238 



Prof. Silvanus P. Thompson on 



centre Q. To a first degree of approximation the arc SBS 

 represents the retarded wave-front, the set of wave-fronts 

 from B onwards being represented by the series of arcs drawn 

 from Q as centre. An eye situated in the medium on the 

 right of SS will perceive the waves as though coming from 

 Q, the (virtual) point-image of P. Accurately the wave- 

 fronts should be hyperbolic arcs, but if S S is small relatively 

 to PM the circular arcs are adequate. Now AM=^ and 

 BM=^/. Hence the action of the plane surface upon the 

 curvature of the incident wave is given by the formula 



W = hW. (2) 



Case (i.) b. Entrant Wave of Positive Curvature, 



The entrant wave (fig. 3) has a positive curvature or con- 

 vergence such as would cause it to march to the point P (the 



incident focus) if its path lay wholly in air. At a certain 

 moment, when the middle point of the wave-front has reached 

 M, the outer portions of the wave-front passing in through 

 S S would have reached positions as far as the vertical line 

 drawn through A had the path lain wholly in air. But being 

 retarded they only reach as far as the line TBT drawn 

 through B; where again BM = A.AM. The circular arc 

 through TBT has Q for its centre ; that is to say, after 

 entry the waves now converge on Q instead of P. In this 

 case also the effect on the wave of entrance into the retarding 

 medium is to flatten its curvature, and the formula is as 

 before, 



<V-=ML ...... (2) bis. 



Case (ii.) a. Emergent Wave of Negative Curvature. 

 Consider the wave emerging (fig. 4) into air from a point 



