FRESNEL OX DOUBLE REFRACTIOX. 247 



to arrive at the same point, we shall have similarly for the three 

 components referred to the instant (/), 



«'.sin2 7r [u' + t — ~J. 



c' . sin 2 vr lio' + t — ~] . 



These three velocities having respectively the same directions as 

 the preceding, it is sufficient to add them in order to have their 

 resultants, which gives — 



a.sm2-!ryu + t -^\ + a' . sin 2 w (u' + / _ l-V 

 b . sin 2 7r(^v + t - ^j + b' . sin 27r(v' + t -~). 

 c . sin 2 TT (^w + ^ - J j + c' . sin 2 w [iv' + t -~). 



If we transform each of these expressions so that it may con- 

 tain only one sine, according to the method indicated in my 

 Memoir on Diffraction {Memoires de VAcademie des Sciences, 

 torn. V. p. 379), we find that the square of the constant coefficient 

 multiplying this sine is, for each of these respectively, equal to 



o2 + a'2 + 2 aa' . cos 2 tt ('m - // + - ' ~ ' V 

 A2 + b'^+2 hb' . cos 2 TT (v - v' + -^-^ — i"). 



c^ 4- c'2 + 2 cc' . cos 2 TT (tv - ic' + '""' ~^'). 



Now it is the square of the constant coefficient of the absolute 

 velocities which represents, in each system of vibrations, the in- 

 tensity of the light, which is al\\ ays proportional to the sum of 

 the vires vivce ; and as these velocities are at right angles to 

 each other, it is sufficient to add the three preceding squares to 

 have the total sum of the vires vives resulting from the three 

 systems of vibration, that is to say, the intensity of the whole 

 Ught. 



Experiment shows that this intensity remains constant, w hat- 

 ever variations are undergone by the difference (.*' - x) of the 



VOL. v. PART XVIII. jj 



