TRANSACTIONS OF SECTiaN A. 629 



I'he cori'espondiDg point (X, Y, Z) can be shown to start at tlie poiut (S, H, /,), 

 •which 19 the inverse of (^, rj, f ), and to travel with velocity v along a straiglit line 

 whose direction cosines are |L, M, N), so that at time 



where 



X = S + Li'T, Y = II + M»T, Z = z + N(T, 



These formulaj establish a correspondence between the rays which are incident 

 at points of a surface / and the rays which are incident at points of the inverse 

 surface F. If (p, q, r) are the direction cosines of the normal at (|, rj, () to the 

 surface /, the corresponding quantities (P, Q, R) are the direction cosines of the 

 normal to the inverse surface F. If (/, m, n) {I', m', n') are the direction cosines 

 of a ray before and after refraction at the surface f, the corresponding quantities 

 (L, M, N) (L', M', N') are the direction cosines of a ray before and after refraction 

 at the inverse surface F, the refractive index of a portion of space being the same 

 as that into which it is transformed by inversion. 



A ray through the origin is seen to correspond to a ray passing throjgli the 

 origin but travelling in the opposite direction. A pencil of rays passing through 

 any given point in space is seen to correspond to a system of rays which meet the 

 line joining th^ given point to the origin. 



It is evident from these considerations that the method can be applied success- 

 fully to the solution of problems in geometrical optics. A geometrical construction 

 for the point (X, Y, Z, T) is obtained by describing a sphere of radius vt round 

 the point (.r, y, s). The inverse sphere is of radius cT, and its centre is at the 

 point (X Y ^, 



4. On the Extension of Optical Ideas to General Electromagnetic Fields. 

 By Professor E. T. Whittaker, Sc.D., F.R.S. 



It was shown in this paper that the resolution of a beam of light into two 

 components polarised in planes at right angles to each other leads to an expres- 

 sion of the state of the luminiferous ether in terms of two potential functions, 

 one of which corresponds to each of the polarised beams. It was then shown 

 that the state of the ether can be expressed in terms of these two potential 

 functions, not only in the case in which the disturbance consists simply of 

 luminous or electromagnetic waves, but also in the more complicated cases when 

 electrostatic charges and voltaic currents are present in the field. It thus appears 

 that tlie ether is a medium with two essential qualities, which are both scalar 

 (like the pressure and temperature of a gas) and the various vectors in terms of 

 which the electromagnetic field is usually specified {e.c/., the electric force and 

 magnetic induction) are merely derived from the variation of these two qualities 

 in space and time. 



5, Distribution of Electricity on a Moving Sphere. 

 By Professor A. W. Conway, 



6. lVt,e Theory oj Solids moving in an Iricompreesihle Fluid. 

 By Professor F. Purser, F.T.C.D. 



Some years ago a paper was published by my brother, the late Professor .John 

 Purser, of Queen's College, Belfast, in the ' Philosophical Magazine ' in wliicli 

 he demonstrated the applicability of the generalised dynamical equations of 

 Lagrange to the case of solids moving iu an incompressible fluid. 



