. The London, Edinburgh and Dublin philosophical magazine and journal of science. ethod employed in this investigation is taken from apaper by Professor H. Lamb, Phil. Trans. Part ii. 1883. (4) Case II. The result for this case may be deduced from the precedingby using a particular case of a transformation due to ProfessorC. Niven (Phil. Trans. Part ii. 1883). He shows that if Pn denote a zonal harmonic of the nth degree, s = sin 6, n — ha, s = — ; then when n and a becomeinfinite, k and p remaining finite,Ynz=J0(kp), r J0 and Ji being Bessel functions. To find the value of O in terms of Besse
. The London, Edinburgh and Dublin philosophical magazine and journal of science. ethod employed in this investigation is taken from apaper by Professor H. Lamb, Phil. Trans. Part ii. 1883. (4) Case II. The result for this case may be deduced from the precedingby using a particular case of a transformation due to ProfessorC. Niven (Phil. Trans. Part ii. 1883). He shows that if Pn denote a zonal harmonic of the nth degree, s = sin 6, n — ha, s = — ; then when n and a becomeinfinite, k and p remaining finite,Ynz=J0(kp), r J0 and Ji being Bessel functions. To find the value of O in terms of Bessels functions. LetP be any point, draw PM perpendicular to the axis of z. Let CD, the radius of the circuit, =/,PM=p, ZPOM = 0, ZCOD = «, OC = c, DM=z, OP=r. n = 27rsin^S^I(-^y1PH(«) .P,(0), r>c. Let n=kr=kiC, A P • f smd=- , sin«=-r c let n, r, c become infinite, k, p, f remaining finite ;/. ultimately k = kx, Spherical Conducting Shell on Dielectric Induction. 163 P«(0)=Jo(kp), P,|(«)=yJl(A/), c = OD = OM-DM= r—z ultimately ; .-. (i J =(l--r Y=e-**m the The successive values of n are 1, 2, .; let k + dk be thesuccessive values of k ; /. n+L = (k+dk)r;.*. rdk = l i .; z (- )= -dh=dk; n + l\r / w + 1 .-. n = 27r/f e-k2J0{kp)^i(kf) result can also be deduced from the equation d2a Ida dm dr2 r dr dz2 remembering that when 2=0, fl = 27r from r = 0 to r= = 0 from r — a to r = oo . 164 The Effect of a Spherical Conducting Shell on thus find ke-kJ0(kp)Jl(kf)dk, j as before. Appendix. It may be useful to add the proof of the transformationused in Case II. If yLt=COS 0, P« satisfies (l-^-2^+n(,i + l)P,, = 0. . (1) Let s = sind. (1) transforms into ao> d rn , 1 — 4S dZn , i\t» /-. -s^ + —r-di+t<n+l^n=°- Assuming ~Pn=a0 + als + a2s2+ .. we find in the usual manner V-„fl (+1) J , (n-2)n(n + l)(n + 3) ,4 V. r* — 0 | X M 6 • 02 Z2 * * J But Vn = 1 when 5 = 0 ; .. a0=l. Let n-=ka. s= -, a and let n and a become infinite,
Size: 1217px × 2052px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1800, bookdecade1840, bookidlondon, booksubjectscience
