to the Longitudinal Expansion in Rods of Spring Steel. 



39 



2C 



l r/io 



+ 8C 2 L\3 



ff = - 



.'/_ 



{-ft- 



«+4(P-«0) (X'-X 7 ) + |s(X''-X'')] }, 



{ 1 -C _ 8C' [(X '- a ' )2+(Y '-^ 2 J 

 8^[(¥ s + 4 (r-«'))(X'-X)+| 4 (X"-F)]}, 



¥'{ 1 + C-At(X"-«T+(Y"-iV] 



2C 



-i[(I s ( X '- X ')) + &° s - 4 ^"- a "))( X "- X ")]}' 



^ --¥-{ 1+ o ~ A [(X "~ a " )S+(Y "~ 4 " )2] 



-*Mf ^'-^ + Q§s-W-a"))W'-X')]}. 



By the aid of these expressions the values of a' 2 , a" 2 , /3' 8 , /3" 2 

 may be formed from equations (6) and (7). I will not draw up 

 these values themselves, but instead of them the values of four 

 magnitudes which I denote by (X f ), (V), (X"), (Y"), and which 

 I define by the following equations : — 



; 



(X0=tf'-2c(a' 2 +4'), 



(X") = a''-2c(a'' 2 +|^ 

 (Y"j=6"-2c(^ 2 -f^) 



(12) 



On the one hand we get from this 



8(X') = -2CS«' 2 , 8(Y') = -2CS/3' 2 , 

 8(X») = -2C5a" 8 , S(Y") = -2C8/3" 2 ; 



and also taking into account the equation (3), 



1 ^ 1 + 26>""8(X")-3(X , )2/ ^ 1<J ' 



On the other hand there is obtained from equations (12), 



