296 



FRESNEL ON DOUBLE REFRACTION. 



Fig. 9. 



surface of the wave at the end of a given time, or in other terms, 

 the equation of the surface of the wave. 



Theorem on ivhich depends the calculation of the Surface of the 

 Waves. 

 Let C (fig. 9) be a centre of disturbance, A R B D the position 

 of the wave emanating from C at the end of the unit of time, 

 which I take sufficiently great for the distance of the wave from 

 the point C to contain several undulations, or in other words, so 

 that the length of an undulation may be neglected with regard 

 to this distance. 



Now conceive a 

 plane and indefinite 

 wave O N passing 

 through the same 

 point C ; at the end 

 of the unit of time, 

 I say, this wave will 

 have been trans- 

 ferred parallel to it- 

 self into the position 

 (on) tangent to the 

 curve A R B D. In 

 fact, let R be the point of contact, and let us seek for the resultant 

 of all the systemsof elementary waves emanating from thedifi'erent 

 points of O N which arrive at R ; it is seen that, for the reasons 

 previously explained, it will only be such rays as cR, c/ R, of small 

 inclination to C R, that will concur in an efficacious manner in 

 composing the oscillatory motion in R. Let c and c' be two centres 

 of disturbance, whence come these rays whose inclinations to 

 C R are small ; at the end of the unit of time they will have sent 

 forth the two waves arbd and a' ?*' y d', absolutely parallel to 

 the wave ARBD, and tangents to the same plane o n in the 

 points r and r'. Hence they will arrive at R rather later than 

 the wave emanating from C ; C R is therefore the path of quickest 

 arrival of the disturbance at R. It is to be remarked, in the 

 first place, that everything is symmetrical on all sides of the 

 minimum throughout a small interval such as that we are con- 

 sidering, and that hence the oscillatory movements which come 

 by the corresponding rays c R and d R, and are slightly inclined 

 to the plane o n, will together form resultant motions exactly 



