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XI. On the "Loss of half an undulation" in Phtjsical Optics. 

 By Professor Challis*. 



BY experiments on light, certain facts have become known, 

 which are described in the language of theory by the 

 terms "loss of half an undulation;" but the theory of light 

 usually adopted gives no explanation of the facts. It is not, 

 however, to be inferred from this circumstance that the Undu- 

 latory Theory of Light is at fault, since the theoiy alluded to, 

 which for distinction might be called the Vibratory Theory, 

 does not really involve the consideration of undulations. The 

 only existing theory of light which is strictly undulatory is that 

 which I have proposed in various pubhshed memoirs. The two 

 views, as I have before urged, are essentially different. In the 

 one the jether is regarded as a continuous medium, and the 

 phsenomena of light are attributed to its motions en masse in 

 waves or undulations : in the other the same phsenomena are 

 attributed to the vibrations of its constituent atoms. The expla- 

 nation of phfenomena in the former view necessarily requires 

 the solution of partial differential equations; in the other, as in 

 every instance of the small vibratory motions of a discrete body, 

 the equations to be solved are common linear differential equa- 

 tions. Now it is possible that the vibratory theory fails to 

 account for apparent instances of the loss of half an undulation, 

 because common differential equations are inappropriate to the 

 explanation of phsenomena of light. At least, it may be shown, 

 as I now propose to do, that such instances admit of explanation 

 by means of deductions from partial differential equations. 



The loss of half an undulation is said to occur when light 

 passes out of a transparent medium into vacuum, the phase of 

 the light reflected at the boundary of the medium being the 

 opposite to that of the incident light. An analogous fact occurs 

 in air, when vibrations excited at one end of an open cylindrical 

 tube are propagated to the other end. A condensed wave is 

 reflected at the mouth of the tube as a rarefied Avave, and a 

 rarefied as a condensed wave. The explanation of this fact given 

 by the mathematical theory of aerial undulations is such as fol- 

 lows : — The density of the air which has issued, or is on the 

 point of issuing, from the mouth of the tube, can be but very 

 little different from that of the surrounding air, on account of 

 the lateral spreading to which the condensation or rarefaction is 

 subject on being propagated from the tube. If we consider an 

 elementary particle; situated at the mouth, and suppose it to be 

 urged from within to without by an accelcrative force due to the 

 fluid pressure, then there must be at each instant contiguous to 

 * Communicated hy the Author. 



