[ 291 ] 



XXXIX. Continuation of the Hydro dynamical Theory of Vibra- 

 tions, By Professor Challis, M.A., F.R.S., F.R.A.S* 



32. "DEFOKE proceeding to the physical explanations which 

 Jl3 are the main object of this hydrodynamical theory, it 

 will be necessary to enter upon the discussion of certain points 

 which were omitted, or only alluded to, in the communication to 

 the August Number. One point in particular, mentioned in 

 art. 18, is the determination of the motion for the case in which 

 the directions of motion are, by the conditions of the disturbance, 

 perpendicular to a given plane. Let the motion be originally 

 produced by a plane surface of indefinite extent, always parallel 

 to the given plane, and oscillating in a given manner. It is evi- 

 dent that under these circumstances, however large the oscilla- 

 tions may be, the motion is wholly in directions at right angles 

 to the plane. As it does not appear to be possible to give an 

 exact solution of this problem on the principles about to be in- 

 dicated, an approximative method will be adopted, and the 

 reasoning will generally be restricted to terms of the first order. 

 The results thus obtained are sufficiently approximate for the 

 applications which will be madt* of them in explaining pheno- 

 mena of light, while at the same time they are necessary steps 

 towards a nearer approximation, if for other purposes it should 

 be required to take account of terms of a higher order. 



33. The exact equation (5) in art. 21, applicable to motion 

 along a rectilinear axis of free motion, becomes, after excluding 

 terms of a higher order than the first, the linear equation (6), viz. 



dt* dz* +0( P- U > 



from which was deduced (arts. 22 and 30) the particular solution 



2tt 



<fr = mcos~ (z—Kat + c) } 

 A 



k being put for the numerical quantity ( 1 -\ ) . Since the 



above linear equation has constant coefficients, the solution ex- 

 pressed in all its generality is 



2ir 

 c/> = 2) . mcos-— - [z — tcat + c), 



A/ 



an unlimited number of terms being included under the sign 5), 

 having different values of the constants m, \ and c. This equa- 

 tion gives, for the general value of the velocity (V) along the axis, 



d6 ^ . 27r . s 



V = -j- =2, ./ism— - (z—Kat + c), 



* Communicated by the Author. 



