402 Mr. O'BRIEN, ON THE PROPAGATION OF LUMINOUS WAVES 



it is clear that it must be very small at the Earth ; for it must vary 

 inversely as the square of the distance from the Sun ; hence we are safe 

 in supposing that there is very little condensation and rarefaction in 

 the undulations of ether which constitute the solar light we have to 

 do with ; and if the vibrations be transversal, as we have every reason 

 to suppose them to be, this is still more evident ; for transversal vibrations 

 cannot be propagated unless the variation of the density of the ether 

 caused by the motion be extremely small. 



If there be very little condensation or rarefaction in the ether, it 

 is clear that the relative motion of any two contiguous particles must be 

 very small, compared with their actual distance from each other. Indeed 

 if this be not true, the principle of the superposition of small motions 

 cannot be applied to the etherial undulations, and the whole undulatory 

 theory must fall to the ground ; moreover, the velocity of light in vacuum 

 cannot be uniform, as we know it to be. Hence I think that there 

 is just the same degree of assumption in supposing that the relative 

 motions of the etherial particles are very small compared with their 

 actual distances, as there is in supposing that light consists in a succession 

 of undulations. I make these remarks because I have heard objections 

 urged against the simplification I am now about to make in the equations 

 of motion, and which has been made by every author I am acquainted 

 with, under similar circumstances. 



Proceeding then upon the assumption, if it may be so called, that 

 the relative motion of two contiguous particles must be very small, 

 compared with their actual distance from each other, it is evident that 

 la, 1(3, ly, and p, must be very small, compared with Ix, ly, 1%, and r\ 

 hence since 



r 2 = lx> + If + lz\ and (r + pf - {$* + lay + {ly + &&> + (j« + l y y, 



we have very nearly, (subtracting, and dividing by 2r,) 



P = -(Ixla + lylp + lzly); 



therefore since fir + p) (Ix + la) = \f(r) + P f(r)\ (Ix + la), we have 

 (neglecting pla and putting for p its value) 



