30 REPORT— 1871. 



We have 



,._ly' s-_y t=lo^y 

 2 00-' X X 



ct--y-, (i=y,, y=--, 8=-: 

 X X- X y 



and - = ^ = -r=— -\ as it should be. 



Ort Douhhj Diamitral Quartan Curves. 

 By F. W. Newmax, Emeritus Professor of University College, London. 

 This paper aimed to detail the /o/-»i of the curves, and point out the simplest modes 

 of investigating their peculiarities. It distributed the general equation into three 

 groups of five, five, and four families, and was accompanied by seventy-six diagrams. 

 If we call 



A.r'+2D.ry-|-B2/'-l-2E.r=+2Fy=-|-C=0 



the general equation, the first group of five families is when A or B or both vanish, 

 the second group when D or F or both vanish, and these together nearly include 

 all the forms. For in the third group, from which either no term of the original 

 equation vanishes, or only C, three of the families are at once reducible to the se- 

 cond group by putting either y--\-f-=y'^, else x'-\-e^=x''; then the proposed curve 

 of {xy) is visibly at most a mere variety of the preceding, being either of the .same 

 species with, or'of a lower species than, that of (.ry'J or of {x'y). In the case of 

 B=/3', D=S2, F=— /'^, this reduction is impossible ; but then by operating on x 

 instead of y, it becomes possible unless also A.=a!^, E=— e'^; that is, the method 

 fails only when A, B, D are of one sign, and E, F of the opposite. The analysis, 

 thus limited, readily yields the same result, that the forms have nothing new. 



A cross division of "the species is into Limited and Unlimited loci. All are Cen- 

 tric, the origin being the Centre. Finite forms are Monads, Duads, and Tetrads. 



Monads are : — 1. A symmetrical oval (say, a Shield), as from 



2. An Oval with undulating sides {Viol or Dumb-bell), as nV = (>«'■'— .r^)(.T2-|-w'^), 

 m'^.^n^. 3. A Lemniscate or Double Loop, a s a'^y'^ = [m-—x'^)x '^. i. A. Scutcheon, 

 with four sides undulating, as i32«/^=/^ + V(w2—a;2)(a:2 +«■'], when m,'^>n'^, and 

 f^<mn. 5. Oials in Contact, as ^'^y* = (tn'^—x^)x^. 6. Pointed Hearts, crossing 

 obliquely at the centre, as 



^^y^ =mn+B^x^±>^ {{m''—x''){x''+n'')}, 



when m^ — w^ > 2m?», and 8* > '" ~" .. 7. Hearts in Contact ; the same equation 



2m)i 



rtw2 AT 2 



as before, only with 6^= — . 8. Inter secttny Hearts, as 



2mn 



when «i^> n^. 9. Litersectiny Ovals : the same equation as in 8, only with m"^ <w^. 

 AU curves are here deemed Monads which can be drawn without taking the pencil 

 of:' the paper. 



Duads are : — 1. Twin Ovals (not singly symmetrical on opposite sides), as 



a'^y^ = (in^ — a;^) (a;^ — w^ ). 



2. Tivin Beans (or Hearts, Dicuamos). 3. Pair of Sandals (Disandalon): this has 

 always two double tangents parallel, yet the disposition of the four points of con- 

 tact is not the same in all cases. (They form a rectangle when D^U ; they are in 

 lines diverging from the centre when F=0.) 4. Pair of unsjTnmetrical Lemniscates, 

 which I call Pour Kites. 



Tetrads can only consist of unsymmetrical Ovals, symmetricallj- disposed. 



Of the infinite curves, one very limited class may be called Parabolic, those in 



