246 Prof. Silvanus P. Thompson on 



wholly in air. The central portion of the wave, which would 

 have reached A, travels backwards to B, an equal distance, in 



Fig. 9. 



the same time. The sagitta B M of the resultant curvature is 

 equal to and of opposite sign to the sagitta A M of the initial 

 curvature ; or 



V=^% (16) 



Case ii. Curved Mirror ; Plane Wave. 



There are two cases, equally simple, of convex and concave 

 mirrors. One will suffice. Consider (fig. 10) a plane wave 



which at a certain instant would have 

 arrived at S A S had its path lain 

 wholly in air. The central portion 

 of " the wave has, however, struck 

 at M, and marches backwards to B 

 in same time as it would have taken 

 to reach A. Hence 



Fig. 10. 



or 



BM = AM, 

 BA = 2AM. 



of 



But AM measures the curvature 

 the mirror, whilst BA measures 

 the curvature impressed on the plane wave 



^=2^. . . . 



Hence 



(17) 



Case iii. Any Mirror ; Any Wave. 



The principle of superposition at once leads to a general 

 formula, expressing the sum of the two actions of the mirror 

 on the wave ; it reverses its initial curvature, and then 

 imprints a focal curvature upon it. In symbols, 



® = -W+p. (18) 



