198 Prof. H. Lamb. [Mar. 24, 



of a uniform field, or to rotation of the shell in a uniform and constant 

 field, is then solved ; and the results are found to agree with the 

 general theory above sketched. 



In the higher normal types the current-function is a Lame's 

 function, degenerating into a spherical harmonic when two of the 

 axes of the ellipsoidal shell are equal. This case alone is further 

 discussed in the present paper ; the persistency of each normal type is 

 found, and various particular cases are considered. Of the special 

 forms which the conductor may assume, the most interesting is that 

 in which the third axis (that of symmetry) is infinitesimal, so that 

 we have practically a circular dish, whose resistance p per unit area 

 varies according to the law 



/= A V{l-»*/«»}, (5.) 



where p ' is the resistance at the centre, a is the radius, and r 

 denotes the distance of any point from the centre. In any normal 

 type of free currents the current-function is of the form 



*«o. d-^^r}-. • • • • W 



where A =' r 2 /a 2 }, 



provided n—s be odd ; in other words, the current lines are the ortho- 

 gonal projections on the plane of the disk of the contour-lines of a 

 zonal (s = 0) or tessaral harmonic, drawn on the surface of a con- 

 centric sphere of radius a. The corresponding persistency is 



{n{n + l)— s 2 }/> ' \n 



f 1.3... (n + s) \* (7 , 

 + 5*12.4... (n—s— I) J ' V ' } 



In the most persistent type of free currents we have n — 1, s = 0, 

 and therefore 



........ 



This result is of some interest, as showing that the electrical time- 

 ccnstant for a disk of uniform resistance p ' must at all events be 

 considerably less than 4" 93 a//> '-* 



* I find by methods similar to those employed by Lord Rayleigh for the approxi- 

 mate determination of various acoustical constants, that the true value lies between 

 na\p' and 2'26 a/p'. For a disk of copper (p = 1600 C.G-.S.), whose radius is a 

 decimetre and thickness a millimetre, the lower limit gives 0'0014 sec. For disks of 

 other dimensions the result will vary as the radius and the thickness conjointly. I 

 hope shortly to publish the details of the investigation on which these estimates are 

 founded. 



