300 Mr. Challis on the Forms of the Arbitrary Functions, Sjc. 



portional to the velocities and condensations, cuts the axis 

 of ^, 



u = fit 5 = ?n X sin - (jr — c if), for the positive direction of 



propagation. 



7; = — as ■=■ — m "Kwix- [x -\- at\ for the negative. 



A similar reasoning may be applied to the equations for the 

 motion in space of three dimensions, to those for vibrating 

 chords, and indeed to the equations which M. Poisson, in a 

 recent Memoir {Acad. Scien. tom. viii.), has obtained, by a 

 very general consideration of the interior constitution of 

 bodies, for the small vibrations which any homogeneous sub- 

 stance whatever performs, in virtue of its elasticity. In all 

 these instances the same primary form of the arbitrary func- 

 tions would be found ; and the universality of the kind of vi- 

 bration that this form indicates, affords some kind of reason 

 (for none has yet been given) why it has been successfully em- 

 ployed in the undulatory theory of light. 



It is necessary to know the primary form of the arbitrary 

 function in all cases in which the vibrations are immediately 

 impressed on the fluid by the fluid itself: for instance, when 

 a uniform current blown across the mouth of a cylindrical 

 pipe, puts the column of fluid in its interior in vibration. But 

 when the vibrations are caused by the motion of a solid in the 

 fluid, it will be true that every elementary portion of the mo- 

 tion may be considered a ver}' srmW portion of a vibration of 

 the primary kind, and will be subject to the same laws and 

 limitations as the primary vibrations ; but the total motion will 

 receive its character from the motion of the solid, which mo- 

 tion determines for the particular case the form of the arbi- 

 trary function : for the equation v ■= a s will obtain with re- 

 spect to the fluid in contact with the solid. All this is a con- 

 sequence of the discontinuity of the motions, which is suffi- 

 ciently proved to exist by the presence of arbitrary functions 

 in the integral. 



It may be remarked that either of the equations = a 5, 

 V =. — as, shows that where s is negative or the fluid is rare- 

 fied, the velocity of the particles is contrary to the direction of 

 propagation, and where it is condensed the velocity is in the 

 dii'ection of propagation. This explains how it is that the 

 same disturbance, for instance, the motion of a small solid for- 

 ward in the fluid, produces propagations in opposite direc- 

 tions. For the solid must condense one part just as much as 

 it rarefies another, but impresses motion on the particles in 

 the direction in which itself moves; therefore the condensa- 

 tions and rarefactions will be propagated in opposite directions. 



The 



