162 OUTLINES OF NATURAL PHILOSOPHY. 



When the section of the beam is such, that the centre 

 of gravity does not coincide with the centre of mag- 

 nitude, as the termination of the fracture, by being 

 brought from the under to the upper side, must 

 change its distance from the centre of gravity of 

 the section, a change of strength will take place al- 

 so on that account. 



241. When a beam is supported at both ends, 

 the weight which it is able to bear at any point, is 

 inversely as the rectangle under the segments into 

 which it is divided by that point. 



A beam, therefore, supported at both ends, and of the 

 same section throughout, is weakest in the middle. 



A beam supported at both ends, and of a given breadth, 

 will be every where of the same strength, if its lon- 

 gitudinal and vertical section be an ellipse, having 

 the beam itself for its greater axis. 



242. From a given cylinder, to cut out the 

 strongest rectangular beam possible. 



Let the circle ACB (fig. 20.) be the base of the cylin- 

 der ; at the extremity A, of the diameter AB, draw 

 the tangent AH equal to the chord of 90, draw HK 

 to the centre K, and let it cut the circumference in 

 C. If CE be drawn parallel to AB, and CD at 

 right angles to it, the rectangle DE, under these 

 lines, is the section of the strongest rectangular 

 beam that can be cut out of the given cylinder. 



For it is easily shewn, that CD 2 X CE, to which 

 the strength of the beam is proportional ( 237-), 

 is greater in the rectangle thus determined, than 

 in any other that can be inscribed in the circle. 



2 As 



