OUTLINES OF NATURAL PHILOSOPHY. 



That is, if W be the weight which a beam can support 



W 



in a horizontal position, - TJ is the weight it can 



COS % 



support when inclined to the horizon at the 

 angle i. 



The proof of this proposition proceeds on two prin- 

 ciples, that the momentum of the weight is dimi- 

 nished in the oblique beam, by its perpendicular 

 distance from the fulcrum being diminished, and 

 that the resistance of the beam is increased by the 

 centre of gravity of the section being carried far- 

 ther from the fulcrum. Each of these being in the 

 inverse ratio of the cosine of the inclination, their 

 joint effect is in the inverse ratio of its square. 



It is implied in this reasoning, that the resistance of 

 each fibre is the same to the oblique and to the di- 

 rect fracture. Though this seems probable, it is not 

 altogether certain, and it would be useful to have 

 experiments directed to the clearing up of this dif- 

 ficulty. If the individual fibres resist fracture also 

 in the inverse ratio of the cosine of the obliqui- 

 ty, the strength of the beam will be in the inverse 

 ratio of the cube of that cosine. This is observed by 

 GUIDO GRANDI, in the treatise already referred to. 

 Opere di GALILEO, torn. in. p. 271. 



245. The theory in the preceding proposition 

 being admitted, the force of a beam to resist the 

 action of any strain, will be inversely as the square 



of 



