J. O. Thompson — Law of Elastic Lengthening. 45 



Thus we shall be in a position to determine the true modu- 

 lus of elasticity, the modulus of the body before it has been 

 subjected to any deformation whatever. 



In order to determine this true modulus we may proceed as 

 follows : 



Let X be the lengthening caused by the initial load P Q * 

 " X " " " " total weight P°+p 



Then the observed lengthening cc=X— X 



Let X = aP + /3P* +/P 3 (1) 



andX =aP +-/5P o 2+ r P o 3 ( 2 ) 



Equation (2) subtracted from (1) gives 



aj=(-aP -/JP «- r P »)+aP+/3P» + yP» (3) 



The results of the measurements were given in the follow- 

 ing form 

 a=a(P-PJ + &(P-P )s+c(P-P )3 



= (-«Po+^o 2 -«Po 3 ) + («- 2 ^o + ^P 2 )P+(6-3cP )P2 + 



cP* (4) 



Equating the coefficients of like powers of the variable P 

 in the two expressions for x in (3) and (4) we obtain 



a=a-2bP n +3cP' 2 



O l O 



/i=5-3cP 



y=.c 



Equation (1) which gives the relation between elastic length- 

 ening and stretching weight when one begins with an initial 

 load zero becomes in the case of the steel wire for instance 



X=34-672P+0-6498P 2 -0-0525P 3 



and (-TFi) =34-6'72 



V c/P/p =0 



With an infinitesimal stretching weight the formula for the 

 modulus of elasticity is 



q ' dX 



where I is the length and q the cross-section of the wire. 

 Therefore in this case 



, ^ 22683 1 ■ 



E= . —=20050 



0-03263 34-672 



The length of the unstretched wire I can be found with the 

 help of equation (2). 



*P represents the weight of the frame with pan and damper, increased by 

 half the weight of the wire. 



