AS EXPLAINED BY THE HYPOTHESIS OF FINITE INTERVALS. 179 



I do not mean, however, to consider the above reasoning as appli- 

 cable to all possible arrangements of particles; it might happen, and 

 very probably is the case, that a differently constituted medium from 

 that which we, merely for the sake of simplicity, have imagined, would 

 lead us to results different from the above. As my object is not now to 

 examme every possible circumstance attending the motion, but merely 

 to lUustrate one particular view, I shall proceed on the hypothesis of 

 the arrangement in cubical forms. We will then endeavour to ascer- 

 tam from an examination of our mathematical results, what assistance 

 analysis affords us in the investigation of the law of transversality of 

 vibration. I shall here assume that the law of the inverse square of 

 the distance has been proved; and shaU adopt the same notation which 

 I apphed to the investigation of that property. Suppose, then, to fix 

 the ideas that the wave is transmitted along the axis of y. 



If V, v', v" be the velocities of transmission respectively of vibrations 

 whose motion is parallel to x, y, %, we have 



= ^.x.^:±5z£3in^z- 



V 5-'-^- r Sin — - 



£ r' X 



= M 2 £+i!n^sin^-'' 

 3 • ■^ • 1:5 sin — ^ — , 





= !^-.2.Il±4^sin^^, 



Z2 



