958 DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



solution. The transitory part being defined by functions of xja, yja, z/a, it follows 

 that diflerentiation as to any of the coordinates lowers the order of any term by one. 

 In the main part this is still true for differentiation as to x or y, but ^-differentiation 

 does not in this case alter the order, the terms of any one order being functions oi xja, 

 yja, and z. 



(c) Thus, away from a critical section all the first derivatives of all the displace- 

 ments, as calculated from 38 (9), (12), are correct so far as they go. 



At a critical section all the first derivatives are still correct except dujdz in (9) 

 and duJdz, dUyjdz in (12). 



{d) It is important to notice, however, that in the case of force parallel to the axis 

 the strains e^^ and e,,^ are not correctly obtained by differentiating (12). The reason 

 is that the terms of order 1 in diijdz and dUy/dz in (12) are of the same order as the 

 terms in dujdx and dujdy arising from the leading neglected terms of the asymptotic 

 solution. 



Hence the stresses zx, zy in the case of force parallel to the axis are not correctly 

 calculable from the approximate displacements in (12). 



(e) For a precisely similar reason the dilatation, and consequently the stresses xx, 

 yy, zz, are not correctly calculable, in the case of force perpendicular to the axis, from 

 the approximate displacements (9), the term of highest order in zz being at fault 

 and XX, yy being incorrectly given, though of the right order, namely 1. 



{/) Excepting the strains e„ and Sy^ from (12), the stresses xz and yz from (12), 

 and the stresses xx, yy, zz from (9), the strains and stresses as calculated from (9) and 

 (12) are correct so far as they go, away from a critical section. 



[g) The strains and stresses from the transitory perturbations are of the order a 

 at the critical sections. The stresses in (10) and (13) are therefore correct even there. 



42. Discontinuities of a more general kind in the applied force. 



It is obvious that the method of the preceding Articles (37, etc.) can be applied 

 when the force or some of its ^-derivatives is discontinuous at any finite number 

 of 2-planes. From 39 (4) it is easy to see that the transitory perturbations will 

 depend on the differences of the values of the force (and of its successive ^-derivatives) 

 on the two sides of the section of discontinuity. 



A similar method can be used when the surfaces of discontinuity are no longer 

 2;-planes, but have a more general form. In such cases it is preferable to modify 

 the process at Art. 37 (G), (7) by integrating over the cross-section first. It may 

 happen that these integrals and their ^-derivatives are continuous functions of z, 

 in which case no transitory perturbations will be introduced. In general, the 

 asymptotic solution would fail at a point where the tangent plane to the surface of 

 discontinuity is perpendicular to the axis. 



