96 Prof. Challis on the Principles of Hydrodynamics. 



other respects the calculation is analogous to that exhibited 

 above. 



This proposition may also be viewed in another manner. Since 

 for small values of r,/=l— er^= cos \^2er nearly, we have the 

 approximate differential equation 



That is, for small values of r, / is given by a linear equation with 

 constant coefficients, which is not the case without that limita- 

 tion. We may therefore suppose the motion in a rigid tube 

 of small transverse section to be made up of an infinite number 

 of normal motions, the axes of which are always in fixed direc- 

 tions determined by the boundaries of the tube. The trans- 

 verse motions will thus be destroyed, and the resulting motions 

 may be in lines passing through centres, or through focal lines. It 

 should here be remarked, that the composition of motion em- 

 ployed above has no reference to any physical circumstance, but 

 is simply an analytical process for representing the motion, the 

 number of components being supposed infinite, and that for this 

 reason it was necessary to prove first that / was given by a linear 

 equation with constant coefficients. Similarly, we found in a 

 former proposition that the general value of / is given by the 

 equation 



/= l^aBd{w cos6-\-y sin 6), 



the sum being taken from ^=0 to 6=2'n; and the number of 

 terms being infinite. But it must not be inferred that the normal 

 motion is physically so composed. 



From either of the two preceding modes of viewing the 

 question we may infer, that as the motion is composed of parts 

 of normal vibrations for which m, X, and c may be regarded as 

 constant through an indefinitely small space and for an indefi- 

 nitely small time, the velocity of propagation is the constant /ca, 

 this quantity being independent of any arbitrary circumstances. 

 We may not, however, conclude that the relation between the 

 velocity and condensation is the same as in the normal vibra- 

 tions. Conceive the axis of a set of normal vibrations to coin- 

 cide with the axis of a cylindrical tube, and the transverse vibra- 

 tions to be every moment destroyed by an impressed accelerative 

 force equal and opposite to the effective transverse force. The 

 free motion will thus be reduced to the motion in the rigid tube, 

 the impressed force having the same effect as the reaction of the 

 boundaries of the tube. This impressed force, being transverse, 

 will not alter the rate of propagation, but the relation between 

 the velocity and condensation will be changed, as appears from the 

 reasoning in Proposition XII. In the equations obtained in that 



