Vibrations in Solid and Hollow Cylinders. 341 



Using in the small term (that containing & 4 ) the first ap- 

 proximation (1), we have 



F /3 =E^{i-^ 2 (« l 2 +«/)}; • • • (14) 



and this, as 



simply reduces to (3). 



That the above proof is as satisfactory in every way as one 

 based on ordinary elastic solid methods I should hesitate to 

 maintain. Unless one knew beforehand a good deal about 

 the problem there would, I fear, be considerable risk of mis- 

 adventure. 



§ 8. In illustrating the method in detail I have selected 

 the case of isotropic material simply because I did not wish 

 to frighten my readers. The assumption of isotropy almost 

 invariably shortens the mathematical expressions, and gene- 

 rally also simplifies the character of the mathematical opera- 

 tions ; and isotropic solids thus flourish in the text-books to a 

 much greater degree than they do in nature. When, how- 

 ever, the mathematical difficulties are trifling, as in the 

 present case, it seems worth while considering some less 

 specialized material. I shall thus briefly indicate the appli- 

 cation of the method to the case of material symmetrical with 

 respect to the three rectangular planes of x, y, z, taken as in 

 the previous example. In this case the stress-strain relations 

 involve, on the usual hypothesis, nine elastic constants. 

 Such quantities as Young's modulus or Poisson's ratio must 

 be defined by reference to directions. Thus let E x , E 2 , E 3 

 denote the three principal Young's moduli, the directions I, 2, 3 

 being taken along the axes of x, y, z respectively. There 

 are six corresponding Poisson's ratios, each being defined by 

 two suffixes, the first indicating the direction of the longitu- 

 dinal pull, the second that of the contraction. For instance, 

 rj 12 applies in the case of the contraction parallel to the 

 y-axis due to pull parallel to the #-axis. The order of the 

 suffixes is not immaterial, but there exist the following 

 relations : — 



WEi=W E »; WEj=WEi.J W E 3 = W*V (15) 

 The three equations answering to (4) are 



vjj fo dz dy dz= k * p Jj] frs-to^-^y}** *9 d ~, ( 16) 



Ei n)s^^^ = ^]Ij ^ ax ~ ri ^~" ni ^ dxd y dz ^ i 17 ) 



