406 



PROCEEDINGS OF THE AMERICAN ACADEMY 



Numerous experiments were also made on pillars with the ends 

 rounded, so that the weights bore on the axis. Most of these experi- 

 ments were made on solid pillars ; the hollow ones had hemispherical 

 caps fitted on the ends. It was found that their breaking weight was 

 about one third that of pillars of the same dimensions with flat ends. 

 Some experiments were made on pillars with one end flat and the 

 other end rounded ; their breaking weight was about two thirds that of 

 pillars of the same dimensions with both ends flat. 



Hodgkinson determined separate formulas for the breaking weights 

 of both solid and hollow cast-iron pillars, with flat ends and also with 

 rounded ends ; for practical purposes, however, formulas (1) and (2) are 

 suflficient, recollecting that in a solid pillar c? = 0, and that the break- 

 ing weight of a pillar with rounded ends is one third of that given by 

 formulas (1) and (2). The breaking weight of pillars with rounded 

 ends is so much less than that of pillars with flat ends, that Hodgkin- 

 son did not find the crushing eflTect sensible in pillars with rounded 

 ends when longer than fifteen external diameters ; consequently, one 

 third the value of W, as given by formula (2), will be less than the 

 true breaking weight, when the formula is applied to pillars with 

 rounded ends of lengths between fifteen and thirty diametei-s. For 

 simplicity, however, I propose to adopt thirty diameters as the limit 

 between formulas (1) and (2), whether the ends are flat or rounded. 

 The errors resulting will be on the side of security, and in ordinary 

 cases, not being large, may be neglected. In a solid pillar with 

 rounded ends, 10 feet long and 8 inches in diameter, the computed 

 breaking weight is about one fourth too small. 



computed by formula (l),the maximum excess being about 6 per cent. This is 



illustrated by the diagram in the margin. 



The curve A B C D represents the 

 breaking weights of solid pillars, 5 

 inches in diameter, by formula (1), 

 and the curve E F C G the same by 

 formula (2), the abscissas being the 

 lengths in diameters, and the ordi- 

 nates the breaking weights. The 

 curves intersect at the point C, the 

 abscissa of which is 25.498. Ac- 

 cording to Hodgkinson the break- 

 ing weights are represented by A B 

 F C G, the abscissa of the points B 

 and F being 30. 



