1883.] On Curves circumscribing Rotating Polygons. 321 



scribe a rotating polygon, that is, the figures on whose sides the angles 

 of the enclosed polygon shall all lie during the whole revolution 

 of the latter in the former. But before proceeding to do this it may 

 be as well to consider how the departure from the circular form 

 originates. 



Let AB, CD, fig. 1, be the tool, which for simplicity may be 

 supposed to have two cutting edges only, AO, BO ; its axis being 

 CD. 



The tool is guided by the edges AC, BC, and the axis may be 

 considered to remain parallel to a fixed straight line, but to be other- 

 wise free. 



In fig. 2 let DEP be the section of the hole, ABO the drill seen in 

 plan, and C the axis of the hole. 



If the hole is circular C and C will be identical, but if by any chance 

 C and C are mutually displaced, one edge (BC in the figure) will have 

 a deeper layer of substance to cut than the other, and the result will 

 be that the instantaneous axis about which the drill is revolving is 

 shifted from C to some point, P, lying between C and B, but by turning 

 about any point between C and B, AC will have to cut through a layer 

 of substance gradually increasing in depth, and when this depth is 

 equal to that along BC, P and C will coincide, but as BC will now 

 encounter a layer of diminishing thickness, P will move through C to 

 some position between A and C. 



These alternate variations of the depth of the cut at the two edges 

 may continue indefinitely, the section of the hole varying at each 

 revolution, but if certain conditions are fulfilled the figure produced 

 will remain constant in character after the first revolution. 



Since P is always on that side of C on which the cut is deepest, and 

 C is on the same side of the perpendicular from C on AB, C will 

 describe some curve round C in the opposite direction to the rotation of 

 ACB. That is, while the drill rotates in one direction, its geometrical 

 axis describes some curve in the opposite direction round the axis of 

 the hole. 



If instead of the centres C and C being mutually displaced, one of 

 the edges meets with an accidental obstruction, or if one of the edges is 

 blunter than the other, the same results will be produced, except that 

 when the last mentioned cause operates the effects tend to accumulate, 

 and this probably is the actual origin in most cases. 



The same sort of result will occur whatever be the number of the 

 cutting edges of the tool, viz., the instantaneous axis will always be on 

 that side of the geometrical axis on which the cut is deepest, or the 

 resistance to progress of the edge greatest, and in virtue of this the 

 geometrical axis will rotate about the axis of the hole in a direction 

 opposite to the motion of the tool. 



This fact at once suggests a method of analysing the figures swept 



