TRANSACTIONS OF SECTION A. 261 



separated by an air-space -002 inch thick, will form very efficient lightning pro- 

 tectors. 



The author is very much indebted to Dr. Warren De La Rue for the performance 

 of the experiments in his laboratory. 



SATURDAY, AUGUST 23,1879. 



The following Reports and Papers were read : — 



1. Report of the Committee for calculating Tables of the Fundamental 

 Invariants of Algebraic Forms. — See Reports, p. 66. 



2. Report of the Committee on Mathematical Tables. 

 See Reports, p. 46. 



3. On some Problems in the Conduction of Electricity. 

 By A. J. C. Allen, B.A., Scholar of Peterhouse. 



The principal object of this paper is to solve the problem of the conduction of 

 electricity in a spherical current sheet, the electricity being introduced and carried 

 off from the sheet at any number of points, called electrodes ; and also to do the 

 same for certain finite portions of a spherical sheet, bounded either by current or 

 equipotential lines, the motion being in all cases steady. 



The method of doing this is summed up in the following theories : — 

 Let v = •>//■ (?•' 8') be the potential at any point (>•' 6') of a plane conducting 

 sheet of any conducting isotropic material and any infinitely small thickness, the 

 sheet being bounded by the curve 



f{r'6')=C, 

 the boundary being either a current or equipotential line, or partly the one and 

 partly the other, and there being electrodes of strengths JS 1 .Z? 2 ...at points r\ & x , 

 ^ 6' 2 ... subject only to the condition 2 U = o: then if we take a portion of a 

 spherical sheet of radius a of the same material and thickness, bounded by the 

 curve 



/ (a tan $) = C 



a 



(6 (f> being the ordinary polar currents on the sphere), and place electrodes of strengths 

 E t i? 2 ...at points 6 t <t> 1} 8. z 2 ... where 4> l = 6\, a tan -^=r', &c, the potential at 



a 



a 

 any point will be v = \jr (a tan — , <£), the boundary on the sphere being a current 



or equipotential line, according to the nature of that in the plane. 



This theorem is then applied to deducing solutions for a number of finite 

 areas on the sphere. The case of one source and an equal sink on a complete sphere 

 is discussed in detail, and the current and equipotential lines shown to be two sys- 

 tems of small circles. 



A similar theorem, though not quite so universal in its application, is shown 

 to hold for a sheet in the shape of a circular cylinder. 



The paper concludes with a solution in singly infinite series of the problem of the 

 conduction of electricity in a plane area, bounded by two concentric circles, and also 

 in that bounded by two concentric circles and two radii, meeting at an angle 



^(n integer). 



