Manchester Mevwirs, Vol. hui.{igi/i^), No. 1^. 5 



spherical polar coordinates, and thus have prepared^ the 

 way for the introduction of the corresponding stress 

 relations. In cartesian coordinates it was sufficient to 

 give two typical relations, each giving the fornn of three 

 others, but this will not suffice in the other cases, and it 

 will be necessary to find the relations in full. 



[In the paper as subnfiitted a statement of results was 

 given here. I have decided to reserve this statement for 

 amplification in view of its importance and to insert an 

 outline of the method of verification in its place. 



In cartesian, cylindrical polar, or spherical polar co- 

 ordinates we have three dynamical equations given in 

 (i), (28), or (37) of Part III.' 



For each system of coordinates we have six relations 

 between the elements of stress derived from the six geo- 

 metrical relations between the strains. 



In Part II.'" combined with Part I.'^ I have shewn that 

 the six relations between the elements of stress may be 

 obtained by writing in the expressions referred to in the 

 pages named from Part I., 



^-^f^l(^+^ + ^) ^""^ ^«^>' ^^^^ 



and ,5 for 2n^\\ , etc. 



The six dynamical relations between stresses only 



may be obtained by eliminating in every way the strains 



from the original dynamical equations and simplifying 



the results by making use of the six geometrical relations. 



In the case of cartesian coordinates the results are 



readily obtained as in (7). The other cases are more 



laborious. The present paper is now limited to the case 



' Manchester Memoirs., Vol. Iviii. (1914), No. 5, pages 6, 16, or 19. 



- Manchester Memoirs, Vol. Ivii. (1913), No. 5, page 2. 



■' Manchester Memoirs., Vol. hi. (1912), No. 10, pages 5 and 9 and 

 Tabic A. 



