1912] ELECTRICAL VIBRATIONS ON A THIN ANCHOR-RING 117 Before we can proceed to an integration there are two other factors to be regarded. The first relates to the intensity of the source situated at adtj)'. To represent this we must introduce cos m<f>'. Again, there is the question of phase. In. eiap we have p = r — a sin 9 cos (<£' — <£); and in the denominator of (4) we may neglect the difference between p and r. Thus, as the components due to adcf)', we have <-« cos wf fr' + y r- with similar expressions for Q and R corresponding to the right-hand members of (28). The integrals to be considered may be temporarily denoted by S, G, where ,> i iSf i i — I rl fr\ r*r\Q wi ft\ 0""^C COS (<p *" (p) / 01 ~r\ /J\ r*r\Q ft\ i i XIM NJ, v ~— I LOIL/ OxJo//t'uJt5 loi-ll UJ « Uwo CLJ y, • •••......lOw/ .' —7T ^ being written for aa sin ^. Here 8 = ijH d</)'e~^cos(*'~fW (sin (m + 1) <£' — sin (??i — 1) ^'}, J —7T and in this, if we write -^ for <f>' — (f>, sin (m +1)0' = sin (m +1) -vj^. cos (m + 1)0 + cos (m + 1) T/T . sin (771 +1) 0. We thus find S = ®m+1 sin (7?i +1) 0 — ®?n-i sin (m - 1) 0, ............(31) where ©n = I d^cosn^e""^003*........................(32) Jo In like manner, C = ©m+1 cos (m + 1)0 + ©,„_! cos (HI -1)0.............(33) f7" Now ©w = d^lr cos ?IA|T (cos (£ cos ^) — i sin (£cos 0)}. Jo When n is even, the imaginary part vanishes, and @— ^n' *)_ /OA\ •n ™~~ ---------_ * i •*•<*••**«•*•,•••««*k«t«****»*\O^jb) cos|-n7r v ; On the other hand, when n is odd, the real part vanishes, and n *TT- / ( (\ 0,^-^Aii............................(35) sin ^ n?r ' Thus, when m is even, m + 1 and TO — 1 are both odd and S and 0 are both pure imaginaries. But when m is odd, S and (7 are both real. As functions of direction we may take P, Q, R to be proportional to r- r- Q- —"~ O zxtic force, and the whole energy of the field T is given by