118 ON THE POWER OF FLUIDS IN MOTION TO PRODUCE 



The case that we have first to consider, is when the force — f arises from 

 the resistance es of an elastic support, whilst y, as ordinarily happens, is a con- 

 stant load. The equations (1 and 2) then become 



2/'<j = es" 



whence we deduce the second of the results already mentioned, namely that 



« = 2^'. 



When the elasticity is imperfect, we may reason by observing, first, that s 

 represents the abscissa, and e s the ordinate y, of the curve of elasticity, or the 

 curve which expresses the relation between the deflection and resistance; and, 

 secondly, that the integral JjTds is equivalent to the area of this curve. Now 

 an area corresponding to an abscissa s is equal to s multiplied into the mean 

 ordinate y'; and when the curve deflects inward from the tangent, and turns 

 towards the abscissa, as the curve in question does when the elasticity is im- 

 perfect, it is manifest that the mean ordinate does not stand in the middle of 

 the total abscissa, but at a point which is nearer to the origin : in other words 

 a being the total abscissa, and s' that belonging to the mean ordinate, we shall 

 have in the equation a = m^', m exceeding 2. But referring to our equations 

 (1) and (2), we observe them to become in this case 



fc = y'^ 

 and 



f=y"; 



which gives y' = y", or the mean ordinate of force to the deflection c, equal to 

 the extreme ordinate of force to the deflection a'. And we have already seen 

 that in the equation a = m a', m exceeds 2; whence we conclude so much of the 

 fourth proposition as relates to bodies imperfectly elastic in the direction of the 

 load. 



To demonstrate the third proposition, let a cord of inconsiderable mass, half 

 length I, and modulus e, be drawn into a horizontal position by the action of 

 an inconsiderable stretching force. Applying to the centre of this cord a force 

 that deflects it through a space s, the half length after deflection will be 



5^ 



V^* + s ^; the extension i j; and the tension in the direction of each branch of 



