[ *87 ] 



LX. Supplementary Considerations to Mr. S. M. Orach's 

 Epicyclical Papers (Phil. Mag. June to July 1849)*. 



SINCE publishing my above-mentioned papers, I have 

 unsuccessfully tried the general solution for more than 

 two circles. However, knowing <p, the polycircloid 

 3S—^{a^ cos g.f + b. cos;?.^ = X.= R. cos ©.), 

 j/=S(a.sin g'.(p + 6.sin;?.<p = Y. = R.sin @.), 



may be regarded as the ultimate locus of a series of bicircloids, 

 the centre of each being on the curve o( its immediate prede- 

 cessor, beginning with the centre of the deferent. With an 

 odd value of/, one ^^=0, and this last becomes a circle. 

 We have also 



w=Xa^cos{q^(^ + t)i y = Xa^%m {q.<p-ht)y 

 giving 



r^ = 2a? + 2Xa.aj cos q.(p — q.fy 



independent of the common constant t. 



r = l,a.cos{q.<p + l — Q), = Xa^sin{q.ip-{-t—$). 



If the cos angles in r^ naturally arrange themselves as the 

 powers of cos \J/= cos p(p — (j<p, so that cos ^ is extractible by 

 quadratic, cubic, or biquadratic equations, the solution is 

 always analytically possible; e. g. the tricircloid 



x=a cos qf + bcospf +fcostf, y— a sin qtp-^-b sin p^ +y sin tf, 



.'. 7-2=a2.|_^2_|_y^_,_ 2abcosp — g(p + 2afcosg—t(p + 2bfcosp — i<p' 

 Let 



.'.r^ = a^+{b~ff + 2ab + 2bf cos i> + 4bf cos- 4>, 



.'. 4:bf cos ^=—ab-a/± \^ ^bjr^ + {a^-^bf){b-ff. 

 And then 



2r^cosA9={a(c + 5)'?^ + &c.}^+{«(c-s)'?^ + &c.}^ 

 X{K-l),.{K-i+\) i{i-l)..{i-j+l) 

 1.2..i 1 . 2 . . J 



rrp;^(>- 0?^ +('-iW +j*^ = (' - ^•''+'')^ + (c - 5)(^ - -•^'+ '^^ 



= 2 cos {q — 2j + i)^}. 

 Substituting the powers of 2 cos xj/ in 2 cos {q — 2j+ #, we get 

 the general equation of the curve. 



* Comnuuiicaled by the Author. 



