42 



Substitute in (2) and put x=z{). 

 M.o?f{at)=:^'Ef\at)- but a^^-, 



Therefore Mf\at)= —^f{ctt)—^^\ 



for initially f{af) — and f{p£)— ; 



Therefore -^^'^""'^ 



(Jb 



= — a. 



fiat 

 .. ,. , MV MV "M 

 ^•^ ^ ^ a a 



^{at — x) 



Therefore ? = aV~^ ^ ) ^^'^® ^^ ^^^ P^^^^^ 



after ^ > - 



Tension =Ey-=: ^ This is greatest when 



at — 03=0, and then=V sIy^u- 



So that for the case of an infinite wire it will break 

 unless the statical breaking force > V Jf^; a limit wholly 

 independent of M. This result is approximately true in the 



case of a very long wire : if F be the force which acting 



F 

 statically would break the wire, velocity necessary = , -— 



Any change then, which increases E will render such a 

 wire more liable to break under impact: cold has this effect ; 

 we arrive then at the apparently anomalous result that 

 though cold increases the tensile strength of iron, yet owing 

 to increasing its elasticity in a higher ratio it renders it 

 more liable to break under impact. 



Now let us return to the case of the wire lenscth I. We 

 have the additional condition that when x^ ^ = for all 

 values of t, and this will introduce a number of discontinui- 

 ties into the solution. Up to the time — we mav deduce 



a "^ 



the solution from the previous case ; from t=^Oio t^" we 



CL 



have as before 



