42 
Substitute in (2) and put x=0. 
M a*f"(af) = —Ef(at); but a 2 =-, 
MV 
Therefore M f'(at) = — j uf(at) — ; 
V 
for initially f(af) = 0 and f(at)= ; 
Therefore . _M g /'( a 0 _ _ a . 
nf(at) 
+ • 
MV' 
a 
’ f A , MV MV ' 
uf(at) H = — £ 
™ v ' a a 
/ua< 
Id 
Therefore £ = — — £ 
fxa\ 
/l i(at — x) 
M \ * 
f true at any point 
after t > 
x 
a 
Tension =E^-=-- £ 
da a 
This is greatest when 
at — x=0, and then=V VE/i. 
So that for the case of an infinite wire it will break 
unless the statical breaking force > V jEjl; 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 length l. We 
have the additional condition that when x—l £ = 0 for all 
values of t, and this will introduce a number of discontinui- 
ties into the solution. Up to the time we may deduce 
the solution from the previous case ; from t = 0 to t — ~ we 
have as before 
