124 PROF. A, E. H. LOVE ON THE TRANSMISSION OF 



noted a difficulty in BYBCZYNSKI'S analysis. It seem to me, however, that there is a 

 serious error in the part of MARCH'S work which is accepted by RYBCZYNSKI. He set 

 out to determine a function, say \[s, which has the property expressed by the equation 



The function ^r satisfies the same differential equation as the Hertzian function IT, 

 the same conditions of continuity, the same condition. at infinite distances, and similar 

 boundary conditions ; and the special form of it answering to the originating doublet, 



say \'A,, is given by the equation 



J v ,*(-]<) 



* 



~ET 



In the region outside the sphere \js has the form ^ + ^,, where ^ in the analogue 

 of II,, in the region inside the sphere it maybe denoted by i//, analogous to II'. Now 

 MARCH seeks a solution in which i//^ and i// have the forms 



Y,, = L f ( a + l) p o (cos 6) ?'"E a (yb-y^'F, (a) da, 

 r J-'/, 



(a + ) P a (cos 0) 7-"f a (k'r) e^'F (a) da, 

 where P a (cos 0) denotes a certain solution of LI-:GENDRE'S equation 



in which a is not in general an integer, K a (&>) and i//- a (k'r) are defined by means or 

 BESSEL'S functions as in equations (19), but not by means of differential operators, and 

 Fj (a) and F (a) are functions of a to be determined. He alsd obtains integral formulae 

 of a similar type to represent \//-,, in the regions >>? and ?>?>. The function 

 denoted by P a (cos (?) is free from singularity at 9 = 0. 



Now these forms fail to satisfy conditions which must necessarily be imposed on 

 V',,, Vo, and \l/. It is necessary that \i/ should be finite at r = ; but when r = 0, 

 and a lies between ^ and 0, r a i/r a (k'r) is infinite. Further it is necessary that ^ , i^, 

 and \ff' should be free from singularity on the axis 6 = -w ; but when a is not an integer, 

 and P a (cos 9) is finite at = 0, it is infinite at 6 = ir. Hence the form taken for 

 T// has a singular point at the centre of the sphere, and a line of singular points 

 extending thence along the radius of the sphere drawn in the direction = TT, and the 

 forms taken for i/r and Y'i bave a line of singular points extending along the 

 continuation of this radius outside the sphere. It appears therefore that no solution 

 of the type sought by MARCH exists. 



