136 



Mr. A. Mallock. [Dec. 8 r 



then suddenly drop to a minimum. It is plain also that a tool with a 

 tendency to dig will call into play forces of a like character. 



Vibrations are in some degree neutralised, and digging entirely 

 avoided, by so shaping the shanks of tools that the centre about 

 which they vibrate is in advance of the normal to the direction of 

 motion through the cutting edge. 



Fig. 12. 



Let /j be the distance of the centre of flexure (s) from the cutting 

 edge (P), a the angle which the line joining the centre of flexure and 

 edge makes with the normal, and ha, the angular distance of the tool 

 from its mean position, the thickness of the shaving removed is t+pda 

 tan «, and if t be the period of the vibration of the tool, cx is propor- 

 tional to sincT, or £a=/csin cr, say, k and c being constants; and since 

 the pressure exerted by the shaving on the tool is proportional to its 

 thickness, pic sin ct tan a. also expresses the variable part of the reaction. 

 The effect of this variable pressure is neither to sustain nor extinguish 

 the vibration, but to increase in effect the rigidity of the tool by a 

 quantity proportional to tan «. 



In tools designed for rough work a. is usually small, but when the 

 quality of the surface left by the tool is of more importance than the 

 thickness of the shavings which it can remove, it may be largely 

 increased with advantage. 



Fig. 13. 



Fig. 13 shows an excellent form of cutter-holder, designed by the 

 late W. Fronde, F.H.S., in which a. is about 45°. Tools held by 

 such a cutter-holder leave a very smooth surface on the substances 



