Permanent Forms of Mathematical Expressions. 1 27 



and writing P, Q, R, S, T, U for the elements of the cor- 

 responding stress, we shall have 



de' da' 



If we retain Cl u etc. as the operator accompanying 6 U etc., 

 when we consider now a function of P, Q, R, S, T, U subject 

 to an infinitesimal variation in the axes, we have 



°- 2S («) + < R - Q) J|- u H' +T ro. 



etc. 



(8) 



From these we obtain readily the invariants and the 

 covariant quadrics of the stress. 

 It is also easy to prove that 



dP dJJ_ dT 

 dx dy dz* 



namely the components corresponding to the body-force 

 due to the stress, are the only possible components of a 

 vector of permanent form which can be formed linearly 

 from their first differential coefficients. 



As an example, we may consider Maxwell's mathe- 

 matical expressions to explain the transmission of gravity 

 or electric forces by a state of stress in the intervening ether. 



The body force is now to be of the form 



df 2 d<p 



dx 'dx 



per unit of volume. 



Let us consider P, Q, etc., to be quadratic functions 

 of the components u, v, w of some vector. 



We must have from 



etc. 



Q>$ = 



, 



Q,S = 



R-Q, 



O a P = 



orp 



HS = 



u . 



& 3 P = 



2U, 



£2 S S = 



-T , 



