to his Epicyclical Papers. 489 



The loops of these curves are in general not analytically 



identical with the lemnoid resembling >0, where q — 2,p = S, 



For in the other loops (wn'=l) R^^'a^ gives 



{2^664^-2 + &c}'''=2X"'"''-R2(«'+ 1)(2X)»+' &c. ; 



so that X does not =X generally. 



The radius of curvature of the syphonoid 



(a?=a cos qfy y=- cosp<p) 



{(fia'-x'^) -^p\b'^-f)]^-^{p^qy Va^-x^-q^px x ^6^-^}, 

 and its area 



fydx = (p \/ (a2-a?2)(62-y) + qxy^ ^{q^ -p^) . 



For the lemnoid 



(a:=a cos 5'(p, y—y'—bs\x\p(p) 



change +g to —q in these two expressions. Their arc- 

 lengths are 



/' / sin^ 



dfy « V sin^ qf + b^ ^^^9. p<p 



respectively. 



The following table gives some results at critical points. 



Thus for the common parabola, 



- abi - ab 

 3 3 



are the respective quantities (the latter for half the area at the 

 axis). Thus every lemnoid corresponds to a syphonoid. 



The paragraphs in the Literary Gazette for March 28, 

 1846, were not penned by me, who had just made Mr. Perigal's 

 acquaintance. Subsequent investigations have shown me that 

 this Kinematic Parabola, being a finite portion of the common 



