120 G. F. Becker — Impact Friction and Faulting. 



form, or in other words, unless the passive mass is infinite. 

 Were this the case w would be diminished by and could be 

 diminished only by wdx ; and dw=wdx, the differential equation 

 of the simple logarithmic curve. But when a portion of the 

 energy imparted to the section at x reappears as kinetic energy, 

 dw must have a greater negative value. The equation might 

 be written dia= — wdx— Qdx, but here Q must be some function of 

 w. Now it is an elementary condition of equilibrium in the case 

 under discussion that for a given displacement at any section, 

 say the one at x, the energy potentialized between this section 

 and the free end of the rod shall be a minimum, so that if a is 

 the value of x for this end of the rod 



/ 



wdx = min. 



r 



This minimum must have its maximum value when the entire 



energy is potentialized or when a = co. But if dw——wdx 



and a (or n) is infinite 



~ a , dw 



loclx— r , 



ax 



so that the infinite rod being merely an extreme case of a 

 finite one, the integral for finite a must be less than —diu/dx by 

 a value which disappears when the kinetic energy of the system 

 at the moment of maximum compression is zero. The effect 

 of the uniform distribution of kinetic energy upon the value 

 of dw has already been traced and the equation 



/ 



1 7 dw Mu 2 1 



wdx= — -T- — 2- 



dx 2 n+l 3 



therefore fully accounts for both the kinetic and the potential- 

 ized energy. For x=a, iu must disappear, and this definite 

 integral may therefore be written G—fwdx. Introducing this 

 value and differentiating 



cfw _ 



dx 2 ~ ' 

 If an arbitrary unit of measurement, c, is adopted in this equa- 

 tion and to and x are each divided by it, 



which leads without difficulty to 



w=Ae~ x / c + Be +x / c > 

 where A and B are arbitrary constants to be determined both 

 in sign and value by the conditions. 



Determination of constants for general case. — For the free 

 extremity of the passive mass w=0 and therefore 



B=-Ae- 2a / c , ' 



