TRANSACTIOXS OF THE SECTIOXS. .j 



A mode of satisfying the differential equations (2), and at tlie same 

 time of representing a large class of the phenomena of light, is to as- 

 sume, 



in which r, v\ C are constants, depending on the extent and direction 

 of vibration : a, b, c, are the cosines of the inclinations of the direction 

 of propagation of a plane wave to the positive semi-axes of x, ?/, z ; v 

 is the velocity of propagation of that wave, and \ is the length of an 

 undulation ; and tt is the semicircumference of a circle, of which the 

 radius is unity. With this assumption (5.), and with a natural and 

 obvious supposition respecting a certain symmetry of arrangement in 

 the ether, causing the sums of odd powers to vanish, it is permitted to 

 substitute in (2.) the expressions 



d- ox /2 ir 7;\„ „ 



ITF - -(-X-)"*^"' (6-) 



A (J,r = — vers. A d. Sx, /y.) 



in which A = — (aAaj+AAy-fcAz); (§,) 



and thus arises a system of conditions of the form 



r (^^y = ^> ^ S. {/(.) 4- ^V (r) } vers. A 



+ n mS ^/' (r) vers. A d 



+ CmS. /' (r) vers. A 9 (9.) 



the masses m^ of the etherial particles, being supposed each = ,«, 

 Three conditions of this form (9.) exist for every particle, and deter- 

 mine, in general, for any given values of a, b, c, X, that is, for any 

 given direction of propagation, and any given length of wave, the value 

 ot V, and the ratios of C, v\ C, that is, the velocity of propagation of 

 the wave, and the direction of vibration of the particle. Accordin^^Iv 

 with some slight differences of notation, they have been proposed for 

 this purpose by Cauchy, and adopted by other mathematicians. Sup- 

 pose now, for simplicity, that the plane wave is vertical, so thatc = o • 

 and let, at first, the direction of its propagation coincide with the posi- 

 tive semi-axis of x, so that b also vanishes, and a is = 1. Then for 

 transversal vibrations, tlie expression for the square of the velocity of 

 propagation is ^ 



'' = (2^.)^ - S {/(O + ''^^f (r) } vers. ^^ , ^,0.) 



which appears to extend not only to the interplanetary spaces, but also 

 to all ordinary transparent media, and contains, for them, the theore- 



B 2 



