﻿subjected to Forces in the Direction of their Axes. 235 



from which 



A (cosh a/— cos/3/) +B(sinha/— -^sin/3/) =Ch 



A(a sinh a/ + /3 sin /3/) -+ B (a cosh al — a cos ftl) = 0> (3) 



A(a sinh \al + $ sin J/3/) + B (a cosh ^a/ — a cos i/3/) = 0j 



The last of these holds only for the fundamental and even 

 harmonics. 



If we put _ a sinh al + ft sin ftl 



' cosh a/ — cos /3/ 



?? = cosh \al — ^sinh ^a/ — cos \$l + ^sin ifil, 

 the constants are 



A- & B=-2», C=-'\ D=X //1 > 



and the solution becomes 



y = I cosh ax — - sinh a.2 1 — cos /3^ 4- sin /8a? ) . (4) 

 The equation of the type used in the previous case is 



and we find that for (4) to be the solution of this the 

 following three conditions obtain 



P / / 4EI/HD/A , ,\ "| 



2EIVV A P 2 //T 



P / / 4E1 / ko^_ \ j 

 ^ 2EIVV x p2 ?/i *>)> j 



atanh W=-£tan-££/. J 



From these equations elimination of « and ft would give 



e Vl 



an expression for '- . 



!h 

 The couples required to fix the ends are each given by 



where -, -. 



v— -sinh al — ~ cosh al— — sin /9/~ -~cos/3Z. 

 a a 1 ftp* 



§ 5. Of equations (5), the first two could have been 



