488 Prof. C. A. Carus Wilson on the Influence of 



varies as the load on the beam ; hence by taking the ordinate 

 at b twice that at a, at c four times, and at d eight times, and 

 so on, the get points on the curve of loading along the normal 

 for the load that gives a difference of phase at a equal to that 

 of one-quarter wave-plate. 



The results are plotted on Plate II. fig. 1 : the observed 

 points are indicated by circles, through one of which an 

 hyperbola has been drawn taking the normal and the upper 

 surface of the beam as asymptotes. 



It will be seen that the six upper circles lie very nearly on 

 the hyperbola. 



It is clear that the upper surface of the beam is an asym- 

 ptote only when the surface of contact between the beam and 

 the roller is a line — making the stress there infinite ; but in 

 practice this cannot be so, the smallest pressure giving a 

 bearing surface — as the roller indents the beam — making the 

 stress there finite, i. e. the asymptote will be at some finite 

 distance 6, say, above the point of contact, and 6 will vary 

 with the load. I have calculated below that with a load of 

 115*3 lb. on this same beam, the value of 6 is 0*044 millim. 



The apparently irregular position of the two lower points 

 observed indicates the amount of error made in the assumption 

 (2) above that the surface-loading effect may be found by 

 substituting a flat plane instead of two supports. 



This assumption would be correct only if the beam were 

 of infinite depth and the surface-loading effect of the support 

 infinitely small ; here, however, the steel frame itself pro- 

 duces a surface effect, and this, added to that due to the load, 

 makes the points observed lie off the hyperbola, which would 

 be the true curve (as drawn) if the beam were of infinite depth. 



The effect of the steel frame must be very small compared 

 with that due to the load for points in the upper half of the 

 beam. In drawing the hyperbola I have considered it as 

 negligible at the centre of the beam ; in other words, I con- 

 sider that the correction of the position of the six upper 

 points, required to allow for the surface effect of the frame, 

 would not make them deviate seriously from the hyperbola. 



It must be noted, however, that when the beam is resting 

 on two supports the surface effect of the frame disappears, 

 yince the beam only touches the supports and surface effect 

 can only be caused by actual contact ; hence I conclude that 

 the surface effect due to loading only is strictly represented 

 by the hyperbola and is as if the beam were of infinite depth *. 



* According to this reasoning there appears to be a shear of finite 

 amount at the bottom of the beam — when doubly supported — due to 

 loading only, but this does not seem to me to be inconsistent with the 

 surface conditions. 



