183] ST. VENANT'S PROBLEM. 479 



^ 



dx 



If the point is in the XY- plane, dz = 0, and as this plane is fixed, 

 w must be 0, necessitating 



If the point is on the X-axis, dy = dz = 0, and as it must remain 

 on the X-axis, v and w must vanish, necessitating 



(TT) = o- 



\#SB/tl 



The six conditions at the origin are accordingly, 



while the conditions 67) everywhere satisfied are 



72) 



Of these the first two give 



x du dv I, fciu dv 3w\ k / du 3w 



fa === jy = ~2^\a^ + ^ + aij = ~2 



^. ^w? dw 



~ Z (I -\-ti~dz = ~ Vjz' 

 with the third 



b) |^+|f = . 



0^ r ^o; 



The equations 144) of equilibrium are 



73) 



