cos 0 + cos 0j sin <£) log ro5<jf>
= — | [smaller of pz/pl and /?i/p2].........................(14)
* Todhunter's Int. Gale. §§ 287, 289. change of fa, fa into TT — fa, Tr — faiven by Maxwell f and Sellmeier, the resonating bodies take their motion from those parts of the sether with which they are directly connected, but they do not influence one another. In such a case the boundary conditions involve merely the continuity of the displacement and its first derivative, and no complication ensues. When there is no refraction, there is also no reflexion. By introducing a mutual reaction between the resonators, and probably in other ways, it would be possible to modify the situation in such a manner that the boundary conditions would involve higher derivatives, as in the case of the stiff string, and thus to allow reflexion in spite of equality of wave- velocities for a given ray.