522 The Rev. Professor O'Brien ofi the 



N and T representing the forces of resistance upon, and v the 

 velocity of any element of the aethereal fluid. 



I also showed that, supposing n and k to be positive, k is 

 essentially positive, which is a result of considerable import- 

 ance. 



I now proceed to examine what consequences result from 

 certain suppositions that may be made respecting the nature 

 and law of the resistance exerted by the particles of matter on 

 the vibrations of the aethereal fluid. 



18. The first supposition we shall make is this; that the 

 resistances do not interfere with the principle of the superpo- 

 sition of small motions, but that this principle holds good for 

 the aethereal vibrations in all cases, whether they take place 

 in vacuum or in the interior of transparent substances. In 

 this supposition we are fully warranted by experiment, and 

 we may proceed upon it with confidence. 



We have already, in article 16, stated two consequences 

 which result from the supposition here stated; the first of 

 them is obvious ; the second we now proceed to prove ; it is 

 this, that 



I=(C,-C3«^+C,n4- ) 



^=.n{C,-C,n^ + C,n^- ), 



where C,, C2, C3, &c. are certain constants depending simply 

 upon the constitution of the tether and of the transparent sub- 

 stance in which it is. 



19. Assuming the principle of the superposition of small 

 motions to hold, we may resolve the vibratory motion into 

 two components, one parallel to the axis of ^, and represented 

 by ^, the other parallel to the axis of j/, and represented by >j. 

 We may determine the laws of propagation of each compo- 

 nent vibratory motion separately and independently of the 

 other; and then, by superposing the two components, the na- 

 ture and laws of which we have thus determined, we arrive at 

 the true result. 



Proceeding in this manner, let us first suppose that the 

 whole vibratory motion takes place parallel to the axis of a:; 

 then jj will be zero, and the resistance will act parallel to the 



axis of .r, and can only depend upon the velocity -7- and its 



differential coeflicients -j^, -j-^, &c. &c. Now it is easy to 



show tiiat the principle of the superposition of small motions 

 cannot hold, unless the forces which are brought into action 



