FRESNEL ON DOUBLE REFRACTION. 293 



division, it is easy to see that by reason of the distance of H, 

 which is very great relative to the length of an undulation, the 

 small parts into which we have divided C A will become sensibly 

 equal to each other for rays which make slightly sensible angles 

 with G H. We may therefore admit that the rays sent by two 

 consecutive parts will mutually destroy each other as soon as 

 they have a sensible obliquity to G H ; or, more rigorously, 

 that the light sent by one of these parts will be destroyed by the 

 half of the light of that preceding it, and the half of the light of 

 that succeeding it ; for its magnitude differs only from the arith- 

 metical mean of those between which it is situated by a very 

 small quantity of the second order. Moreover, the rays sent by 

 these three parts must have sensibly the same intensity whatever 

 be the law of their variation of intensity round the centres of 

 disturbance, since, being sensibly parallel to each other (by reason 

 of the distance of H), they are in the same circumstances*. 



Moreover, it results, from the nature of the primitive vibratoiy 

 motion which gives rise to all these centres of disturbance, and 

 the oscillations of which are necessarily repeated by them, that the 

 elementary waves which they send to H will carry to that point 

 absolute velocities alternately positive and negative, which will 

 be the same in magnitude, and will differ only in sign. The 

 same will be the case for the accelerating forces resulting from 

 the relative displacements of the molecules, which will be equal 

 and of contrary signs for the two opposite movements of the 

 primitive wave. Now this equality between the positive and 

 negative quantities contained in each complete undulation, is 

 sufficient in order that two systems which differ in their route 

 by a semi-undulation may mutually destroy each other when 

 they have besides the same intensity. Hence all the rays sensibly 

 inclined to G H will mutually destroy each other, and only those 

 which are almost parallel to it will concur effectually in the 

 formation of the resultant system of waves. 



They may then be considered in the calculation as having equal 

 intensities, and the integration be made between the limits of 

 positive and negative infinity in the two dimensions, employing 



* We may make the same observation with regard to the intensities of these 

 rays as with regard to the extent of the portions of A C which send them, by 

 remarking that the rays of the two consecutive portions differing only in in- 

 tensity by an infinitely small quantity of the first order, the intensity of the 

 rays of an intermediate part differ only by an infinitely small quantity of the 

 second order from the mean between the intensities of the rays of tiie two ad- 

 jacent parts. 



