296 Prof. Challis on the Hydrodynamical Theory of Vibrations. 



and if this quantity be called f-r-, the motion transverse to the 



df . 



axis is represented by $-j- s r being any distance from the axis. It 



is particularly to be observed that, by the demonstration in art. 

 28 (Phil. Mag. for August), the free motion is exclusively that 

 which is symmetrical about the axis, and the condensation and 

 transverse velocity are consequently functions of r. The con- 

 densation at any point of the axis at any time being ~- } 



that at any position in the same transverse plane at the same 

 time is — ~ • -~. From these expressions, and by taking 



account of the properties of the function f, it will be seen that 

 there are positions of maximum condensation, positive or nega- 

 tive, and of no condensation, and intermediate positions of 

 maximum transverse velocity, positive or negative, and of no 

 transverse velocity, in fixed cylindrical surfaces about the axis, 

 and that the number of such surfaces is unlimited. The maxima 

 both of condensation and of transverse velocity continually de- 

 crease with the distance from the axis ; and the intervals between 

 their consecutive positions, as also the intervals between the con- 

 secutive positions of no condensation and no transverse velocity, 

 go on decreasing till they reach a certain limiting value, which 



by the reasoning in art. 30 was ascertained to be — =. Now, 



Ve 



although this be the character of free motion, it by no means 

 follows that the motion relative to an axis is necessarily such. 

 In fact, the partial differential equations (10) and (11), by which 

 the values of/ and a are determined, prove that the motion may 

 be subject to arbitrary conditions. But as the motion in every 

 instance exists originally in the form of free motion, the arbi- 

 trary motion is subject to the limitation of being such only as 

 may result from modifications of that motion. For instance, 

 conceive that the free motion relative to a given axis is by some 

 disturbance separated into two parts; which is an allowable sup- 

 position, because the equations determining the motion are linear 

 with constant coefficients. The parts may or may not subse- 

 quently follow the same course. The resolution must take place 

 in such a manner that the sum of the condensations and the 

 sum of the longitudinal motions, at corresponding points of the 

 two parts, must be equal respectively to the condensation and 

 longitudinal motion at the corresponding point of the original 

 motion ; and the sums of the resolved parts parallel to the axes 

 of x and y of the two transverse motions, must be equal respect- 



