An easy "Rule for Formulizing all Epicyclical Curves. 44-5 

 is to be put equal to 



2 cos {p - q)d' = SB A cos fl' = ^ ) . 



Ex. 7 



»«=— 3> p—q'P + gi''^:lO:2:5, 



2 cos 5Q = —r ^ + — =2 cos 25' = -5 — 2. 



Case 3. a and 5 unequal, p positive. For brevity, put 



cos«+sin«. -/— l = (c + s)«=(c + s)*, 

 .'. rcosfl + rsin^. ^^^=a(c±s)q^ + b{c±s)p^z=a{c±sy^ 



+ %±5)J=r(c±5/, 

 .*. 2rP-9cos {p-'q)^ = rP-'t{c + s)^"^ + rP-i{c—sYf'' 



-{a{c+s)l + b{c + s)l}P-i+ {fl(c-5)'+6(c-s)5}^-» 



h'aP~'i 

 = 2aP-? cos ( j05'(p — q^<^ = y^|/) -+• S r— . 



l.2..i V + ((;— 5)?^+'^ = 2cosg\I/ + i\I// 



=2^>p-^cos«4.+S2a'^-^-'ig::ig-^i-P::ig~^^--(^"g~^'+^) 



1 . 2 .. 2 



COS {p—i)'^, 

 agreeably to the binomial development. Hence 



2 cos (j3— g)a=22*A/cos 5= -j 

 put equal to the sum of 



2cosxf/= — -^ =^j 



is the equation of the curve, and easily expansible by the for- 

 mula for expressing the cosine of a multiple angle in powers 

 of the cosine of the simple angle. Thus, for 



p—q=i ; 2^=:2ar cosj94' + 2arcos {p — l)4f 



p—q^^; ^x^ — 2r^ = 2b'^r^ cosp^ + ^abr^ cos' [p—l]'^ 



-f2aVcos(^ — 2)^^ 



^=5, (7=3; a*J3(4^2_2r2)=Q5-5a2^,2Q3_,_5^4^4Q^2a2Q4 



Case 4, a and i still unequal, but p negative ; 



