54 FRESNEL ON THE COLOURS PRODUCED IN 



and recollecting that the i in the characteristic is equivalent to 

 the sign minus, we obtain, — sin^ . O and cosp . O^. But the 

 resultant of these two systems of waves differing by the qu^ter 

 of an undulation, is x^O^sin^p + O^ cos^p, or O. Hence the 

 waves arising from the ordinary refraction of the first particle . 

 suffer ordinaiy refraction in the second, because, both in the one 

 and the other, the principal section is turned towards the same 

 side as regards the plane of entrance. 



This principle may be further verified by calculating the in- 

 tensity of the polarized light in the plane Ej E/ perpendicular 

 to the principal section Oj O/. We then obtain for the four 

 constituent pencils, — 



--smp.Of^, or 4--^sinj!>.0, +-cosp.O^, 



O.T.R,.E/. O.T.T,.E/. 



— -cosp.Oi, —-smp.U. 



O.R.Rj.E/. O.R.T/.E/. 



The expressions having the same characteristic are equal and 

 of contrary signs, so that these four systems of waves mutually 

 destroy each other. Thus no one of the ordinary rays issuing 

 from the first particle can suffer extraordinary refraction in the 

 second. If the latter be turned in such a manner that the plane 

 of exit becomes the plane of entrance, it is evident that it will 

 still be placed upon the same side in relation to the principal 

 section, and consequently the rays will still be refracted in the 

 same manner. 



It should be noticed that the calculations which have just 

 been made, and the results to which they lead, are independent 

 of the relations of intensity of the double refractions exercised 

 by these particles, and that vre have only supposed them to be 

 constituted in the same manner ; that is to say, that their axes 

 wei'e turned towards the same side in relation to their plane of 

 entrance. Hence, whatever may otherwise be their inclinations, 

 or even whatever the nature of the particles successively tra- 

 versed by the incident light, the rays which have in the first in- 

 stance suffered ordinary or exti'aordinary refraction continue to 

 undergo the same kind of refraction throughout the whole extent 

 of the fluid. The hypothesis which we have adopted will ex- 



