ACTED UPON BY THE VIBRATIONS OF AN ELASTIC MEDIUM. 385 

 or, putting v'p for vp + —7— Sr, 



dr 



m*St. 



1 



r~v p — rvp 



Now if any point be selected between P and P, the radius to which at 

 the time t is r t , by what has been already shewn, the transverse section of 

 the cone through this point at the time t + St is with sufficient approxima- 



»«V, 2 



tion — -'- , and is therefore independent of St. Hence at any instant 

 during the interval It the content of the conical frustum is • f—jr dr, 



'ill* 



{from r t = r to r t = r'\, or — (r' 3 - r»). The increment of density (Sp) 

 in that space in the time St is consequently, 



m 2 St(r'°p'v - r'pv) 3r 



s 



r~ m? (r' 3 — r 3 ) ' 



Hence, fe + ? (^ ~f> ) - „ 

 d£ r 3 — r 3 



and passing from differences to differentials, 



&-33£*«~r f <"■ 



It is plain that since i 3 has been assumed to be a fixed point of 

 space, the differential coefficients here are partial. The above equation, 

 with 



P = <*P (2), 



and that derived from D'Alembert's Principle, viz. 



£♦<&-■ » 



are the three eqviations which determine the circumstances of the motion. 

 As the velocity (a) of the centre C in no way enters into them, we 

 may conclude that the same equations apply to motion tending to or from 

 a moving centre as to motion tending to or from a fixed centre. 



002 



