ihe Vibrations of an Elastic Fluid. . 141 



Regard being had to the meaning of the suffix n, this result 

 shows that we may suppose that Y(/jl) =^m cos (kfi-\- c) : whence 

 it will be found that 



<£ = mcos-| kl/n — p J +c>; 



6 



or putting q for k — j, and substituting the values of fi and v } 



<f) = m cos q ( z — at \J \-\ — ^ + °' ) • 



The consideration of the function G by itself would similarly 

 conduct to the same form of <£, with the difference only of a po- 

 sitive sign before at, indicative of propagation in the opposite 

 direction. We may hence infer the possibility of the coexistence 

 of two series of waves propagated in opposite directions. 



Thus we have been led, step after step, by the indications of the 

 analysis, to a special and definite expression for the function (/> ; 

 and as this result is antecedent to any supposed disturbance of 

 the fluid, it must be taken to be expressive of a law of the mutual 

 action of the parts of the fluid. Further, it may be remarked 

 that every finite expression involving trigonometrical functions 

 is to be regarded as an exact algebraic expression, because, as is 

 known, these functions all admit of being expressed in terms of 

 exponential quantities. Thus the particular solution of (6) which 

 we have obtained is finite and exact, and in this respect stands 

 out from all particular integrals of the same equation, which, as 

 the foregoing reasoning shows, can only be expressed in infinite 

 series. Upon the general principle previously enunciated, this 

 analytical circumstance is significant, and may be interpreted as 

 meaning that the fluid is susceptible, by its own action, of a 

 special kind of motion of which this unique result is the exponent. 



23. Another inference of a remarkable character may be drawn 

 from the above particular solution, viz. that there is no other solu- 

 tion of equation (5) which gives a constant velocity of propagation. 

 This theorem admits also of direct proof as follows. Let us sup- 

 pose for the moment that cj) = ~F(z — a l t), a x being some constant. 

 Then we have 



g-PC-.O. J=- fll P(*-M. 



But for the motion under consideration along the axis of z 3 the 

 general equation (1) becomes 



«3Nap.logp+f+|g=0 ; 



whence it follows that p is also a function of z-*- a x t. Thus the 

 above supposition is equivalent to assuming that a given state of 



