104 Prof. Challis on the Direction of the 



In the matliematical theory of vibrations given in two papers 

 contained in vol. viii. part 3 of the Cambridge Philosophical 

 Transactions, and in a communication to the Philosophical Ma- 

 gazine for December 1852, I have shown that the free vibrations 

 of an aeriform medium in their simplest form, consist of longi- 

 tudinal vibrations parallel to an axis, and transverse vibrations 

 which at each instant are the same in a given transverse plane 

 at all equal distances from the axis. The longitudinal velocity 

 V, the transverse velocity to, and the condensation cr, at any 

 ])oint whose coordinates along and perpendicular to the axis arc 

 z and r at the time t, are expressed by the equations, 



^=4 "=4' «'»'— /f. 



m\ 277 . , , , 



9=— jr— cos-— [z—Kat + c), 



(2,f {2rY {2rf 



Respecting the function /, it is only necessary to remark for 

 ray present pui'pose, that the vibrations that cause the sensation 

 of light are only those for which r is very small compared to \ 

 so that only the first two terms of the above series are significant. 

 The motion considered is, therefore, that along and immediately 

 contiguous to the axis, which is distinguished from the rest of 

 the motion by the analytical circumstance that the difi'erential 

 function ^dz-\-ivdr is more nearly an exact difi'erential in pro- 

 portion as r is less. (See Proposition X. in the communication 

 to the Philosophical Magazine of December 1852.) If, now, an 

 unlimited number of such vibrations, having the same values 

 of X, be propagated with their axes very close to each other 

 and all parallel to the common direction of propagation, the 

 transverse motions will destroy each other, and the result is 

 simply a series of waves of longitudinal vibration and alternate 

 condensation and rarefaction. For the sake of distinction, let 

 us call the simple component vibrations wave-rays. By the 

 priiiciple of the coexistence of small vibrations, the wave-rays do 

 not lose their individual character by being compounded in the 

 manner just supposed. Hence if we woiild inquire what takes 

 place when the series of compound waves impinges on a refract- 

 ing medium, we must first ascertain what happens to each wave- 

 ray considered by itself, the total efi'ect being the sum of the 

 eff"ects on the separate rays. This question appears to admit of 

 the following answer. 



It will be assumed that the axis and all the parts of a wave- 

 ray suff'er refraction, on entering a medium, according to the 



