MATHEMATICAL PAPEBS ffiotrtxm: 0. J. CLAY AND SONS, CAMI3HIBGE UNIVEBSITY PBESS WABEHOUSE, AVE MAMA LANE. ! DEIOHTON, BELL AND co. j p. A. BBOOKHAUS. Jforfti MAOMILLAK AND 00. THE COLLECTED MATHEMATICAL PAPERS ARTHUR CAYLEY, So.D., F.E.S., BADLKUIAN PHOPHSBOK OP PURE MATHEMATICS IN THE UNIVERSITY OP CAMBRIDGE. VOL. VI. CAMBRIDGE : AT THE UNIVERSITY PRESS. 1893 [All Mights reserved.] ADVEETISEMENT. THE present volume contains 33 papers numbered 384 to 41G published for the most part in the years "1865 to 1872; the last paper 416, of the year 1882, is inserted in the present volume on account of its immediate connexion with the papers 411 and 415 on Reciprocal Sur- faces. The Table for the six volumes is Vol. I. Numbers 1 to 100. II. 101 158. III. 159 222. IV. 223 299. V. 300 383. VI 384 416. CONTENTS. 384. On the Transformation of Plane Curves ..... 1 Proo. London Math. Society, t, i, (186566), No. in. pp. 1 11 385. On the Correspondence of Two Points on a Curve ... 9 Proc. London Math. Society, t. i. (1865 66), No, vn, pp. 1 7 386. On the Logarithms of Imaginary Quantities .... 14 Proc. London Math. Society, fc. n, (186669), pp. 6054 387. Notices of Communications to the London Mathematical Society . 19 Proo. London Math. Society, t. u. (186669), pp. 67, 2020, 29, 6163, 10310']., 123125 388. Note on the Composition of Infinitesimal Rotations . . 24 Quart. Math. Jour. t. vm. (1867), pp. 710 389. On a Locus derived from T^VQ Conies ...... 27 Quart Math. Jour. t. vm. (1867), pp, 7784 390. Theorem relating to the four Conies tuhich touch the, same two lines and pass through the same four points Quart. Matli. Jour. t. vm. (1867), pp. 162167 391. Solution of a Problem of Elimination .... Quart. Math. Jour. t. vm. (1867), pp. 183185 392. On the Conies which -pass through two given Points an tivo given Lines ...... Quart. Math. Jour, t. vm. (1867), pp. 211 219 v l CONTENTS. 393. 0/i the Conies which touch three given Lines and pass through a given Point ....... , . Quart, Math. Jour. t. vm. (1867), pp. 220222 304. On a Locus in relation to the Triangle . . . L/ * * " Quart Math. Jour. t. vm. (1867), pp. 264277 305. Investigations in connexion with Casey's Equation . , Quart. Math. Jour, t, vui. (1867), pp. 33d 841 3DC. On a certain Envelope depending on a Triamjk imcribcd in a Circle ........ . Quart. Math. Join-. t, ix. (1868), pp. 31-41 imd 17fi 170 397. Specimen Table M , <M (Mod. N) for any prime or composite. Modulus . ......... Quart, Math, Jour. t. ix. (1868), pp. 96-90 ami pl,vte 398. On a Certain Sextic Developable and Sextic Surface conned t/ieremth . ' ' ' * Quail Math. Jour. t. ix. (1868), pp. I20-.ua and 373-370 890. On the Cubical Divergent Parabolas ..... Q^rt. Math. Jour. t. ix. (1868), pp. 18G-189 W. On tlu, Cubic Ct. m inscribed in a given Pencil of flfe .^,, Quart. Math. Jour. t. ix. (1868), pp. 210-221 401. A Notation tf tjle foints md ^ ^ pasmts Qa.t Math. J 011 , t Ix . (1868)| , )M68 __ 274 2. On a Singularity of Surfaces Quart. Mail, Jom . t . , x . (1868)| pp jM!uas8 ' ' ' ' ia.1 3. On Pascal's Tlieorem Quart. Maft . Jou , , I18 . . . , lg|l 404. Reproduction of Euhr's Memfr of ,,,. rt Solid Sody. . J 758 on the Rotation of a th. Jou , t . IX . - ias otvi , fe 1 p> HJ ^ . 147 CONTENTS. IX FAQE 406. On the Curves which satisfy given Conditions .... 191 Phil. Trans, t, OLVIII. (for 1863), pp. 75143 407. Second Memoir on the, Curves which satisfy given Conditions ; the Principle of Correspondence . . . . . 263 Phil. Trans, t. CLVIII. (for 1868), pp. 145172 408. Addition to Memoir on the Resultant of a System of two Equations .......... 