166 Prof. Airy's Remarks 



the fringes become narrower as the prism is turned so as to 

 increase much the angle of emergence, — is a direct consequence 

 of the theory of undulations. In fact, the light after emergence 

 comes from two virtual images more widely separated than 

 the first images; and the breadth of the fringes is, cateris pa- 

 ribus, inversely as the distance of the radiant images. 



I have now made the remarks which I proposed to make 

 upon Mr. Potter's experiment. I cannot, however, conclude 

 without noticing an incidental expression, " the unfortunate 

 half undulation which has continually to be asked for by those 

 who adopt the undulatory theory of light." And this I do, 

 partly because I have heard an objection something like Mr. 

 Potter's, and partly because from Mr. Potter's way of stating 

 it, I conclude that he must have derived it from some very 

 imperfect or erroneous statement. I know of wo case in which 

 " half an undulation has to be asked for." 



It happens sometimes unfortunately for a theory, that the 

 words of its original proposer, which were necessary when the 

 theory was new, are retained when they are not only unneces- 

 sary, but even mischievous. The propositions which in the 

 early stages of a theory are necessary to point out the distinc- 

 tions of different cases, add in no small degree to its obscurity 

 when it is so far advanced that every case can be included 

 in one general process. This has happened in regard to the 

 propositions to which, I suppose, Mr. Potter alludes here. 



The change of half an undulation is, in fact, a change of 

 sign of the vibrations of which the undulation consists. The 

 only thing to be explained then is a change of sign ; and the 

 only cases in which it occurs (so far as 1 know) are, certain 

 cases of the interference of polarized light; and the interfe- 

 rence of light forming Newton's rings. Perhaps lean best 

 explain the apparent difficulty by referring to simple geome- 

 trical cases. 



In describing to a student the relation between the versed 

 sine and cosine, we might say, " the versed sine is formed by 

 subtracting the cosine from the radius when the arc is less 

 than a quadrant, and by adding it when it is greater than a 

 quadrant." These are to him two distinct cases; but the ac- 

 complished mathematician considers them as one, connected 

 by the theory of the change of sign. Thus it is with the in- 

 terference of polarized light : in certain cases two resolved vi- 

 brations are added; on approaching a certain limit one of 

 them disappears ; on passing that limit it reappears, but in 

 such a way that it must be subtracted from the other. But 

 all these changes take place with as great regularity as the 



