368 PROFESSOR CHALLIS, ON A THEORY OF LUMINOUS RAYS 



The equation (8) is satisfied by (p = \l/{t)cos iw^ + xWi ; and the equation (9) by (j) = (p{nx -na't). 

 Hence \^ (<) = constant — , and -^(t) = - na't. Consequently (p = — cos n {x - a't), and the 



velocUy — ^ = "' sin n (a t - x) = m sm — (a t - z) suppose. 

 dz X 



It results from the foregoing reasoning that if the small vibrations of the fetlier in the direction 

 of propagation follow the law expressed by the equation last obtained, the condensation in any plane 

 perpendicular to an axis of rectilinear propagation may vary at a given time from point to point, 

 and at the same time the propagation be uniform. A consequence of this result is that a very 

 slender cylindrical portion of the aether may continue in agitation while the contiguous portions are 

 at rest ; and since the law above obtained is that which has been found by experience to apply to 

 the pha?noniena of light, the existence of rays of light, which was proved experimentally at the 

 commencement of this paper, is accounted for theoretically. 



As far as we have hitherto proceeded, the function / has remained indeterminate. The con- 

 siderations I am now about to enter upon will serve to ascertain its form. Take a plane perpen- 

 dicular to the axis of x, in which the velocity parallel to the axis of a; is a maximum, and in which 

 consequently ii = 0, and v = 0. As the motion at any point of this plane is parallel to the direction 

 of propagation, and as the velocity of propagation is uniform, it follows that an equation like (5), 

 applicable to this point, is obtained by simply substituting f(j) for <p. Substituting also a" (1 + k) 

 for a'-, we have, 



/l?f-L.i.)-'/S (.»)■ 



dz \R R'l •' dz" 



■ ^1 ' • 

 At the same time the general expression for „ + "S' g'^es, 



\R '^ R-J <p' dz da^ "^ df /■ Uw' dfl 



•' \dx^ df 



Hence by substitution in equation (10), 



dz- k 



Consequently, by comparison with equation (8), 



1 



^ + f.i_4 + _4/d, = o (u). 



,(12). 



The function / must consequently be such as to satisfy this equation. 



Again, as the phaenomena of light shew that a ray of common light has similar relations to space 

 in all directions perpendicular to its axis, the function /, which is arbitrary, to apply to this kind of 

 light, must be assumed to be a function of the distance from the axis. That is, if r' = .r" + y , 



