20 -A-rt. 7.-K. Terazawa: 



different for different materials; the otlier which was recommended 

 by G. H. Darwin asserts that, as mere h^'drostatic pressure can 

 hardly affect the case, the maximum difference of the greatest and 

 the least principal stresses should be less than a certain limit. ^^ 

 These two hypotheses lead, in general, to different results. Eitlier 

 of them will give w^arning that danger is l)eing approached, and in 

 any case a certain factor of safety must, in practice, be adopted. 

 Here we shall calculate the limits following from these two 

 h^^potheses and compare corresponding results. 



For tins purpose we liave, in tlie first place, to find the 

 distril)Ution of principal stresses throughout the body concerned. 

 Let ^Y;, jVo, Ns denote the values of principal stresses at the point 

 (/', ^, z). Owing to the hypothesis of symmetry round the axis of 

 z, the component 66 is one of the principal stresses, say J\\, as is 

 to ])e seen from the formulae (31); and any plane passing through 

 the ;i-axis is one of the princi])al planes of stress. The other two 

 principal stresses will be found by 



^1 = -J- {>r + 2z) + -^J{rr-zzy + ^rz\ 



N:. = ^{^■r + zz)-^J(f?-^f + 4fP. 



(48) 



2 ^ ^2 



At the surface, since rz = (), the stress components Pr,66,zz them- 

 selves are the principal stresses. 



§19. Now, to apply these formulae to this example, let 'us 

 assume"'^ that the pressure modulus >^ is so great compared with the 

 rigidity /^ tliat the material may be considered to l>e incompressible. 

 Thus, substituting tiie values of the stress-components found in 

 (4-4) in the formulae (48) and making ?. = x, we have simply 



// Sz + a 



N,= 



Qrr 



N, = N. = 



J J ^^ a 



~2^' -(r'+iz + afy- 



1 There is another view often adopted, in which a limitation on strain is taken as the 

 measure. 



2) This supposition is not at all necessary, but it makes the calculation extraordinarily 

 simple. 



