G. F. Becker — Impact Friction and Faulting. 201 



If this last case is compared with another in which the coeffi- 

 cient of restitution is not zero, w' for the same value of w will 

 evidently be smaller than the above ; and, by a similar argument 

 to that adopted in determining the total expenditure of energy, 

 it follows that 



-^= — (1— e ) w. 

 ax 



Here then the subtangent instead of being unity is 



_ W _ 1 



W 1 — 1 - e" 



so that c is a function of e which can become unity only when 

 e=0. The unit of measurement in this equation is the value 

 which the subtangent would have were e=0 or the system 

 wholly institutional, but it is also the length of the subtan- 

 gent in terms of the thickness of a sheet and consequently if 

 the system is institutional the length of the subtangent is the 

 thickness of a sheet, and in general if the thickness of a sheet is £, 



c(l-e 2 )=:d\ 



Now from an experimental case it is extremely easy to deter- 

 mine c and d so that the coefficient of restitution may readily 

 be found for a given case from the equation 



= \/i- 6 -=Vi- 



■ Injj. 



Fore=l or for a perfectly restitutional system c=oo and ^=e°=l 

 while for e=0, c=3, or if d is the unit, ju=e, and these are the 

 limiting values of /u. 



Friction in terms of elasticity — The energy equation for an in- 

 finite system of sheets may therefore be written 



2 



and if the functional relations of e and the velocity were 

 known w could be immediatel}" stated in terms of velocit}^. 

 On the other hand, experiments on friction are capable of yield- 

 ing values of e, though whether this would be convenient and 

 expedient method of investigation is another matter. Be this 

 as it may, the above equations establish a simple and rational 

 relation between friction and restitution ; they show that fric- 

 tion may be removed from the anomalous position it has hith- 

 erto occupied, and brought under the great problem of elasticity. 

 Geometrical differences between systems under friction and impact. 

 — There is a formal difference between the behavior of a finite 

 system of sheets resting against a rigid obstacle and that of a 



Am. Jour. Sci. — Third Series, Yol. XXX, No. 177. — Sept., 1885. 

 13 



