344 Dr. C. Chree on Longitudinal 



§ 10. If a and b are the radii of the outer and inner cylin- 

 drical surfaces respectively, then from the conditions which 

 hold over these surfaces we must have 



*+%-*> ™ 



(m-n)B + 2np=0, .... (34) 



when r = a, and when r=b. 



As regards the terminal conditions we should have, follow- 

 ing the ordinary view of longitudinal vibrations, 



M? = over a fixed end, .... (35) 

 a free end. (36) 



zz = (m — n)$-\-2n—=0, 



> over a 



^(S+f)=° * 



We have no means of satisfying these terminal equations 

 by means of the present solution save by selecting suitable 

 values for p and e. Clearly if both ends e = and z = l be 

 fixed we accomplish our object by putting 



e = 0, p = iirjl. 



If, however, z = l be a free end, while ^ = is a fixed, we 

 must have e = to satisfy the conditions of the fixed end ; 

 and this leaves us with 



zz cc cos pi, 

 zr oc sin/>Z 



over the free end. This is the difficulty we have already 

 indicated in § 4 ; and it is in no respect peculiar to hollow 

 cylinders, and need not further concern us at present. 



§11. In dealing with the surface conditions, brevity is 

 effected by the use of the notation 



\=>v/a 2 -p 2 , /xeee V73 2 -;? 2 , . . (37) 

 whence 



a 2 _/3 2 = \ 2 -Ar. 



After simplifications, into which I need not enter, the elimi- 

 nation of A, A', C, C from the four equations holding over 

 the cylindrical surface supplies the determinantal equation 



