Equations for Material Stresses, and their formal solution. 235 

 Thus we can write finally 



where Gr(.c) is nearly unity, and more accurately log Gr(#) 

 _*V1 1\ ( l^/l _1 \ _i^Yl_ 1 H 



~ 2nV a + ft / + 1 6 n* \a 2 /? ) 2 n \a b) J 



1 # 4 /l 1\ la 2 /l 1 \ 1 /l 1 ,\ ) /1A , 



The terms are arranged so that when # is of the same order 

 as ^/n, the first term in (10) is of zero order, the next { } is 

 of order 1/a (or ra~*), and the last { } is of order 1/x 2 (or n~ l ). 

 This remark may assist workers in selecting the order of 

 approximation required in special problems. 



The formula (10) reduces to (6), as it should, on writing 

 a = ^, &=i (and allowing for the fact that x here corresponds 

 to \x in Lord Rayleigh's work). It will be noted also that 

 (10) contains only terms corresponding to the first and second 

 cor recti on- terms. 



XX. The Equations for Material Stresses, and their formal 

 solution. By R. F. Gwyther *. 



IX this communication I offer statements of the Stress 

 equations based on the " Doctrine of Material Stresses" 

 contained in your issue for June 1918 ~\, and the formal 

 solution of those equations in Cartesian coordinates. In the 

 Cylindrical Polar system, and especially in the Spherical 

 Polar system, the expressions become long and complex, and 

 I have not dealt with those cases in full. 



The special Stress equations of which I treat are derived 

 from the three Mechanical Stress Equations by eliminating 

 separately all but one of the separate stresses by means of 

 the six stress conditions arising from the doctrine which I 

 have proposed. It is to be noted that this mode of treat- 

 ment includes the Elastic Stresses, although not necessarily 

 confined to these stresses only. 



As is indicated in the paper referred to, T propose to 

 include among the forces acting per unit of volume of the 



* Communicated by the Author. 



f Phil. Mao-. V ol. xxxv. p. 490, June 1918. 



R2 