292 Phil, Trend, t. OLVIII. (for 1868), pp. 173180 409. On the Conditions for the existence of three equal Moots or of two pairs of equal Roots of a Binary Quartic or Quintic . . . . . . . . . . 300 Phil, Trans, t. OLVIII. (for 1868), pp. 577588 410. A Third Memoir on Skew Surfaces, otherwise Scrolls . . 312 Phil. Trans, t. CLIX. (for I860), pp. 111126 411. A Memoir on the Theory of Reciprocal Surfaces . . , 329 Phil. Trans, t. CLIX. (for 1869), pp. 201229 412. A Memoir on Cubic Surfaces ....... 359 Phil. Trans, t. OLIX. (for 1869), pp. 231326' 413. A Memoir on Abstract Gfeometry ...... 456 Phil, Trails, t. CLX. (for 1870), pp. 5163 414. On Polyzomal Curves, otherwise the Curves -77+Vl 7 -t-&c. = . 470 Trans, R. Soo, Edinburgh, t. xxv, (for 1868), pp. 1110 415. Corrections and Additions to the Memoir on the Theory of Reci- procal Surfaces 577 Phil. Trans, t. OLXII. (for 1872), pp. 8387 416. On the Theory of Reciprocal Surfaces ..... 582 Addition to Salmon's Analytic Geometry of Throe Dimensions, 4th ed. (1882), pp. 593604 Notes and References ......... 593 Plates to face pp. 52, 122, 190 Portrait ......... to face Title. CLASSIFICATION. GEOMETRY : Abstract Geometry, 413 Curves, 114 BeciprocHl Sm-faces, 411, 416, 416 Cubic Surfaces, 412 Skew Surfaces, 410; Developable, 398 Singularity of Surfaces, 402 = Curve., ! pencil of aix K MS , ANALYSIS : Hotatio, ls , 388 M cmoh . f of eight,, M enioi ' ' > 8, 40D 384. ON THE TRANSFORMATION OF PLANE CURVES. [JTi'om the Proceedwcjn of the London Mathematical Society, vol. I. (1865 1866), No. in.. pp. 111. Eead Oct. 16, 1865.] 1. THE expression a "double point," or, as I shall for shortness call It, a "dp," is to be throughout understood to include a cusp : thus, if a curve has S nodes (or cloTuble points in the restricted sense of the expression) and K cusps, it is here regarded as having 8 4- K dps. 2, It was remarked by Cramer, hi his "Theorie des Lignes Courbes" (1750), that; ft curve of the order n has at moat (?&-l)(ji 2), =s -J- (n 9 3n) + 1, dps. 3, For several years past it has further been known that a curve such that the coordinates (fa : y \ z) of any point thereof are as rational and integral functions of tlie order n of a variable parameter 6, is a curve of the order n having this maximum mi rnboi 1 |(?i !)( 2) of dps. 4 The converse theorem is also true, viz. : in a curve of the order n, with (ra l)(?i 2) dps, the coordinates (so : y : z} of any point are as rational and integral functions of the order n of a variable parameter Q or, somewhat lesa preciselv. him coordinates are expressible rationally in terms of a parameter Q. OS THE TJUHSFOKlttTION OF PLANE CUBVBS. fgg. series of curves of the order n-1, given by an equation V+9V = (\ eontaininrr ,,, rbitrary parameter 6; any such curve intersects the given curvo in tl.o ,1,,, ,,| >.mt,, lg as two points in the S.-S points, and in o e other point; hone,, w ' tl,,,, ' only one vamble pomt of intersection, the coordinates of HUH point, vii! ., ,,,', ,,, :.tes of an arbitrary pomt on the given curve, are expressible mtio mllv in t, r "*" * = the foregoing theorem is tha o expressible rationally in terms of a parameter 9 S fet po ints ^succeed each other in ^^ AR ^" ^ T" V " ',"" tamed by g lv i llg to the parameter it s different real Un fron , " ,'"' rve may be termed a , curve. "^ '"' ' lllui inciation it , s necessary to refer to nae-e l<f! , Tr PP ' , " <mi ' lll!to th " ; Abelschen Functioned Or*, t J "( 8 7, 1 'TT-- '""' ^P' " Thll '' rio I lei L1V - ^ a7 >: PP' Ho-],,,,, Vlai> th(J ( ,, mil(!i(l |. ilm of any order form (1, f)(i, , ) = . P aiamete " (6 ';), connected by nu (!