﻿subjected to Forces in the Direction of their Axes. 241 



When a = a the two expressions for f- are the same, 

 whence "-' 



c h 



— 2 + riE cos na = — 2 + nD cos na + nC sin na. 



(14) 



Similarly the two values of y are equal at the point x — a. 



c h 



•'• ^ 2 a + Bsinna=^(a — /) + Dsin?ia — Ccosna. (15) 



Equations (13), (14), and (15) give 



B= — ., (sin na . cot ?il— cos na) 



n d ' 



rH c — h . 

 U= — r sin na, 

 n 6 



D = •> sin ?ia . cot nl. 

 n 6 



The equations of the elastic central line are therefore 

 for x<a, 



m Va \ l—a 1 , . _ N "} 



^ = P ) •^~ a? +-( smwa • cot??/ — cos na) sin ra# 5-, (16) 



and the expression when x>a is obtained by writing as for a 

 and a for d* in the above. 

 By putting x=a we find 



y a p r#fr 1 • / • 7 x "I _1 



— = — -y- + -sin na(sm na cot nl — cos n a) (17) 



y a ?/i |_ t n J 



§ 13. As a particular case, if the mass m be at the centre 

 of the span, 



y a _4PA 2(l-cos^A -' 



y a ??i/ \ nZ sin nl ) * ^ ' 



and vibration is impossible when 



sin??Z = 0, 

 agreeing with the condition of instability of § 3. 



PA?7. Mag. S. 6. Vol. 12. No. 69. &?/?* 1906. R 



