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LXV. An easy Rule for FormuUzing all Epicyclical Curves 

 with one moving circle by the Binomial Theorem. By S. M. 

 Drach, Esq., RR.A.S.'^ 



I REFER to the monography " Trochoidal Curves " in the 

 Penny Cyclopaedia for the various forms, but which recent 

 article does not mention the following generalization, extend- 

 ing the use of Newton's theorem to these curves as well as to 

 the interpolation series. 



Origin is at the deferent's center, x positive towards one 

 apo-center. 



;r=rcos 9=a cos q<p + b cos p<p 

 y=r sin 9 = a s'm qip -\- b sin p<p 

 i^szx^+y^—a^-\-b'^ + 2ab cos {pf — q<p:=^) 



The resulting function (r% x) shows the general symmetry 

 as regards y ; when w = -, interchange a and b. 



Case 1. a=bi p positive. 



,: a;=:2a cos^—-^ <p X cos^--^<p, r=2acos^—-^<p, 



fl--^i— 2^, _=2cos(p-g').— — , -=cos(;? + g') 



qn=:p 



<pn—(p of Pen. Cyc. 



2 ^ a '^ ^' P + T '' P + q 



Hence 2 cos {p—q)Q developed as 



22'A.(cosS = ^y 



is to be put equal to 2cos{p + q)d' developed as 



2B.(2cos«'=^y 



for the general equation of the curve. 



^*^- n=L p^q:p + q::^:l0::2'.5, 



*i2 f^ ^f& f 



2 cos 26 = 4.-^-2=2 cos 56'= —^ g- +5- 



?-2 a^ a-* a 



is the equation. 



Case 2. a=6,;? negative; change sign of 5' in first case, 

 and 



2 cos (;)-|-2')d=22'AYcos d= ^j 



* Communicated by the Author. 