(I , llUi , m >0 - rationally in terms of the p a mn rn IK \ a certain form, viz.: ' te ' S (fi ') O ' mn *<l by an o,,,,atioii of fl = l, the equation is (1, t\t n ~f-n , , j^'-^erootof;^;,^: oT "' tho """" "-4 the equation is (1, py> a x,_ n v i. the square root of i 6 'exU 'fltbn O f P" to " 10 * mo " > 2, viz. ; ' 5 odd, = 2/1 -3, the equation is a tw, , that treating (f ^ , r,^' f r(1> '" =. and is besides such represented Lyi 2 )f d J rteSmn CM '*^s, the curvo thoT' that treating (f^' 1?' P" (1 ' '' ) " = . a "d is besides sue -" ^ & of this, f , . , f coordmates (. s , ; , ^ ,- f. place of (f, ,), Possible rationaUy and hooge neously i n teni 384] ON THE TBANSFOKSfATION OF PLANE CURVES. 3 of (f. % D connected by an equation of the form ( )* ( ^ = 0. Such an aquation, treating therein (, 1;, f) as coordinates, belongs to a curve of the order 2/t, with a /i-tuplc point at (f = 0, =0), a /*-tupIe point at (77 = 0, = 0), and which has besides (M-2) 3 or (/*- !)(/*- 8) dps, according as D = 2/i-3, or 2/*-2. The coordinates (x-.y.s) of a point of the given curve are expressible rationally in terms of the coordinates ($''>}' ) of a point on the now curve ; and we may say that the original curve is by means of the equations which give (to : y \ z} in terms of ( ; 7; : ) transformed into the new curve. 11. A curve of the order 2/a may have ^(2/4-1) (2/t- 2), 2^ s -3/t+"l dps; hence in tlie new curve, observing that tho /t- tuple points each count for Kffft we have In the caso jD = 2/i ~ 8, Deficiency = 2/t fl In the case D = 2/j, ~ 2, Deficiency = 2/i 3 8/t + 1 -/*"+ 9,i 9 n *""* fffM ' ' >Wj " J^/ Moreover for 7J = 0, tho transformed curve is a conic, with dps, and therefore wibh deficiency =0; in' the case J)~l, it is a quartic with 2 dps, and therefore deficiency = 2; in the caso D = 2 it is a quintic with a triple point =3, and a double point = 1, together 4 dps, and therefore deficiency = 2. Hence in every case the new curve has the wamo deficiency as the original curve. 12. The theorem thus is that tho given curvo of the order n, with deficiency D, may bo rationally transformed into a curvo of an order depending only on the deficiency, and having the same deficiency witli the given curve, viz.: D = 0, the new curve is of tho order 2(=D + 2); = 1, it is of the order 4 (=. + 8); JD2, ifc is of the order 5 (= + 3); and D>2, it is for D odd, of the order D + 3; even, of the order .0 + 2. It will presently appear that these are not the lo which it is possible to give to tho order of the now curve, Riemanu' r ON THE TRANSFORMATION OJT PLANE CUIWES. - of ( " i * ): H is con to iew - and [384 1-1 the s toV* with deflcie 7 A which is by the '^z^^^Htf eacl i i / -r, 1 w> 7 ?> 6; ~u. ine transformed <leficiency D ^ ^ ^^ ^^ equations f f-P - n . /? f i ^ ' J * -, t0 each of fch i ^ / ' " 5 ' * : ^ theie C01 'responds a ainde set of of tlie nf /fc . M ' IT. If, however, the curves Po = ff-n - curve is equal to this number. I assume that we have k < n : P though t , le same ii(*8)-l pia i"t," ' ' f "* " ler sections common, the equations bein/f fee onms have all their inter the tan P + ^ + Jlo To mX t ***, "T" ' ^ " Manti<lal relati pebble, TO ^ tike the c , v f P -* (P^'o' f T*"? m ' Ve " ' W a as many poiuta ns possiblej and it ""'3^ from T]^ m89 ' .r "" on > ff = groateat possible number i s =*/,% + 31^2 i,f? , , ""J"* , J " St " lade ' that the number of points m the eSvt ^-0 rflTff'' T ^T'' ""^ 8 -- 3, K'^- 3^) -2, respectively. " """" ^ to 7 e ; 00 f de S ^e curve ff = with deficiency D, or with i(tf -lh,)- i Ai = ?t 1, we mav asmmn flinf I-K i. r ^^ ' of the order -!, ^ pas f *** ^ '"-fa* Curves P-0 ( -0, J2-0 through the * (m - 3n) - .D + 1 dps, and through 2 + JD - 4 other poiiitB, together j^j^ T^ points of thfl 384] ON THE TRANSFORMATION OF PLANE CUEVES. This being so, each of the three curves will meet the curve U=0 in the dps, counting as ?i a 3?i 2.D + 2 points, in the 2)i + D~4> points, and in D + 2 other points, together ?t a )i points ; whence the order of the transformed curve is D -I- 2, 20. In precisely the same manner, secondly, if k = n~ 2, then we may assi that the transforming curves P = 0, Q = 0, R - 0, of the order n - 2, each pass through the 4( a -3w)-D + l dps, and through n + D-4 other points, together ^(?i a ~?i) 3 points of the curve 7=0; and this being so, each of the three curves will meet the curve U = in the dps counting as n* - 3n W + 2 points, in the n+ -D 4 points, and in 7) + 2 other points, together a - 2??, points ; whence the order of the transformed curve is also in this case = D + 2. 21. I was under the impression that the order of the transformed curve a not bo reduced below D + 2, but it was remarked to me by Dr Clebsch, that in case JD > 2, the order might bo reduced to D + 1. In fact, considering, thirdly, ease & = w-3, we see that the transforming curves P~0, (2=0, .# = of the o , 8 may bo made to pass through the (w 9 - 3) - + 1 dps, and through JD - 8 other points, together (n a - 8w) - 2 points of the curve U = ; and this being so, each of the three curves meets the curve U= 0, in the dps counting as (?t a - 3n) -2.0 + 2 points, in the -8 points, and in > + ! other points, together n* - 3 points ; whence the order of the transformed curve is in this case =/)+!, 22. The general theorem thus is that a curve of the order n with deficiency can be, by a transformation of the order n-I or n-2, transformed into a curv 6 ON THE TRANSFORMATION OF PLANE CUEVttS. [8g the order + 2; and if D>2, then the given curve can bo by a transformation of the order 3 transformed to a curve of the order .Z) + l: tho trans formwl curvo having iu each case the same deficiency D as the original curve. 23. In particular, if jD-1, a curve of the order n with deficiency 1, or with ^ (n s - 3) dps, can be transformed into a cubic curve with tho aamo doluuoiioy, thud is with Odps; or the given curve can be transformed into a cubic, Thin WIBO is discussed by Olebsch in the Memoir "Ueber diqjenigou Curvcn doron G'ooi'diiuiton elliptisehe Functioiien eines Parameters sind," Grelle, t. r.xiv., pp. 210271. And lu< has there given in relation to it a theorem which I establish as follows: 2-1). Using the transformation of the order n 1, if besides tho 2H + .H 4(=: g w ;j) points on the given curve U~0, we consider another point on tins curve, thon wn may, through the (w 3 -3) dps, the 2w-3 points and tho point 0, draw a wrios of curves of the order n-l, viz., if P 0t Q 0t R , are what tho function* P, Q, M, become on substituting therein for (ts, y, z\ the coordinates fa, y 0i *) of tin; givuu point then the equation of any such curve will be aP + bQ + olt^Q, with tho relation aP + bQo + cll between the parameters a, b, c; or (what ia tho same thing) olimimitiii"' o, the equation will be a^R*- 2\R) + b(QR,- Q ^) = 0, which contain* tho Hind?, arbitrary parameter a : b. In the cubic which is the transformation of tho givuu ourvi- we have a point O' corresponding to and if (f a , %] ,) bo the Gi.ordinatoH of this point, then coi-re spoil cling