[ 63 ] 
IV. On an Aj^j^roximate Solution for the Bending of a Beam'of Rectangidar 
Cross-Section under any System of Load, luith Special Reference to Points of 
Concentrated or Discontinuous Loading. 
By L. N. G. Filon, B.A. [Cantab.), M.A., B.Sc. [Land.), King's College, Cambridge, 
Felloiv of University College, London, and 1851 Exhibition Science 
Research Scholar. 
Communicated by C. Cheee, F.R.S. 
Eeceived June 12,—Read June 19, 1902. 
Index and Table op Contents. 
Part I.—Establishment and General Soi.ution of the Equations op the Proei.em 
Discussed. 
Page 
§ 1. General sketch of the problem proposed.. . 65 
§ 2. Object of the investigation. 66 
§ 3. Establishment of the equations..67 
§ 4. General solution of the equations in arbitrary functions.69 
§ 5. Solution involving hyperbolic and circular functions. 71 
§ 6. Determination of the arbitrary constants from the stress conditions over the faces y = ±h. 72 
§ 7. Expressions for the displacements and stresses. 74 
§ 8 . Conditions at the two ends x = ± a . 77 
§ 9. Part of the solutions corresponding to the terms ao, /So, Co, do.78 
Part 11.—Discussion of the General Solution when the Forces on the Beam are 
Purely Normal and are Symmetrical about x = 0. 
§ 10 . Expressions for the stresses and displacements.. . 79 
§11. Approximate values to which the expressions of § 10 lead when “ 6 ” is made very small . 80 
§ 12 . Analysis of the approximate expressions for the displacements. Shearing deflection , . 82 
§ 13. Value of the deflection when h is not small and the beam is doubly supported.84 
§14. The doubly-supported beam under central load. Expressions for the strains and stresses 
when we remove the supports to the two extremities. 86 
§ 15. Definite integrals to which the expressions of the last section tend when we make 
very large . 
VOL. cci.—A 334. 
. 88 
3.4.03 
