TRANSACTIONS OP THE SI'CTIONS. 23 



If, however, C'=0 and E=0, the Polar equation becomes simply r=|B' cos 3\fr, 

 which is a Starry Trijuga, admitting r=0. 



In general, the equation to rect. coords, falls under the class 



which is the highest form of those which I call Quartotertian. 



The Polar equation may be presented in the fonn cos 3\|/'= -^ . 



The curve is evidently in every case finite, and the species must apparent^ change 

 according as the equation admits the forms ar^ cos Sx//'^ (?•"— 6^)(r^— c-),fO"^cos3\/f = 

 ('•'+6')(»-'— C-), «/-'cos3\//=(?-2 + 6^)(»--^+e^), or finally rt/-^ccs3\|/-=(»-^+i2)2+c*. 



Evidently y =0, when sin 3\//'=0. 



If y^ = \2^''-\-6, cos 3\^=cos 3(9. Hence the figure is Equilateral. 



On Quartan Curves tviih 3 or 4 Diameters. By F. W. Newman. 



This Memoir proposes and solves the Problems, in what case Cui'ves of the Fourth 

 Degree have 3 or 4 diameters. 



It briefly analyzes the forms of the Trldiametral Curves, under the heads which 

 rise out of the general equation 



2«r' cos 3f =r'+26r= + c=R : 



1. when E=r*, or 2rtcos3A/r = r; 



2. when E=:r'— /3V,or 2«/-cos3>/' = r=— /3-j 



3. when E^r'/SV^; 



4. when E=:r'— y ; 



5. when E = ?-*+y^, and generally when E is essentially positive; 



6. when E=(r^ — /3^)(?'- — y-j, which has 3 remarkable forms; 



7. when E= (r- 4-/3'-) (?•'—•/-), which has 2 forms, according as /3- is>-).= or <:-y2. 



'•1 



On Monodiametral Quartan Curves. By F. W. Neavman. 



This Memoir is a continuation of the paper laid before the Association last year 

 on Doubly Diametral Quartan Curves, and follows upon a notice now presented on 

 Tridiametrals and Quadridiametrals of the same degree. 



Employing X„ to mean an integer function of x, of degree n, it is proposed to 

 digest ali the Monodiametral curves into five Groups, twenty-one Classes, as 

 follows : — 



1. y'-|-2A!/^=Xj, or x an integer function oi y'^ [Quartic Parabolas]. 1 



2. ?/' + Xi?/"=Xi, or X rational in y'^ (Conic Parabola for asj'mptotes). f 



,' V ^Zv f (Horizontal and Vertical asymptotes). 



5. «/^=X^ (two equal and opposite Conic Parabolas for In these, y^ 

 II.<( asymptotes J. S- is rational 



6. xy-=Ki ; or, the Semicubical ; with Tertian asymptote. in x. 



7. X,j/- = X3 (Conic Parabola for asymptote). 



8. X,?/- = X^ ; Quartohyperbolic. J 



9. y-= y/X.^ ; Quartotertian of 1st Branch. 



10. y''+A'-= v'X, ; Quartotertian of 2nd Branch. 



11. _«/=-t-Xj= \/Xj (Epiparabolic asymptote). 



12. 2/'- + Xj= ^^2 (Unequal Conic Parabolas for asymptotes). [_y'^= V^ 

 is omitted as Doubly Diametral.] 



[ 13. «/-+Xi= \/^3 j Quartotertian of 3rd Branch (jE/«parabolic asymptote). 

 ri4. ?/^-|-X2= \/Xi (-E/j/hj'perbolic asymptote). 



I 15. 7/^-{-X2=VX2 (admits two as3'mptotic hyperbolas, with their recti- 

 IV. <; linear asymptotes prtw/Ze/, set to set). 



I 16. 2/^+X2= /y^Xgj Quartotertian of 4th Branch. {Tridiametrals vd^XsthQ 

 ( excluded.) Epihyperbolic asymptotes, 



in.<^ 



