278 Proceedings of the Royal Irish Academy. 



It is easy to see that tlie complete integral of the equation of 

 motion, 



must be of the form 



X = «(?"" cos nt + he""^ sin nf, (4) 



where a and h are arbitrary constants, and where m and n have the 

 values 



/. 



m = -^^, 



(5) 



-J'"-'t 



If we reckon the time from the commencement of the oscillation, 

 equation (4) reduces to 



X = ae'"* cos nf. (6) 



If T denote the time of a complete double oscillation, we find from 

 the above 



fnT 



On^e^e'' (7) 



where 



On - amplitude of the {n + 1)** vibration; 

 Bo - amplitude of the first vibration. 



Prom (7) we obtain the following working equation, for use in 

 the calculations to determine the coefficient of friction : — 



f'>'-^6 w 



Also, we have. 



T ~V 4' 



from which we obtain, after some reductions, 



T=-=====^- (9) 



If we introduce into this equation the value of / determined by (8), 

 we obtain h, which depends on the torsion only. 



