208 



It maybe shewn that tlic ililVcrcntial c<>< i'.icicnt of , 



/,: - 



with respect to (> .; this result will 



(k - cos BY ' 



we procc 



I. Suppose Wless than 7', so that /; is less than unl 



Now fi may have any value from zero up to a certain limit, 

 M by experiment, which we shall denote by tane. Thus 

 at the limit 



sin0 

 tan e = . -^ , 



k - cos 6 



fore cot e + cot 6 - -^^ , 



sin 6 



therefore sin (9 + e) = k sin e. 



Let a be the least angle which has k sin e for its sine, so 

 that 



sin (6 + e) = sin a. 



1 as a is less than e the only solution admissible here is 



The expression ^ is not positive unless 6 is greater 



than the value which makes cos 6 k ; and f< .r values 



of 6 the expression decreases as 6 increases, and has its least 



value with which we are concerned when #=-; its value 



1 P 



then being , , that is, .... 



I \ nee we have the following results : 



If the coefficient of friction is less than -vp. there is no posi- 

 tion of equilibrium. 



