598 Dr. C. Chree: Applications of 



acts, at the distance x from along the " neutral " line, or 

 line of centres of the cross sections. What principally con- 

 cerns us at present is the inclination or " slope " and the 

 curvature ; so I give the values of dyjdx and cPy/dx* as well 

 as y ; the results of course are well known. They are as 

 follows, y heing taken positive downwards. 

 Between and A : — 



y=( l v/EG>K*){il(il-a)x* + ^x*\, ) 



dyjdx = (iv/EcDfc 2 )\l{±l-a)x + ±rX*}, |. . . (69) 



Between A and B : — 



dyldx={iolEG> f e' 2 )±{l(l 2 -3a*)-(l-xf\, (70) 



tfyjdx* = (ztf/Ew/e 2 ) J (/- x) 2 . I 



§ 22. On the Bernoulli-Euler theory the elastic extension 

 in an element ds of a longitudinal u fibre " at a distance h 

 above the neutral plane is (Ji/R)ds, where R is the radius of 

 curvature of the bent neutral line. But if yjr= t&n- l (dy/dx), 

 we have 



l/R = dylr/ds, 

 and so 



(h/R)ds = hd4r. 



Thus the total stretching a x in the portion of a longitudinal 

 iibre — distant h from the neutral plane — extending from the 

 central section x = to the section where the slope is i/r, is 

 given by 



a x = I hdy]r = hyjr 

 Jo 

 = h dyjdx, (71) 



when (dy/dx)* 2 is neglected compared to unity. 



In fig. 2. dyjdx and d 2 y/dx 2 are positive for all values of x 

 from to I; thus a longitudinal fibre is stretched at every 

 point of its length, or contracted at every point of its length, 

 according as h is positive or negative, i. e. according as the 

 fibre lies above or below the neutral plane. 



In fig. 3, a longitudinal fibre above the neutral plane would 

 shorten in its central portion, while lengthening near its ends. 



The ordinary expression for the curvature, viz.: 



1/R= {(Pyjdx*) (1 + (dyjd.x)*) -*, 

 is on the Bernoulli-Euler theory replaced by 

 lin=d 2 yjdx* ; 



