146 The State of the Medium in the Electrostatic Field [OH. vi 



As a preliminary we must find the values of the stress-components f^, 

 Py~y, ... referred to fixed axes Ox, Oy, Oz. 



The stress- quadric at any point in the ether, referred to its principal axes, 

 is seen on comparison with equation (83) to be 



Here the axis of f is in the direction of the line of force at the point. 

 Let the direction-cosines of this direction be l lt m l} n l . Then on transform- 

 ing to axes Ox, Oy, Oz we may replace f by lx + m^y -\-n^z. 



Equation (85) may be replaced by 



and on transforming axes f 2 + if + f 2 transforms into x 2 + i/ 2 + z 2 . Thus the 

 transformed equation of the stress-quadric is 



^ {2 (^ + rn.y + M) ! - (of + f + ^)) = 1. 

 Comparing with equation (82), we obtain 



P = ^W-1) ........................... (86), 



P xy =^(1l 1 m l ) ........................... (87), 



and similar values for the remaining components of stress. 



Or again, since X = ^R, F = m 1 .R, Z nJi, 

 these equations may be expressed in the form 



_XY 



Xy ~ 47T ' 



In this system of stress-components, the relations P xy = P yx are satisfied, 

 as of course they must be since the system of stresses has been derived by 

 assuming the existence of a stress-quadric. Thus the stresses do not set up 

 rotations in the ether (cf. equation (80)). 



In order that there may be also no tendency to translation, the stress- 

 components must satisfy equations of the type 



3 ^ + ^ + ^ = ........................... (88), 



dx dy dz 



expressing that no forces beyond these stresses are required to keep the 

 ether at rest (cf. equation (79)). 



