183, 184J ST. TENANT'S PEOBLEM. 483 



Now since the beam is in equilibrium, this integral Z- traction must 

 be the same for all values of , hence & = 0. We have then finally, 



[2-2 _ ^,2 

 ax + i g-^- 



a 2 2 b^z* , /8fl\ 



-v - ^ + y - ^^ 



- y 



From these we deduce the non- vanishing stress -components, 



Z, = E |f = E [a + a 



100) 



y dw . du 



and changing x to i/, a lf a s to & t , 6 2 , j3 to /3 ; 

 100) 2i- 



The condition 69) at the cylindrical surface gives 



-IAIN d& , s , da ' '" : " 

 101) cosM + c 



184. Determination of Function for Particular Cases. 

 Torsion. We may easily show that a function harmonic within a 

 certain plane contour, and on the contour satisfying the condition, 



is uniquely determined. For if there be two functions l and i 2 

 satisfying the conditions, the difference V= fl * 2 is harmonic, 

 and on the contour 