to the series of curves of the order n-l, wu havu a HOHCB of lines through the point 0' of the cubic, viz,, the linos f + * + < with thu relation a D + 6^ + cft = between the parameters; or, wluit is tho samo tliinif, w, have the series o lines <&- ffi) + 6- ^o) = 0, containing tho Hmnn inglo parameter - I By determining this parameter, tho CUVVOB of tho ordor n~l, will vlicMoiich .h * rder ^7 h t dp3 ' th 2?1 - 8 P 01 ^ ^ * 1*** ^ hich touch the pven curve ^=0; and the lines will bo tho tangent* to tho onbu nm 1 ' tane " tS *" ' theorem, viz.: p0m; mt 18 ' ATO have the following * cency V 1 wo curve, four curves of the order Ul - vifTl ^ draw S M to touch tho where the ratio ft : 6 is determined bv .' -T-^ 9 ^ by ftn equation ~, ttmi ULUVBS oi uiue order n i' viV tl vuu\,n unu the ratio a : 6 is determined by a certai^ 9 ^7" 7 ^ eqUati n ^ + ^'"0, ( wn the absolute invariant J*-.J O f tho '1,^ f 8 4 ( ! ua1a ? 1 (*!* & ) 4l=0 i then 01 the quartic funchon, is independent of the 384] ON THE TRANSFORMATION OF PLANE CURVES. 7 positions of the "2 2 points on the curve U 0, and it is consequently a function of only the coefficients of the curve U = 0, being, as is obvious, an absolute invariant of the curve U= 0. 26. And, moreover, if the curve U= is by a transformation of the order n l, by means of 2 3 . points on the curve as above, transformed into a cubic, then the absolute invariant I s -s- J- of the quartic equation which determines the tangents to the cubic from any point 0' on the cubic (or, what is the same thing, the absolute invariant S 3 -=- 2 lJ of the cubic, taken with a proper numerical multiplier) is independent of the positions of the 2 3 points on the curve U=(), being in fact equal to the above-mentioned absolute invariant of the curve U=Q. The like results apply to the transformation of the order 11 ~ 2. 27. Suppose now that we have D>2, and consider a curve of the order n with the deficiency D, that is with (n a 3n) D + 1 dps, transformed by a transformation of the order n 3 into a curve of the order D-|-l with deficiency D', then, assuming the truth of the subsidiary theorem to be presently mentioned, it may be shown by very similar reasoning to that above employed, that the absolute invariants of the transformed curvo of the order JD + 1 (the number of which is =47) G), will be independent of the positions of the D 3 points used in the transformation, and will be equal to absolute invariants^) of the given curve 7 = 0, 28. The .subsidiary theorem is as follows: consider a curve of the order D + l, with deficiency J), that is, with ^J) (D - 1)- D = $(&- 3-D) dps; the number of tangents to the curve from any point ()' on the curve is -(JD + l)D-(7> -3D) - 2, =3 4kb -2, (this assumes however, that the dps are proper dps, not cusps,) the pencil of tangents has 4>D - 5 absolute invariants, and of these all but one, that is, 4>D 6, absolute invariants of the pencil are independent of the position of the point 0' on the curve, and are respectively equal to absolute invariants of the curve. 