134 Mr. A. Mallock. [Dec. 8, 



The sliding along ED per unit advance of tool is — 



cos0 + cot(0 + 0) sin0 (3). 



The pressure under which the sliding takes place is to the normal 

 pressure on the face of the tool as 



cos (0 + 0) sin (0 + 0) (4). 



Thus the work expended in internal friction is proportional to 



#-i-{(cos0 + sin0 cot (0 + 0)) (cos (0 + 0)+/* A sin (0 + 0))} . (5). 

 sin 



The work done in friction against the face of the tool is for the same 

 travel proportional to 



n t sin6? (6). 



m sin (0 + 0) v } 



Collecting these results, the total resistance will be made up as 

 follows : — 



(1.) Bluntness=A/9 where p— radius of edge and A = constant. 

 (2.) Elastic and permanent distortion— 



cos 



(3.) Internal friction = - — -{ (cos + sin cot + 0) 

 v > sm0 lv ^ J 



(cos + + /i-j sin + 0) 



(4.) Friction against tool =^ 1 -t SU 



sin + 

 (5.) Elastic bending=B^ 3 . 



Considering these terms in order : — 



The first ought always to be small if the tool is sharp. 



Q in the second term is proportional to t, and is probably a function 

 also of and 0, but as observation shows that is independent of 0, 

 that is to say, that for a given material any form of tool that can be 

 employed causes sliding to begin in the same plane, it seems likely 

 that the reaction due to distortion should not vary much with 0, and 

 that for the present purpose Q may be regarded as t X constant. 



The internal friction vanishes whan 



cos0 + 0=— sin0 + (A), 



that is, when the resultant force through the face of the tool is parallel 

 to EC. 



When this is the case, however, there is a large component tending 

 to make the tool dig into the substance, so that the form indicated by 

 A is not one which can be used in practice without certain precautions. 



