﻿subjected to Forces in the Direction of their Axes. 243 

 For x — a we find 



P 



Va 



1 .-_^, ./__ ixl- ■ (21) 



and 



^ mj~(&-y)(l — isinfia) + a(coswa-l)| 



§ 15. If the load be at the centre, a = b= 2 in this result, 



- sin — 



• (22) 



vibration ceasing when 



I 2 2 



sm- = U. 



Section VII. 

 Deduction of Solutions to some Static Problems. 



§ 16. The calculations contained in Sections V. and VI. 

 are the same as those involved in the corresponding statical 

 problems of the deflexion of a bar of negligible mass carrying 

 a concentrated load and subjected to the axial compressive 

 force P. 



If both ends are supported, instead of my a we must put 

 the magnitude of the load in the formulas of Section V. and 

 so obtain an expression similar to (16) for the curve of 

 deflexion of the central line, and to (17) and (18) for the 

 displacement at the point at which the load is concentrated. 



Making the same change in notation in Section VI. we 

 get equations (19), (20), (21), and (22) for the case of the 

 bar clamped at both ends. 



An expression equivalent (except for a numerical error) 

 to (22) has been given hj Kirchhoff*, but I am not aware 

 that the more general solutions for any position of the load 

 have hitherto, in either case, been recorded. 



University College, Bristol, 

 March 1906. 



* Vorlesungen ilber mathematische Physik, Bd. i. Lecture 29, § 3, 

 See also Toduimter and Pearson's • History of Elasticity,' vol. ii. pt. 2. 



R2 