29. To establish it, I observe that a curve of the order D + 1 with deficiency D, or with |(J9 3 -3Z)) dps, contains ^(D + l)(D + ^)^^(D^-W) 1 ~4D+2 arbitrary constants, and it may therefore bo made to satisfy 4D+ 2 conditions. Now imagine a given pencil of 4-Z* - 2 lines, and let a curve of the form in question be determined so as to pass through the centre of the pencil, and touch each of the 4.D-2 linos; the curve thus satisfies ID - 1 conditions, and its equation will contain 4D + 2 - (4-D - 1), - arbitrary constants. But if we have any particular curve satisfying the 4.D - 1 oonditu then by transforming the whole figure homologously, taking the centre of the pencil as pole and any arbitrary line as axis of homology, ao as to leave the pencil of linos unaltered (analytically if at the centre of the pencil 0, y = Q, then by writing <uu + (3y -\- yz in place of z) the transformed curve still satisfies the 4>D-l conditions, and wo have by the homologous transformation introduced into its equation 3 arbitrary constants, that is, we have obtained the most general curve which satisfies the conditions in question. The absolute invariants of the general curve are independent of the 1 It is right to notice that the absolute invariants spoken of hew, and in what follows, are not in gaucral rational ones. OE P LANE . 3 arbitrary constants introducer] 1 *], i consequently functions of on y the cn.ffi 'T 8 '7 te sfo >*tion ; and they are "'-being eo.it is obvious that Lr^'t 8 ^ f" ^ f * D ~* "- of the pencil of _ 2 ,. at ^ " be respectively equal to absolute invariant, curve of the order D + l i, =J(/+ nTfl ' f, he j* 50 '"' 6 ln * of the general each of the dps, honee i,, the prLt oL *n, , ~ i^n '^ ^ " MdaoHon =1 ' for -40-6i and there are th, 4^-7 a h !,7 ' * ( ^ + !) ( ^ + ^-H*-SI))-8, oql to absolute invariant the pet l^TT* * %' ' ^ f ">"> of the pencil, there a re 4D-6, each o he ', t ^~ 3 abs Ulte """* cm-ve, and consequently indepe, <l e f O f H ?" '" " absolllte "variant of the d,ffe out planes, thail joilli ^ ^ e ^ d ransformed curve as situate in point on the transformed curve we hnvf g " Wl m ' Ve with the oorrespondi lw *): if the t,vo curves : %r;.; ol , s S 7 S , of "-. Arming a scroll" see ,on by the plane of the oriri lm l "! vn ; ' " res P eo "ly, then the eon pleto d of ' Derating En.. J'ti , y l^! ,"? f ^ of the order transformed curve i s made of ^^ &l* sect.on by the plane of the inc.. Conversely, given a sc ^,, of ' *e order ', and of genei ^ tic od . , c o be.ng ln ,,. Olu , ves rf the tic od _ + sMtona ^ e _ of the other; but for t !lo ^ scroll of ,1 ' , "^ ta 8f otion e the one . breaking up as above th <ndar + ' ; ' i> not possible to find UN TIIK ('uKUKHl'nNUKXi'K OK T\Vi) I'nlNTrt ON A (IIMtVK. A.ii .l/Mf/mdM/t't-if ,S'.,,!V/y. tul, 1. {iMli V, II* i<l A|'iil Itl, iNiiii } VII. fiiul lllf '! 111 Mtiriii M ti J*| I h .'I.' t** 6?.*'"!i r hit'.*' im l'|6Utl"l| (I s ', I) 4 '* II ul lll' 'lrl)M' I * 1 at I ', I- thcit t-f 10 OX THE CORRESPONDENCE OF TWO POINTS ON A OUHVK [{J8. r J HirvL' (-) (the equation of the curve @ will of course contain the coordinntGH of J* us |m-aim:ters, for otherwise the position of P would not depend upon tlmt of ./>), I iiml that if the curve has with the given curve k intersections ut the point /*, hon in the system of (P, P) t tlie number of united points is '.vhenc. in particular, if the curve does not pass through tho point 1\ thon "ibvr oi united points is = + ', as in a unicursal curve. * The foregoing theorem is easily proved in tho particular liasu whom th X- - y, -') m,d of the order in u . 'Tf fu oto " t tho ordur iu ' rf the,,, vanish, id.nBc.ny. o, b^* 1P '7 T BU h "^ "'"^ <lu '" y.) = fc ,. 4 ' "* " = , of ft- .8 o the n ooo,,, . have ff=o, an el similarly if' rv / J" 1 *' ' } ' m tho !"'. J- wo have ff' = . The eo Z -n ( !' ^ *? a '' " 10 "'''i^ of tho wnluiatcs of the given point P ^7 ' COH8ld( g therein (,, ,, s ) , t ,, n /, as L ters " = have ft,, a the case above SUHM8 .j"^^ a havmg a /{4l , 1)lo iu|i 1 "f h , but i T ? 