282 



Mr Vegard, On some general Pi'operties 



or putting for brevity's sake 



^ dx j=o dug dx 

 and remembering equation (9a) 



\QMq + Qihn^ + ... Qr^nr = 0, 

 [yLto S/Zo + jx^ hh + ... fi.,. Sn,. = 0. 



The equations (19) must be fulfilled for any variation Sni con- 

 sistent with the two equations, then 



(19) 



(20) ^' = ^=... = ^ = XQ,n, 



H'O M'l f^r 



Forming by means of the expression for Qg the sum to the right, 



^^ dU 



AQsns = p- 







dx' 



The equations (20) and (9b) then give 



'd<f)o duo d(f)i d7ii dcf)). dn,. d U 



duo dx 9wo dx " dnQ dx dx 



(21) 



{pfi,-M,) = = E„ 



dn-^ dx dui dx 



dn,'d^rd^^P^-^^^^^=^- 



d^dno d^drh 

 dn,- dx dn,. dx 



dno 

 dx 



d(f),. dUr 



dn^. dx 



dui 

 dx 



'^(pf.r-Mr) = = E,, 



/*o V7 + /ii-Tf +.../*. -^ = 0. 



dUr 



dx 



We have here apparently (r + 2) equations between the (r + 1) 

 unknown -j-^, ~, ... -j-^; but they are not all independent, in fact 



(ajiA/ ijjOG 



dx 



the following linear relation exists 



EoUa + j&\wi + . . . E,.nr = 0. 

 The solution of equations (21) gives 



and for the other axis 



dui 

 dx 



dn,- 



Di 



= A 



dU 



dx ' 



dU 



dy' 

 dU 



and 



dy 

 dn^ 

 dz 



(22) dni = DidU. 



^ ^ T) 



dz ' dz ' 



