4(50 PROFESSOR KNOTT ON THE STRAINS PRODUCED IN IRON, 



Now we may write 



Su/v = \ + 2fi (1) 



where m represents the transverse dilatation of the core. It may be regarded as 

 measuring the elongation at each point of the inner surface of the tube in the direction 

 perpendicular to the axial plane passing through the point. If we suppose 8 to be the 

 cubical dilatation at this point, we have the equation 



S = X + v + v (2) 



where v is the elongation in the direction of the radius. 



Similarly, ^(SV/r' — X) gives //, the " tangential " elongation at the outer surface of 

 the tube ; and then the radial elongation is given by 



„' = 8-X-V (2) 



In calculating these strain coefficients I make the two assumptions : — First, thai 

 S, which really measures the average cubical dilatation throughout the metal, also 

 measures the cubical dilatations of the elements at the surfaces ; and second, that \ 

 has the same value at every point of the tube. There seems no way of testing the 

 truth of the first assumption ; but the second was tested by direct experiment, and no 

 indication was found of A having different values at the outer and inner walls. 



The precise significance of the ratios X, n, p } //, v' may be thus indicated. Imagine 

 a small spherical element of diameter 2e at the inner surface of the tube. After the 

 application of the magnetic stress this sphere becomes an ellipsoid, whose principal 

 axes are 2e( 1+ A) in a direction parallel to the axis of the tube, 2e( 1 + v) in a radial 

 direction, and 2e(l +m) in a direction at right angles to these — that is, tangential. The 

 ratios 1+X, l + i/, l+//have similar meanings for an originally spherical element at 

 the outer surface. 



Again, if r, R, are the inner and outer radii of the tube, r/x and R// represent the 

 outward displacements of the corresponding surfaces. 



Although it is not possible to calculate accurately these strain coefficients in the 

 case of the large tubes, we may obtain an approximate estimate of their values on the 

 further assumption that the cubical dilatations are the same for all tubes of the same 

 metal. Thus, since V + v = v', we have SV + Sv = S?/ in any given field. Hence 



SV Sv v SV v' 

 T v V " V- V 



or, 



^|s + (X + 2^}=A + 2m' ... (3) 



Now, in the case of the large tubes, X and X + 2ju, arc measured by direct experiments, 

 and the volumes v, r', V are known. Consequently, if S be assumed, the value of 

 X + 2/x' at once follows. Hence m and // may be calculated. The values of v and v 

 arc then found from the equations (2) above. 





