THERMAL CONDUCTIVITIES OF SOME ELECTRICAL INSULATORS. 441 



where K is the function to which BESSEL'S function of the second kind reduces for n 

 pure imaginary value of the argument. 



Since the limit of 2ir/>X-^=, when p = 0, is pH.coK^f.-, i.e., H.cos^, we see 



t'p '21 p 'II 



that the expression for satisfies the necessary conditions. 



If the origin of the r co-ordinates be taken at a point distant c 

 from that of the p co-ordinates, p = v/r*+r* 2ercos 6, where 6 is 

 the angle which the co-ordinate makes with the line through the 

 source. 



Hence, in terms of r, c, and s, the temperature due to the source 

 placed at a distance c from the new axis, is 



To expand the K function in a Fourier series in terms of 0, we have, if Y,, is the 

 Bessel function of the second kind and zero order, and /><, 



Yo(av/< 4 +r > -2erco80) = J (ar). Y (ar) + 2 i J f (*c)Y ,() cos 00 

 (HEINE, ' Kugelfunctionen,' I., p. 342). Therefore 



Yo (ta v/c* + r* - 2cr cos 0) = J (tar)Y ffl (mr) + 2 2 J,(mr)Y,(/ar)cos00. 



where i = \/ 1. 

 But 



(GnAY and MATHEWS, ' Bessel Functions,' p. 66), and 

 Therefore 



) = I (r)K(ar) + 22 (-l)T,(-)K,(r)oo8^, 

 for r>r. 



Hence the temperature v m in an infinite cylinder due to a source H. cos -^ distant 

 r from the axis is, for points at a distance from the axis greater than r, given by 



For points nearer to the axis than the source, r and must be interchanged. 

 Since cos o.rlj(ar) . cos fid is also a solution of equation (1), we may add to i\ terms 

 of the form cos otxl ft (or) cos f30; which, since I, has no singular points, do not 



the strength of the source H, cos -^- 



VOL. OCIV. A. 3 L 



