Formulizing all Epicyclical Curves. 447 



and the right-hand member is 



,P„P-.(K..i£rM=2.^,K3.c.). 



divided into odd and even angles as cos Pd= funct. (cosfi) 

 decreases by ex)en exponential differences. Hence the first 

 { } is represented by 



1 ,, y(!y--i-i)(g-i-2)..(g-2;+i) ' 



1.2.. 3.. 7 

 ._.^^ P(P-l) ,, (g-2)(g-i+i)(y-jf)..(g-2;-+3) 



o^* 2 ^ 1.2..(j-l) 



&V P(P-l).(P-3) (g-4)(<y-i+3) . (g-2; + 5) 

 a* 1.2.3.4 1.2.. 0-2) 



_^ P(P-l)..(P-5) (g-6)(g-y+5) . . (g-2;-+7) 

 a^' 1.2. .6 1.2.(j-3) 



&c. &c. 



and the second { } is replaceable by 



(y-l)(!7-i)(2'-i-l) . . (?-2/) 



1 X 



1.2.3..^ 

 - ^ (P-^) y (g-3)(9-i+2)(g-i+l)..(g-2;+2) 



fl^* 2 1.2. .(7-1) 



^ (P-l).(P-3 ) (g-5)(y-i+4) . . (g-2;-+4) l> 

 "^ a^- 1.2.3.4 1.2. .(7-3) 



__66 (P-l)..(P-5) (g-7)(g-i+6) . . (g-2i+6) 

 a^* 1.2. .6 1.2. .(7-5) 



&c. &c. 



the sum of these two 2 is the right-hand member developed 

 as SQ^. 



When a=bf w=— 3, there results a three-looped curve, 



t^ = 4a V — 1 Gr^a^w^ + 1 Qx'^a^ ; 

 hence 



^=4a2(/r2-j/2)2, r4=4aV^-j/^), r2=4flV-2/^)^> 

 rO= 4a2(a?2 -/)- \ r'^ = 4^2(^2 _y )-2, 

 are this three-looped curve, two-looped lemniscate, one-looped 

 circle, equilateral hyperbola, quadrilateral equal hyperbola, 



