PRESNKL ON DOUBLE REFRACTION. 275 



of the aether by the direct and the reflected waves, are respectively 



equal to «.sin27rJ^ — ^ + ^ j and to —a.sm2-!r{j - J^ — ^)- 



This second expression has necessarily the negative sign, since 

 the aetherial molecules remaining immovable against the reflect- 

 ing plane, the luminous vibrations also change their sign by re- 

 flexion. Consequently the absolute velocity resulting from the 

 superposition of the direct and reflected wave is, at the instant [t), 



«[sin2.(/-J + 0- sin 2.(^-^—0]; 



which expression may be put under the form 



2 a. sin 2 TT (^ j.cos27r. It — -). 



Such is the general expression of the absolute velocity which 

 animates at the instant {t) an aetherial molecule situated at a 

 distance {z) from the reflecting plane. It teaches us, in the first 

 place, that at certain distances from this plane, for which 



sln2T(-) =0, the aetherial molecules remain constantly at 



rest ; now sin 2 tt I - ) becomes nothing when ^r = 0, or a whole 



number of times - X ; hence the nodal planes, that is the planes 



of rest, are separated from each other and from the reflecting 



surface by intervals equal to -X. On the contrary, the bellyings, 



that is the points where the vibrations have the greatest ampli- 

 tude, have intermediate positions and at equal distances from the 



nodal planes ; in fact, sin 2 tt {-) attains its maximum when 



{z) is equal to an uneven number of times — \. 



The above formula may be used also to represent the mo- 

 lecular displacements by merely changing {t) into {t — 90^), or 



cos 2 IT (/ — -) into sin 2 TT y —V)\ i^ becomes then 



?/ = 2 6.sin2 7r. f ^j.8in27r ( ^ — 3^ }• 

 If (y) be taken as the ordinate corresponding to the abscissa (z). 



