620 

 and Basset: W^6:^^av -, where ^ is the coefficient of external 



friction. The second expression is also equal to: 6nL,av 



(-èJ 



so that the difference between the results is: G:rz^v 





'+4 



A mistake suggests itself in the indirect method and seeking for it 

 we are led by the remark that the two values become equal when 

 /i = and when i? ^ oo viz. when there is no friction and when 

 there is no sliding. 



In both cases the friction does no work : in the first case because the 

 force becomes zero, in the second because the way becomes zero. 



Probably we shall therefore have to add the work of the friction 

 at the surface to the heat, calculated from the dissipation- 

 function with the formula 1 1 Ï Fd.v di/ih, wUeve F=^ — i^{^-\-^-\-(^y 



+ 2S {a' -\- b' -\- c' -\- 2/' + 2(/' 4- '>'/i') ')■ This proves indeed to give 

 the right result. 



At a point determined by the angle ^ the force of friction is: 



A B ) -iva* 



•■(■'3 



U=^ + r 5171 xV), where --1 = — -~ and B — ^ ^ 



1 + 3— 1 f 3 



3gv 



2|3a . , 



80 that u = ^ s'l/i i). This friction is exerted on the surface 



2na*smih di)^, while the relative velocity, opposite to the force, is 

 —; sm^. 



1) Notation of Lamb. 



~) See Lamb p. 230 and Basset p. 270. 



