on the Undulatory Hypothesis of Light* 481 



tions. In two directions at right angles to each other, corre- 

 sponding to the axes of the ellipse, the effective elasticities will 

 be at a maximum or minimum, and through a small variation of 

 direction may be considered to be uniform. Consequently in 

 these two directions the transverse motions of a polarized ray 

 can take place without disturbance; and accordingly rays whose 

 transverse motions are in, and parallel to, either of the two planes 

 which pass through the centre of the surface of elasticity and the 

 axes of the elliptic section, can be transmitted in the crystal. 



It remains to show that the velocity of propagation of each 

 ray depends exclusively on the effective elasticity in the direction 

 of the elliptic axis which coincides in direction with its trans- 

 verse motions. We have seen that fjX is constant for a ray of 

 given colour. Any cause, therefore, that augments or diminishes 

 X, will diminish or augment /x. Also since the quantity of 

 matter in the condensed part of a wave, exceeding that which 

 would occupy the same space in the quiescent state of the fluid, 

 must under all circumstances be constant for a given wave, any 

 cause which augments the condensations along the axis of the 

 ray will proportionally diminish X and increase fj,. If the effec- 

 tive elasticity in the direction of the transverse vibrations be the 

 same as that along the axis, the rate of propagation of the ray 

 will plainly be determined by that elasticity. But if the trans- 

 verse elasticity be altered in any proportion, the effect on the 

 transverse condensations must be to alter them in the inverse 

 proportion. For instance, if the transverse elasticity be increased, 

 the transverse condensations will be proportionally diminished, 

 the accession of elasticity being equivalent to the accelerative 

 force lost by the diminution of the gradations of density. But 

 the transverse condensations cannot be diminished without a like 

 diminution of the condensations estimated along and parallel to 

 the axis. Hence, from what is said above, X will be propor- 

 tionally increased, and ft proportionally diminished. Thus fi 

 depends wholly on the transverse velocity. 



It is unnecessary, after arriving at this point, to go through 

 the analytical process for finding the equation of the wave-sur- 

 face, as I have given it in a paper " On Double Refraction " 

 contained in the Cambridge Philosophical Transactions (vol. 

 viii. part 4), and the process is otherwise well known. Respect- 

 ing the theory of double refraction attempted in that paper, I 

 may here state that it rests fundamentally on the same principles 

 as the present one, but in the details of the argument is not so 

 complete. I will now only add two deductions from the theory. 



(1) An optical axis is defined to be such that the section of 

 the surface of elasticity by a plane at right angles to it is a 

 circle. Hence, by the foregoing theory, the effective elasticities 

 in all directions perpendicular to an optical axis are equal. Con- 



