[186] H. COX, ESQ., ON THE DEFLECTION OF IMPERFECTLY ELASTIC BEAMS. 



For, as in the last diagram, let A, B represent the positions of the upper and lower 

 surfaces of the beam respectively, C of the neutral surface, a b of a trans- 

 verse vertical section, so that the sum of the tensions developed between c 

 and b is equal to the sum of the compressions between c and a. Also, c' 

 at a small distance above the real neutral surface, let another, C'c, be c 

 taken. Let T be the sum of the forces of tension developed between c 



and b, and C the sum of the forces of compression developed between c' b 



and a. Then C is nearly equal to T, and C - T is a small quantity. 



Also, let m T be the moment of the forces T about c ; then it is clear that their moment 

 about c is (jn + cc')T. If nC be the moment of the forces C about c, (ra - cc')C is their 

 moment about c. Hence the difference of the sum the moments of T and C taken about 

 c and c respectively is (C - T')cc. Also cc' is a small quantity ; hence (C - T)cc' is the 

 product of two small quantities. 



Next with respect to the moment about c of the tension which must be supposed to exist 

 between c and c, if c be taken as the neutral axis, it is obvious that the moment of this 

 tension is still smaller than the above. 



Hence, on the whole, the difference of moments due to a small error in taking the position 

 of the neutral axis too high is two quantities small with respect to that error, and therefore 

 small in a higher degree with respect to the total moment. A similar conclusion would be 

 arrived at if C'c were taken a little below the real neutral axis. 



Assuming then for the purpose of this computation the neutral axis to coincide with the 

 centre of gravity, we have for a rectangular beam R - r = r t — R = d, where 2 d is the 

 thickness of the beam. 



Substituting for the values of (A') and (B h ) in the equation of moments 



«P a + 7 | _ d_ 3 q/3 + 7 3 d« 3 a/3 2 + 7^ 1 



" R 3 ( R' 4' a + /3 + R* 5' a + 7 '"}' 



Now this series is always very convergent, both because the fraction — is small, and 



because /3 and S are small in comparison with a and /3 respectively. Hence, we may substi- 

 tute for the above equation the following, without considerable error, 



y = R~3~[ 1 ~(i'R' a + 7 / + \i'R' a + 7 j '") R 3 \ + *' R' a + 7 i 



The last series coincides with the former in its first two terms, and differs from it by small 

 fractions of each of the subsequent terms, if to a, /3, 7, and 8 be given the values hereafter 

 determined from experiments. 



