64 
ME. L. N. C4. FILON ON AN APPROXIMATE SOLUTION FOR BENDING A 
Page 
§ 16. Consideration of the stresses in the neighbourhood of the point where the concentrated 
load is applied. 90 
§17. Expansion in integral powers about the point of discontinuous loading. 93 
§18. Coiwergency of the series of the last section. 94 
§19. Transformed expressions for the displacements. 96 
§ 20 . Expansions about other points. Expansion about the origin.98 
§ 21 . Expansions about the point ( 0 ,- 6 ).101 
§ 22 . Effect of distributing the concentrated load over a small area instead of a line .... 103 
§ 23. Case of a beam under two equal and opposite loads, or resting upon a rigid smooth plane. 106 
§ 24. New form of expansion for the pressure on the rigid plane.108 
§25. Justification of the procedure employed in the last section. 112 
§ 26. Deductions as to the rapidity with which the local disturbances die out as we leave the 
neighbourhood of the load ... .114 
Part III.— Solution for a Beam under Asymmetricaj^ Normal Forces : Special Case of 
T\vo Opposite Concentrated Loads not in the same Vertical Straight Line. 
§27. Expressions for the displacements and stresses in series.117 
§ 28. Integral expressions ■when a is made infinite.119 
§29. Series in powers of r. 122 
§ 30. Distortion of the axis of the. beam.124 
§ 31. Distortion of the cross-section x = 0 and shear in that cross-section.125 
§ 32. Practical importance of this problem.128 
Part IV.—Solution for a Beam whose Upper and Lower Boundaries are Acted 
UPON BY Shearing Stress only. 
§ 33. Expressions for the displacements and stresses in series and integrals.129 
§ 34. Expressions for the displacements and stresses in series of powers of the radius vector 
from a point. 134 
§ 35. Distortion of the beam.136 
§ 36. Case where the shear is spread over an area instead of over a line.139 
§ 37. Application of solutions of § 33 to the case of tension produced by shearing stress applied 
to the edges.142 
§ 38. Correction to be applied in this case to the stretch along the edges as we approach the 
points of application of the load.143 
IArt V.—Solutions in Finite Terms; Special Application to the Case of a Beam 
Carrying a Uniform Load. 
§ 39. Solutions in finite terms. 145 
§40. Case of 71 = 4. 147 
§41. Determination of the constants for a beam uniformly loaded. ... 148 
§ 42. Remarks on the above solution.150 
§43. Historical summary; remarks and criticism.151 
§ 44. Recapitulation of results and conclusion.-. ... 154 
