1894.] on Ellipsoids and Anchor-Rings. 157 



journals. My object is to shew that the currents in the sheet 

 and the state of the magnetic field surrounding it may be com- 

 pletely determined in all the above-mentioned cases by the solution 

 of a general equation connecting the current function with the 

 potential due to the current sheet and the external magnetic 

 field, which equation holds for the most general case and is 

 theoretically capable of solution as soon as the attendant cir- 

 cumstances are known. 



In order to solve this equation we assume the external field 

 of magnetic force to be given by the product of e'^* and a function 

 of the co-ordinates used. This is sufficiently general since any 

 function of the time t can be expanded in powers of e^'^*. We 

 then assume the magnetic potential due to the system of in- 

 duced currents to be of a similar form, and determine the 

 arbitrary constants involved in this assumption by means of the 

 general equation above referred to and the necessary conditions 

 of the problem, viz. that the magnetic potential is discontinuous 

 at the conducting surface by an amount 47r<^ where <l> is the 

 current function, while the rate of variation of the potential in 

 a direction normal to the surface, i.e. the normal differential 

 coefficient, is the same on both sides of the surface. 



The co-ordinates used throughout are those known as ortho- 

 gonal, and are defined by the equation 



ds' = A'da'' + B'd¥ + G'dc\ 



where a = const., b = const., c = const, are the equations of three 

 families of surfaces cutting each other orthogonally, as is easily 

 seen from the equation itself ; A, B, G are here functions of a, h, c 

 and ds is the distance between two points 



(a.b.c) and (a + da, b + db, c + dc). 



In Section (1) the general equation connecting the current 

 function with the magnetic potential is found. 



In Sections (2) and (3) the well-known results for the infinite 

 plane and the sphere are deduced and found to be in agreement 

 with those of Maxwell and of Larraor (Phil. Mag., 1884). 



Section (4) deals with the infinite right circular cylinder. 



Section (5) contains the solution for the ellipsoid. I am not 

 aware that any result has been obtained previously for the case 

 when the ellipsoid has three unequal axes except that by Prof. 

 Lamb for free currents of the type <l> = Cz. 



In Section (6) an attempt is made to deduce results for the 

 anchor-ring, but here we are met by difficulties which did not 

 present themselves in the solution of the previous cases, and it 

 is only for a simple form of magnetic disturbance that a complete 

 solution is arrived at. 



