AND THE HYPERBOLIC LAW OF ELASTICITY. [185] 



Substituting then for the area and extension in the formula (A), the total tension at a 

 distance r - R below the neutral surface is 



~~ ^ a' 



r,-R 



Similarly substituting in (B), the total compression at a distance R - r above the neutral 

 surface, supposing n to remain constant (as the transverse section of the beam is rectangular) is 



/xdry 



R-r 



where r is the radius of curvature of a filament above the neutral surface. 



The total moment about the neutral axis of the tension developed in any transverse ver- 

 tical section of the beam is found by multiplying the first of the two preceding expressions 

 by r — R and integrating between the limits defined by the lower surface of the beam 

 and the neutral surface. In this way it will be found that if r f be now taken for the radius 

 of curvature at the lower surface of the beam, the total moment of tension at the vertical 

 section here considered is 



j (r t -Rf /3(r,-.B)« ff(r, - Rf 1 ,,, 



And similarly the total moment of compression is, r being the radius of curvature of the 

 upper surface of the beam, 



f (R - rf _ $(R-rf &(R - rf 

 M7 l SR 4R* + 5-K 8 



-....} ..... (#)• 



Equating moments about the neutral axis for the equilibrium of the portion of the beam 

 between the transverse section in which the above moments are developed and the nearest 

 fulcrum, we have the sum of these moments equal to the moment of the pressure on the 

 fulcrum. This pressure is equal to half the deflecting pressure applied at the centre of 

 the beam. Also, if a/ be the perpendicular distance of the centre of moments from the 

 direction of P, the moment of P is Poc and 



Px = {A') + {B'). 



The relation of r t — R to R - r determines the position of the neutral axis. Now, if for 

 equal degrees of longitudinal compression and extension the elastic forces of a rod of iron were 

 the same, we should have, by known principles, the neutral axis at the centre of gravity 

 of the transverse vertical section of the beam. But that the neutral axis is near the centre of 

 gravity is evident from the fact that the ordinary formulae for the deflection of beams, in which 

 the position of the neutral axis is so assumed, agree well with experiment. The same hypo- 

 thesis also renders the numerical results accompanying this paper consistent with themselves. 



It is important, however, to shew that a small error with respect to the position of 

 the neutral axis does not induce a large error in the equation of moments, but, on the 

 contrary, produces an error which is small in comparison with the original error, and is 

 therefore of a second order of small quantities. 



Vol. IX. Part II. 48 



