322 



Mr. A. Mallock. 



[June 21,. 



out by the angles of the tool. Let e be the distance between C and C 

 and the direction of the line joining them, then e can always be 

 expressed by the following equation : — 



e=6 o -t-26isin(*0 + d5i) (1), 



and since the tool is a rigid body, its angles are displaced from the 

 position which they would occupy were the hole circular by an amount 

 n in a direction 9. 



Let A 1; A 2 (fig. 3), be the adjacent angles of the tool, which is sup- 

 posed to be a regular polygon of n sides, P 1? P 2 the paths along which 

 they move, c the centre of hole, and p x p 2 the path described by c' y 

 the centre of the polygon. 



It is clear that if P 1? P 2 , . . . are to form parts of a single curve, 

 . . . A g , A 2 must occupy the same position when the polygon has 



turned through — that . . . A 2 , A A occupied at first ; thus, while the 



n 



polygon rotates once, its centre will move n times round the centre of 

 the hole, and hence if be the angle which AC makes with a fixed 

 line passing through c, and if also be measured from the same line, 



.'. e= e + 2ei sin (in(fi + ai) (2); 



hence e goes through n complete cycles, while increases by 2ir ; but 

 as e revolves in a direction opposite to that of the polygon, it will go 

 through n + 1 cycles if the be measured from a line fixed in the polygon. 

 Let C A be this fixed line, then A goes through n + 1 cycles of displace- 

 ment in virtue of the first term of e, viz., e , and in+1 cycles in virtue 

 of the ith. term, while CA revolves once ; hence the symmetry of the 

 curve P 1? P 2 , . . . will be n + 1-fold as far as it depends on e ; but 

 the only other terms which will give symmetrical curves if present 

 with e are those for which i has such values that while changes 



from < ^ 7r to — , in<j> changes through ^ + a complete multiple of 

 n+1 n n+1 



27r, or for which 



i=l+p(n + l), 



where p is an integer, and if the symmetry of P 1? P 2 , ... is to be com- 

 plete — that is, if each interval, P x P 2 , P 2 P 3 , &c, is to be composed 

 of similar halves — ati must be either or a multiple of ir. 



If the sum of the coefficients e$ in e is small compared with CA, the 

 equation of the curve P X P 2 referred to its centre is approximately 



/3=/) + e cos O + l)0 (3), 



where 



/>=CA and /> =CA. 



