186 Mr. Horace Lamb on the 



dhv g _ ") 



(8) 



(9) 



Let us write for shortness 



4 3(1 -a 2 ) 



rw == — 



4 Ay 



The proper solution of (7) is then 



w -f <r p = A cos mx cosh m«£ + B sin mx sinh mx ; . (10) 



and the conditions (8) to be satisfied at the edges #= + b 

 give 



— A sin mb sinh m& + B cos mb cosh wi& = — -5- 



wrp I 



A (cos ??iZ> sinh ??i£ + sin wi& cosh mfr) 



+ B (sin mb cosh mb — cos wi6 sinh ra&) 



-J 



leading to 



. <t sin mb cosh ?n& — cos mb sinh ?nfr 



A = 



B= 



ra 2 /o ' sinh 2m& + sin 2mb 



a- sin ?7i& cosh mb + cos m& sinh mb 



(12) 



m 2 /o ' sinh 2mb + sin 2m& 



The condition that w = when #=0 gives 



*oP=A, (13) 



and the value of w is thus completely determined. 



The distribution of applied force and couple over the ends 

 (supposed straight) necessary to produce the strain in question 

 is given by 



2(3\ + 2/a)/* 

 \ + 2fju 



*(«-.+ ^), • • • (W) 



This distribution is somewhat artificial, but the theory of 

 " local perturbations " developed by St. Venant, Boussinesq, 



* Since <r=X/2(X+/*). 



