280 



Mr Vegard, On some general Properties 



^ sf ^' Wd.v = = 1I\S (Ax) + I ^'S Wdx, 

 Jo Jo 



g md^c = = ni'S{Aa') + I Siiida:, t - 0, 1, 2 ... r. 



Jo Jo 



fAX fAx 



S pd.r =0 = p,S{Aiv)+ Bpdx. 

 , J Jo 



(13) 



Wi, Ui, pi are the values of TF", ui and p at the end (1) of the 

 element. We develop the functions W, ni and p after Maclaurin's 

 formula, and, forming the variations, we get 



SF=STFo+s(^) .^• + etc., 

 'dn^ 



Sn; =8 nf + 8 i ~ ] x + etc., 

 8p =8po+8(-~] w + etc. 



These values put into (13), and integrating 

 (14a) F.^-^> + SFo4-iSf^) A.+ ...=0, 



Ax 



dx Jo 



(146) n/^~^ + 8nf + ^8(p) Ax + ... =0, i = 0,l,2...r. 



(1.0) ,.«-^) + 8,. + p(gU.+ ... =0. 



To find 8 (Ax) we shall apply equation (146); we multiply 

 with fi/ and add 



(i5„) ?^>i ,; ,,' + f ./(*..« + p (^■) A. ... ) = 0. 



fii' is the value of fXi at the eud (1), 



Putting the expression for fii into (15rt), and using the equa- 

 tions (86), (9«) and (9c), 



Ax KdxJoX \dxjQ 



Using equation (146) 



(15.) ^>(l-A.i^-„,')=«A..+ ..., 



0. 



