298 Prof. Challis on the Hydrodynamical Theory of Vibrations. 



41. From the above results it may be concluded that, if the 

 disturbance be such as to separate the original condensation into 

 two unequal parts, there will be equal transverse motions in 

 rectangular directions corresponding to the less, and a portion 

 of the larger equal to the less, while the other portion will remain 

 symmetrically disposed about the axis, just as in the case of free 

 motion. 



42. The identity of the values of of for the original motion 

 and the components is found to extend only to the squares of x 

 and y. Hence this resolution into rectangular motions in fixed 

 directions applies only to motion at distances from the [axis so 

 small that the equation /= 1 — er 2 gives the value of/ with suf- 



4r 2 

 ficient approximation. Hence er 2 , or -r-o, must be a very small 



X 



fraction, and consequently r very small, compared to. ^. 



Each of these components might also, under certain circum- 

 stances, be resolved into two parts. The laws of this second 

 resolution I have treated of in a paper " On the Theory of the 

 Polarization of Light," in the Cambridge Philosophical Transac- 

 tions, vol. viii. part 3. The mathematical reasoning is too long 

 for insertion here. 



43. In art. 34 it is supposed that the oscillating plane which 

 produces the motion is of unlimited extent, in order to avoid the 

 consideration of the lateral motion which would take place near 

 the borders if the plane were finite. Let us now suppose that 

 the parallel axes of the component motions are included within 

 a limited space, for instance a cylinder of given radius, and 

 endeavour to ascertain the kind of motion which prevails at and 

 near this boundary. It is evident that, since at these parts the 

 transverse motion is only partially destroyed, the motion must 

 there be compounded of transverse and longitudinal vibrations. 

 This motion, however, as the following considerations will show, 

 does not spread laterally to an indefinite extent, but is always 

 confined within certain limits. Having regard to the applica- 

 tion proposed to be made of these hydrodynamical researches, 

 let us conceive the values of X to be very minute, and the velo- 

 cities of the fluid particles to be extremely small compared to 

 the velocity of the propagation of the waves. Then taking 

 account of the characteristics of the component vibrations 

 described in art. 39, it will be seen that, although the vibra- 

 tions relative to each axis individually are not limited as to di- 

 stance from the axis, a limit to the compound motion is imposed 

 laterally by the composition of the vibrations. Admitting that the 

 number of the axes of the components within a given space may 

 be as* large as we please, since the vibrations are by hypothesis 



