60 FRESNEL ON THE COLOURS PRODUCED IN 



rhomboid of calcareous spar is parallel or perpendicular to the 

 plane of double reflexion ; in fact, the expressions in which the 

 characteristic is a function of e become then equal to those in 

 which the characteristic contains o. 



It is easy to perceive also that the rotation of the rhomboid, 

 that is to say the variations of i, ought not to alter the nature 

 of the tint. In fact, if we consider the resultant of the two first 

 systems of waves, the variations of i, affecting only the common 

 factor cos {i — r) + sin (i — ?•), cannot change the position of that 

 wave, but only its intensity. For the same reason, these varia- 

 tions do not change the position of the wave resulting from the 

 union of the two other pencils. Hence the interval between 

 these two resultants, which alone determines the nature of the 

 tint, suffers no change during the rotation of the rhomboid. 



It is not the same with the variations of r; as they affect 

 unequally the two first pencils, the one of which is multiplied 

 by sin r, and the other by cos ?*, they cause the position of their 

 resultant to change. They likewise change the position of the 

 other resultant, and in a contrary direction, in consequence of 

 the opposition of sign between the first and the third pencil. 

 But this becomes still more evident on calculating the total re- 

 sultant of these four systems of waves. The general expression 

 for the intensity of its vibrations is found to be — 



F A / — + — [cos^ (i — r) — sin^ {i — r)] cos [2 r — 2 w (e — o)]. 



F./1 + 1 



V 2 2 



cos 2 [i — r) cos [2 r — 2 tt (e — o)] . 



It is clear, from this formula, that the variations of i only affect 

 the intensity of the tint*, whereas those of r change its nature. 

 When r is equal to 45°, for instance, cos [2 r — 2 w (e — o)J be- 

 comes cos 2 u — — (e — o) , and the colour of the image is that 

 which corresponds to a change of a quarter of an undulation in 



* The maximum intensity of the tint corresponds to i = r, as had been al- 

 ready observed by simple inspection of the constituent pencils. The formula 

 then becomes 



F \/ — -)- — cos [2 »• — 2 TT (p — o)], or F . cos [_r — tt (e — o)]. 



Thus the tint is precisely that which was observed before the interposition of 

 the glass parallelopiped, with the same position of the rhomboid of calcareous 

 spar. 



