in the Theories of Light and Sound. 363 



ticular state of phase, we shall know the form of F for all 

 varieties of phase. 



Now, when the two rays are in the same phase, we have 



a = intensity = F |« cos a), (a sin a) J- ; 

 therefore 



Fj(flcosa), (« sina)} =\/(«cosa) 2 + (a sin a) 2 . 



Hence, in the case under consideration, the intensity will not be 

 measured by the sum of the amplitudes, but by the square root 

 of the sum of the squares of the amplitudes of the component rays. 

 It thus appears that the argument which Mr. Bosanquet puts 

 forward as decisive against the simple power of the amplitude 

 being taken for the measure of intensity in plane-polarized 

 waves has, in fact, no bearing upon the subject. 



The measures of intensity which I have proposed as appli- 

 cable to plane and elliptically polarized light coincide in a re- 

 markable manner. 



For, when the component rays are represented by (1), the 

 resulting ray will be elliptically polarized — the magnitude] and 

 position of the axes of the ellipse depending on a, which mea- 

 sures the intensity of the incident light, D which represents the 

 difference of phase of the component rays, and a, the inclination 

 of the principal plane of the crystal to the plane of polarization 

 of the beam originally incident upon it. 



The absolute magnitude of either axis will always be propor- 

 tional to a, while the position of the axes and their ratio to each 

 other depend on D and a. Hence, so long as its form is unal- 

 tered, the circumference of the ellipse (J. e. the length of path 

 described by a particle in a single undulation) will vary as a ; 

 and the intensity of the overlapping rays for the same form of 

 vibration will also vary as a. 



It thus appears that when from the consideration of plane- 

 polarized we turn to that of elliptically polarized light, length 

 of path of the particles is not sufficient to determine the inten- 

 sity ; the form of vibration must also be taken into account ; 

 but for a fixed form of vibration the intensity varies directly as 

 the length of path of the particles. 



The single argument I adduced against Mr. Bosanquet' s view 

 of the relation of the amplitude to the intensity was that, accord- 

 ing to the latter, two equal vibrations in the same phase will 

 give four times as much illumination as either separately. This 

 argument, which appears to me irrefragable, Mr. Bosanquet 

 regards as so utterly trifling that it is only from the considera- 

 tion that it is " sometimes felt as a difficulty by learners " that 

 he is induced to "just touch upon it." 



