162 Mr. C. S. Whitehead on the Effect of & 



where 



.*. real part of e" lA,, 6 t ^ = real part of e~ qYl e^-i^ 

 = e-M cos (pt—qrj), 



.-. v = e-2"27r7sin 2 a V~y +1 p,[(«)P«(^)cos {pt-qrj). . (19) 



Let v be the maximum value of v, U that of U ; 

 .'. from (16) and (19) 



w =e "" < 20 > 



The method employed in this investigation is taken from a 

 paper by Professor H. Lamb, Phil. Trans. Part ii. 1883. 



(4) Case II. 



The result for this case may be deduced from the preceding 

 by using a particular case of a transformation due to Professor 

 C. Niven (Phil. Trans. Part ii. 1883). 



He shows that if P n denote a zonal harmonic of the nth 



degree, s = sin 6, n — ha, s = — ; then when n and a become 

 infinite, k and p remaining finite, 

 Y n z=J (kp), 



r 



J and Ji being Bessel functions. 



To find the value of O in terms of Bessel's functions. Let 

 P 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 (-^y'" 1 P H '(«) .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 = k x , 



