Manchester Memoirs , Vol. Iviii. (19 14), No. 5. 19 



By equating P and P\ Q and Q\ R and R\ we get 

 equations of condition of the same form as in (12) where 

 V" now stands for 



3- I 9 I 5- 9- 

 dr'- r dr r'- dd'- 9s- 



Spherical Polar Coordinates. 



13. In the case of these coordinates, I pursue the 

 course which I have followed in that of cylindrical 

 coordinates. 



In a previous paper read before the Society, I pro- 

 posed for the case of spherical polar coordinates the 

 notation u, v sin 0, w sin Q for the components of dis- 

 placement, and in the previous Parts of this series of 

 papers I have prepared the expressions for the different 

 quantities which will enter on the supposition that this 

 notation is used. The notation is made use of here also. 

 Throughout x will be used to replace cos 0. It is 

 necessary to give the equations in full. 



The Dynamical Stress Equations. 



In these equations the stresses are /*, Q^ R, S, Ts'm 6, 

 and Us'mO, the elements of strain being e,/,£; a, (^sin 0, 

 c sin 0, so that T, U, b and c have not their usual 

 significance. 



The resulting equations are 



W 2P-Q-R I 9., ■-^m^I^^ I 

 — - + ^^ _ _ _^ // 1 _ X-) 6/} + - ^— = p?' . 



or r r dx r om 



df r rXZx \-x- ) r{i-x-)d(p 



f .3l_ . 3 j(,.,.)5) + _L^|f-p. . (37). 



dr r r{\-x-)dx r{i - x') d(p 



