Surface-Loading on the Flexure of Beams. 485 



The value of the integral between the limits a = and a=='ao 



2 

 is, as has been stated, -5, or 0'667. For a = 5, ?. e. for w = 5# 

 o 



as the upper limit, the integral = 0*666, and for u = 2x the 



integral =0'656; so that this solution is approximately correct 



for elements lying at a distance of J the width of the beam 



from the point of contact. 



Hence for a beam where the length A A' is 5*5 millim., 

 this solution would be applicable up to points lying at a 

 distance of about 1*4 millim. from the top surface. 



I have investigated the law up to within 0*5 millim. of the 

 top surface, and find it to be 



^=0-726-- 



3d 



The investigation of the state of strain in glass beams by 

 means of polarized light was first suggested by Sir David 

 Brewster*, and his experiments are usually quoted as proving 

 the truth of the Bernoulli-Eulerian theory of flexure. It is, 

 however, easy to show experimentally that these experiments 

 must have been made under conditions where the surface- 

 loading effect was inappreciable ; though very accurate reason- 

 ing on this point is impossible, as the drawings accompanying 

 Sir David Brewster's paper are not to scale, and the span of 

 the beams and the precise method of application of the loads 

 are not indicated. 



M. Neumann developed a theory of the action of strained 

 glass in the polariscope t, and found that the velocity of light 

 in a medium is increased by compressing it. He bases his 

 calculations on the measurement of the deflexions of glass 

 beams supposed to obey the Bernoulli-Eulerian theory ; the 

 beams are doubly supported and centrally loaded, having the 

 proportions 66 x 8*5 x 2, the latter being the depth. It is not 

 in all cases stated what spans were employed, so it is impos- 

 sible to say how far the results were influenced by surface- 

 loading. 



Professor Clerk- Maxwell f has examined the state of strain 

 in pieces of unannealed glass of various shapes, the lines of 

 equal intensity of strain being deduced from the isochromatic 

 lines. 



The lines of Principal Stress are found from those of Equal 

 Inclination in the manner described later on in this paper. 



* Phil. Trans. 1818, p. 156. 



t Abhandlungen der k. Akademie der Wissenschaften zu Berlin, 1841, 

 vol. ii. pp. 50-61. 



X Trans. Roy. Soc. Edinburgh, vol. xx. (1853) p. 117. 



