FRESNEL ON DOUBLE REFRACTION. 297 



parallel to this plane, like the oscillatory movement which comes 

 from C. The same may be said of any other two corresponding 

 points situated out of the plane of the figure ; therefore already 

 the oscillatory motion will have the same direction as it must 

 have in the plane o n. With regard to the position of the re- 

 sultant wave, it will be found in arrears of the point R by a 

 quarter of an undulation, on integrating parallely and perpen- 

 dicularly to the plane of the figure ; but in a calculation where 

 we have considered the length of an undulation as a quantity to 

 be neglected with regard to the distance C R, we may say that 

 the wave O N has in fact arrived at R at the end of the unit of 

 time. By going through a similar reasoning for each of the 

 other points of o n, it might in the same way be proved that the 

 disturbances resulting from all those which start from O N arrive 

 there also at the end of the unit of time, and that consequently 

 the entire w^ave is found at this instant transported to o n. We 

 might demonstrate in the same way that every other plane wave 

 P Q passing through the point C would, at the end of the unit 

 of time, be in the parallel position p q, tangent to the same curve 

 surface A R B D ; therefore this surface must be a tangent at 

 the same time to all the planes occupied at the end of the unit 

 of time by all the plane indefinite waves which have started from 

 C. Now we know their relative velocities of propagation mea- 

 sured in directions perpendicular to their planes, and we may 

 consequently determine their positions at the end of the unit of 

 time, and obtain therefrom the equation of the surface of the 

 wave emanating from the point C. In this manner the question 

 is reduced to the calculation of an enveloping surface. 



Calculation of the surface of waves in doubly-refracting media. 



Consequently the equation of a plane which passes through 

 the centre of the surface of elasticity being z =^ mx -\- ny, that 

 of the parallel plane to which the surface of the wave must be a 

 tangent, will he z =■ m x -\- ny + Q, G being so determined that 

 the distance of this plane from the origin of coordinates may be 

 equal to the greatest or least radius vector of the surface of 

 elasticity comprised in the diametral plane z = mx ■\- riy. 



The equation of the surface of elasticity, referred to the three 

 rectangular axes of elasticity, is 



y^ = a^ . cos^ X + 62 . cos^ Y + c^ cos* Z. 



Lot X = a. . z and y = |3 . 2" be the equations of a straight line 