0+ii the total "umber of 7,,, 7 ' W ' UOh m ' of un e ;"v;d,tha t upon writing (- , are ~ iv ' " ni vJii- Is, *'. 'i. -i !< *,( < ( s oj I * /. , f MtlVr. / " It, Julcl ()( tmtf I "1 .Mn]si):i'i -. S'i tie- [!. tfi"it "I n rune i-l ill- ini,i m" lli- mid i i n.'H M! llii" IMIV- v, ifh <h !!l" nmd'l J"mf i S'i !hl',' (D (.,i I t/,, /' (Sti .-. d in * w i'i|"i, - 12 ON THE CORRESPONDENCE OF TWO POINTS ON A CURVE. [385 tangents from F to the curve, that is, ct - n - 2. Hence the number of inflexions is (-2)+4J} I =m + n-4 + 2(-2ro+2), =3(w-m), which is right. 8. For the purpose of the next example it is necessary to present the fundamental equation a = a + a'+2/ i ;D under a more general form, The curve may intersect the given curve in a system of points F each p times, a system of points Q 1 each 7 times, &c, in such manner that the points (P, P'), the points (P, Q'), & c ., are pairs ot points corresponding to each other according to distinct laws; and wo shall then have the numbers (a, a, a'), (b, ft 0'), &c, belonging to these pairs respectively j via. (1,1) are points having an (a, a') correspondence, and the number of united points is =a; similarly (P, Q') are points having a (ft ff) correspondence, and the number ot united points is = b ; and so on. The theorem then is 9. Investigation of the number ef double tangents :-Take P', an intersection with the curve of a tangent drawn from P to the curve (or what is the same thing P P' "* "" Cm ' Ve); th Unikd P^ ta hero the "o nts "of c ero the po nts of ontact of the several double tangents of the curve; or if T be the number of double " r r - s^wKvTsv 1 ^ , v;> ana ^T, /y, ) to the points (P, P'), 2(a~a-') + 2T-/3- / 3'= Moreover, n,m the last example the value of .-._, is ,, D , ftnd fchfl 2r-^-^2(n-0)JDj but from above it appears that we have fi-ff- (n - 2)(m _ 3)) whence ON Til 10 milWWI'ONI>l'JN(!K Oh 1 TWO PO1NTM ON A (HWVIO. henre, il' (/i, ;>) he tlm iOmm<!lerinl,i<!M n|' Mm nyHtein of (lOiiinn (4#), Mm number of tlm nonien through ./' in '/ A l l!IU ' n ' these I'"-" 1 with Mm tfiveu <mrvn I iiitnrriiutlJnn ut /', mid (KinmtipmnMy k.--fi, Mnrnover, eaeh ni' 1,1m (umii-H bi^idus nmntH Mm e.urve in (2m -1) pointN, and (uiimnmmnMy a t .; a.' /L (2w, - I), jlnncii Mm rnnniila ^iviiH l'.ho ininilici 1 of nnili'il or, HM tliis may bn ': IUI -I- l'l -I- ill (tyi I'}. llulp Mm HyHlinu !' rnnicM (-l-^) rniidiiim (iljt - 1/) |i(inlr|iiurH (t'Ofw/Hfls wfnmnuiit 1'iinli M|' whir.h, vi'Kimlril us a ptiir nl 1 r.iiiii(!)(lonf> lini'M, nmnln tlm givnn r.nrvn in nuiuniilt'iil, pninUM; thai, SH, UIM ]Hmil,-]tuir in l.n bit niiimidiiniil IIH IL i- hlit! tfivi'ii c.iirvn in HI puiulH; mid Mmm in on thin lumouiit n t'nliu^ () in tlm number nl 1 Mm unil.i-d pniuln ; whounti, Ilimlly, Mm niinibm 1 f Mm H (-I./) (!) in M/WC -i-nm. It- in liardly nrnwary l,u remark Mini. il> IH iwmnimtl Mm nmdiMiniH {$%} \\\\\ ntntUl>mim having im Hjmdiul ri'lal-imi do M II. An H linal example, HiipptiHi- that Mm puinl, 7' uu a ^ivnu curvo of Mm inilnr nt. and t.lm point. (,? nu a K' v "" nill ' Vl ' "' )lln nnll<r m '' ' iavn au t. ') nim'iiHpinHlounn, ami ln|, il, In- ivipiinid hi linil t-lit- fla^t ul' Mn> mirvi' cnvolnpcd by Mm linn i'Q, Take un iirbitravy pitinl O, jnin fUJ, and Ittl. Miw meet llm uni've m in /", then (/', /") iti-t* puintN nu Mm eurvn m, Imvin^ a (Vet, ma') nurn i Hpniitlitin!o; in limt., l.n a ^iven pciHitiinu ul' /' I here (tuiTi'Mpuml ' piwilimiM "!' Q, and t,o eiL^h id' Lhese. ? puKiMuiiH til 1 /*', lhat in, I,M 1'iieh pusitinn ul' /* there mnvNpund waf pnMiMntin nl' /"; and Himiltivly In each piiHiMnn nl' /'' there mm'spttiiit