272 Mr. Girard on the Resistance of Cast-iron. 



friction to the annular extremities of the pipe. In these 

 conditions the vertical elementary rings into which we might 

 conceive a pipe to be decomposed would each support the 

 same pressure ; we can therefore make the leiigth as small as 

 we please, and confine our researches to the effect on one of 

 these elements. 



We conceive, he says, that that element ought to yield at 

 two points in its perimeter, that is, in the case of rupture it 

 would be divided into two indeterminate sectors. He further 

 conceives the sepai-ation to be in consequence of those forces 

 only which are perpendicular to the plane of fracture ; and 

 that the surfiice of fractm-e must be the one of least resistance ; 

 and, consequently, its breadth equal to the thickness of the 

 ring. Hence the only forces it is necessary to determine, are 

 those which are parallel to tangents to the ring at the points of 

 rupture. 



These conditions settled, if the tangents to the ring be pro- 

 duced till they intersect, and a straight line be drawn from 

 die point of intersection through the centre of the ring, that 

 line will divide the sectors each into two equal parts ; and it is 

 sufficient to determine the effect of the forces sought, on one 

 of the parts of one of the sectors *. 



It is easily shown that the pressure on any point whatever 

 of the arc, is equal to the perpendicular pressure on an unit of 

 surface, multiplied by the sine of the arc comprised between 

 the plane of fracture and the point considered, when that pres- 

 sure is considered to act in a direction parallel to the tangent; 

 whence the sum of the forces on the semi-sector is equal to the 

 product of the versed sine of the semi-sector, multiplied by the 

 perpendicular pressure on an unit of surface, antl by the in- 

 terior radius of the pipe. Consequently the force is greater as 

 the semi-sector is greater, and is most when the vei'sed sine is 

 equal to the radius ; or when the whole sector becomes equal 

 to the semi-circle. The thickness of the pipe being supposed 

 uniform, one such as we have considered ought to break at 



* If Mr. Girard hail been a practical engineer of the same ability as he 

 i« a geometer, he would liave found this to be one of the most favourable 

 cases. Pipes are often of very irregular thickness, and in consequence the 

 resistance is unequal which causes the strain on the surface of fracture to be 

 unequal, and the direction of the resultant should be considered in respect 

 to the distribution of resistance, because it determines the position of the 

 axis of equilibrium or neutral axis. This is only a particular application of 

 a principle for which we are indebted to Dr. Thomas Young, and may serve 

 to show that Mr. Girard has been somewhat more than anticipated in this 

 part of his inquiry. And a reference to Prop. XCV. of Emerson's Me- 

 chanics will further show that the English reader has for some time had 

 the advantage of an investigation leading to the same result as he has now 

 obtained. 



the 



