Manchester Memoirs, Vol. Ixii, (191 7) No. 1 7 



It is usual to require that the stresses 4^, ^F 2 , U'V, are non- 

 existent ; it should, however, be pointed out that if all nine (of ihe 

 elements of stress are existent, we may either express the element? 

 of stress in terms of first differential coefficients!, or vice-versa. 

 but if the ^-stresses are nonexistent, the methods are not 

 reversible. We can express the remaining six elements of stress 

 in terms of the nine differential coefficients, but cannot express 

 the nine different coefficients in terms of the six elements 

 of stress. 



There appears to be two' cases to mention. Firstly, that in 

 which we omit all reference to' the rotations £, ??, 4, as. well a? 

 to the stresses ^^^o, ^F 8 . This would lead to' a simplification 

 but would find no' justification, either on physical or geometrical 

 grounds. The second remark is that if we confine ourselves to 

 terms of the first order of differential coefficients, Lame's state- 

 ment of the elastic force takes the form 



8x\8x 8y 8z J \8y 8z 

 etc. 



and that the inclusion of a force due to the elastic -^-stress, 

 would not alter the form of the expressions, although it would 

 affect the .determination of the values of the constants. 



It is also worth while to note that no reasons are given for 

 excluding the ^F -portions, of the stresses from the tractions which 

 can be applied to the surface of the body under consideration. 

 The grounds for omitting these portions of the stresses are that 

 the body would be unable to sustai'n the elemental couple which 

 would act throughout the body as a mathematical consequence 

 of such stress components. In other words, that fracture would 

 result or in some way the ordinary mathematical methods would 

 become inapplicable. 



APPENDIX B. 



On the Formation of the Fundamental Equations. 



If we follow Newton's statement of the Laws of Motion, the 

 formation of the equations fall into two parts— the estimate of the 

 forces causing a rate Of change of momentum, and the expression 

 in proper terms of the corresponding rate of change of 

 momentum. 



In the case of changes of internal momentum, due to 

 tractions exerted on the surface of a portion of a material 

 body, we need to make preliminary assumptions : — that the 

 ordinary mathematical processes, such as those of differ- 

 entiation and integration apply to the problems, and also 

 that the conditions applicable to special theorems, such as 

 Green's Transformation, may be freely made use of. The 



