58 Mr. J. H. C. Searle on the Effect of 



2. Free end. — At a free end we have 



«jXi)' 



<%= 



" d d z f p dfp d d 2 f p 

 ! v dh~df ~~ "dfUdp 



Now f p = A cos 71 J + B siii7!f +Cco?h7 2 f + Dsinli7 2 |; 



= <£(«) +t(/?); 

 where « = 7l £; 



/3=7 2 f. 



J = -7i¥+72 3 t'; 



<A, 



5| = 7i'<£' + 7sty' ~ 7i7i'f + 7278^ 



Jig = -2 7l7 /(/> + 2 72 7 3 't-7iV^ / + 72V^' 5 

 ^ = - 3?i V*' + 372 2 72>' + 7i V» + 72 Vft- 



= (* + +)[" 3?i V*' + ?7s Vf + 7iV W> + 72 3 7 2 '^ 

 + X 4 J7i'</>' + 7s>' ~ 7i7i'^ + 7272 7 f^} 



+ 4\ 3 { 7l (/>'-}-7 2 t / }] 

 - (71*' + 7 3 f ')(- 2 7i7i'4> + 2 73r2 't-7i 2 7i'^ / + 7s VM 

 = </>£' [ - 37! V + *V + ^ 3 7i + 27iV] 



4- -v/r^' [3y 2 2 7i' + X V + 4x3 72 - 2y 8 2 y 8 / ] 

 + <t> 2 [7i V - *W] f + ^ 2 [72 3 72 / + ^W] f 

 + <£^ [7i 3 7i' + 72V — ^ 4 7i7i' + ^ 4 7^72 7 ]? 

 + <£f [3y a V + X 4 7 3 ' + 4\ 3 72 + fytfiYi'] 

 + ^/[- 3.7iV + *V + 4X 3 7l - 2 7l 7 2 7 2 '] 

 - (71^ ^ 72^')( - 7i 2 7i ,< £ / + 72 2 72'^ / )f • 



