96 Mr. 0. Chree on Bars and 



5 

 however, the rigidity n alone vary, and m be greater than ^ n 



tn ^~7t 1 

 (i. e. — | > k), the pitch is highest when the least rigid 



end is fixed. For the special case q=p we have 2H=</ +p ; 

 this signifies that, to get the highest pitch, we must fix that 

 end for which the product of the elasticity into the density is 

 greatest. 



Again, the pitch of the fundamental tone is higher or lower 

 than that calculated from the material at the fixed end ac- 

 cording as 7r 2 K + 4H is negative or positive. When q=p, it 

 is easily seen that 



2(7r 2 K + 4H) = (7 , (7r 2 + 4)- i 9(7r 2 -4) ; 

 so in this case the calculated pitch is too low or too high 

 according asp(7T 2 — 4) — cr / (7r 2 + 4) is positive or negative. 



The torsional vibrations of aright circular isotropic cylinder 

 are obtained by putting u = = w, and assuming v independent 

 of 6. Thus (1) and (3) are identically satisfied, and (2) 

 becomes ^ I^_ji.^_£^ 



dr 2 + rdr r 2 + dz 2 ~ n dt 2 ' ' ' ' [ °> 

 If we assume v x r, this becomes 

 dh__p_d?v 

 dz 2 ~ndt 2 > ^ L) 



dv 

 while the accompanying stresses all vanish except S = w-r-. 



Thus the conditions over the cylindrical surface are identically 



d/V 

 satisfied, while by taking -=- — over a free terminal section 



we leave no stresses there. In this case, then, the conditions 

 in the interior and at the surface may be simultaneously 

 satisfied, and the theory of the vibrations so deduced is per- 

 fectly satisfactory. 



If we suppose n substituted for M, the internal equations 

 for the torsional vibrations are identical with those we assumed 

 for the longitudinal, while the conditions at a fixed or a free 

 end are exactly the same, interchanging v and w. Again, at 

 a surface z = l, separating two different materials, the surface- 

 conditions are v^ — v^ and ^i-t^ =^2^- Thus the methods 



to be employed in the corresponding cases are identical ; while 

 the occurrence of only one elastic constant n, the rigidity, 

 simplifies the case of the torsional vibrations. We shall 

 therefore merely state the results for the several problems 

 analogous to those of the longitudinal vibrations. 



