2/8 FRESNEL ON DOUBLE REFRACTION. 



On differentiating twice the value of y, we find 



Hence the accelerating forces, and consequently the velocities 

 impressed at each point of the curve A B C, at the instant ■when 

 the oscillation recommences, are proportional to the correspond- 

 ing ordinates ; therefore the small spaces described during the 

 first instant will also be in the same ratio and will not alter the 

 nature of the curve ; hence, after the first instant d t the new 

 accelerating forces will still be proportional to the corresponding 

 ordinates ; and since the acquired velocities are so likewise, the 

 spaces described during the second instant will still preserve 

 amongst each other the same ratio. The same will hold true 

 after the third, fourth instant, &c. Consequently all the points 

 of the curve A M C will arrive at the straight line ADC toge- 

 ther, from which they will afterwards deviate by quantities equal 

 to those of their primitive deviation, to re-commence afterwards 

 an oscillation in the contrary direction. We see that the law of 

 these vibrations will be similar to that of the small oscillations 

 of a pendulum, since the accelerating force which urges each 

 material point is always proportional to the space which remains 

 for it to describe in order to arrive at its position of equilibrium. 

 Hence the duration of the vibrations will be in the inverse ratio 

 of the square root of the elasticity of the medium, an elasticity 

 which is measured, in the case we are considering, by the energy 

 of the force resulting from the relative displacements of the 

 parallel strata of the medium, supposing them equal to a small 

 constant quantity taken for unity. 



It is easy to see also that the duration of the oscillations of 

 the point M will be proportional to the length (A) of an undula- 

 tion. In fact, to compare the durations of an oscillation corre- 

 sponding to different values of (^), we must always suppose dz 

 constant, in order that, the distances being the same, the mole- 

 cular actions and the masses to be moved may be similar on one 



part and on the other. On substituting for sin ( 2 x . - j its value, 



in the expression for d'^y, we have 



always supposed the sphere of activity of the elastic force to be infinitely small 

 with regard to the extent of the disturbance. [No such note to the memoir.- 

 Translator.] 



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