AND THE HYPERBOLIC LAW OF ELASTICITY. [187] 



Solving the equation with respect to — , 



2_ Px 



ucP '-- d-— '—Px 



3 4 a + 7 



x ., _, j3 a + 7 j 3 afi +y$ 



if p ' = nd 3 ~ and »=d,-.— '—, 



p~ l — nx 3Jf 4 a + y 



(«♦£)' 



Now if y be the co-ordinate of the curve perpendicular to x and referred to the same 



origin of co-ordinates, namely, the extremity of the neutral axis, 



1 fig (^ df 



R = ~dx 2 



dy . dy 2 



Also, since — is always very small, -— may be neglected as compared with unity, and 



lliV Q>W 



* l d2 y 



then — = -r— „ . 

 R dx* 



Substituting — - for — in the above equation, and expanding the right side of it, 



— - = px + p 2 x 2 n + p 3 x 3 n 2 + ... 

 a Or 



du 

 Integrating twice and remembering that when x = a (the half length of the beam) — = ; 



CL3C 



and y m when x = 0, 



dy a 2 — x 2 a 3 — x 3 a* — x* 



dx r 2 * 3 * 4 



p i x\ p 2 n ( , x*\ p 3 n 2 I . afi\ 



The integration might also be easily effected without expanding the expression for — , in 



R 



which case the equation to the elastic curve would be exhibited in finite terms. 



If/ be the central deflection or the value of y when x = a, 



pa 3 3 3 



/= (1 +-pna + -p 2 n 2 a? + ...) 



3 4 5 



This series is rapidly convergent, and if we assume 



i.3 19. \ /« \2 



, pa 3 . (3 \ (3 \ 2 , 



The latter series will coincide with the former for the first two terms, and differ from it by 

 about 2*g-th part of the third term, ^gth part of the fourth term, &c. We may therefore 

 consider the last equation very nearly accurate : and we have 



pa 3 I 3apny l 



or, giving p and n their values, 



f = pa 3 Ud 3 (a + y)-—Pd ^ T \ ; 

 I lo a + 7 J 



48 — 2 