m' pnsiiiunH "I" I". The dill've H in the- nyHleni ul' Mm lines drawn IVmu i-twh of llm ' puMitioiiH nl' Q to Mm pnint U, henen Mm IMIIVC H ilne-i imt JIUHH Mtniit^b 7 1 , mid we have /,:-'(). Uniirm the uiLinht'i' nl 1 tlm united puinls (/', ]"). Mind in, Mitt mnnher !' the liimw ./'^ whicih thrungh tht pninl- f>, in i 7itV -I- ?it', r thin ! the rliiHM til' tlm niii've envnlnped by 1JJ, 1 1. may bit remarked, I hut ii' Urn two enrve.s are tiurvi-H in Mpatie (plann ni- nl' dimhln rnrvatiirc). tlmn the liki? retmoning H||(IWM that Mm numhor if tlr '" ..... '"' whinli nittub a given lino in ja'H-Ht'o, tliitt. in, Mm tirdnr ol' Mm HCIM by Mm Un /'(,| in MHI' + ?'. ON THE LOGARITHMS OK IMA<;|,\.\ | !V 10* O.f tlm ./.< ," [From the PIWQBC 1'P- ^0 H Knit | y ol' (ihu low but wo may ah) fl p ca i f Writing tluis and similarly wo have of course We have _ _ P KJ j' fl rt moreover bo ^ a,. HO that wo Jmvd a^t^^y i( > ON THE LOGARITHMS 01? IMAGINARY QUANTITIES. mid similarly log P = log r' + iff ) and logp=:logp + f Hence log P - log P = log + i (0 -ff- ,/,), so that, by what precedes, log P- log P', if the chord 1"P, coiMuloral H drawn I,, P' to P, cuts the negative part of the axis of . upward.,, fc = l (lg .,. 2lV . if ^ chord out, the negative part of the axis of , downwa,*, iO is = log- 2i7r , a,,<l in every other case it is = IOET * Consider the integral path of the variable , ; in order fco ? , Iimi J *~ P > <' <V it -nay dopcmd on bho * 1 p rt vAvi\j4. uu yivti IjiiO llrtrnrii\ r. & ^ therefore fix the path of the variable ,- and I -r T^ H ^ mlicaliion ' must the right line P'P w -f ' j 7 y tlUt " lff tho l )ath fco bo gw nne j P. Wnte Mw ^ p/> ^ du PI " ilUIVU , ! fflrtss / J (Vllr,M, ... . 'I . -1, ^ w ******>> ** * is a, on g t , le right lino , , P (fclwt is| from uio coordinates whereof are 1 *-o M H *-> y-u, to the pomt We have thus '*"* ^! U the path in each case beW a ririlt ,- fl n , p g UG ^ ab Ve ' The indefinite integral ^^lo,^ and as u pa8ses f rom j t P . ^ / w Io tt ' P" there 18 no diseontinuity in the value of lo gw; tho 386] ON THE LOGARITHMS OF III AGINARY QUANTITIES, 15 It is to be observed that 6 has always a determinate unique value, except in the single case y 0, us negative, where we have indeterminately Q ir, It is further to be remarked that, taking A for the origin of coordinates, we have 6 angle asAP, considered as positive or as negative according as P lies above or below the axis of is. Starting from the equation. P = re ie , we have similarly P'-rV ', and P ' u ~ ft 1 ? P' r ' p where </> is derived from - in the same way as Q from P t or & from P'. Consequently e i<0-0'-<) l f and therefore 6' <f> a multiple of 2??-, say 0-6'-$ = 2flwr, and in this equation the value of in is determined by the limiting conditions above imposed on the values of 6, &, <$. To see how this is, suppose in the first instance that the finite lino or chord P'P, considered as drawn from P' to P, cuts the negative part of the axis of as upwards ; P is then above, P' below, the axis of as ; that is, 6, 0' are each positive ; and drawing the figure, it at once appears that the sum & + ( Q'}> that is Q 6', is a positive quantity greater than IT. And in this case the angle </> will be equal to 2?r (Q &') taken negatively, that is, $ ~ - {2?r (0 0')}, or ~ & iJ3 = 2vr. Bub, in like manner, if P'P out the negative part of the axis of so downwards, P will be below, P' above, the axis of CD ; 6 and &' are here each positive, and the figure shows that the sum + 6' is greater than ?r; and in this case the angle $ is = 2?r ( + 6'} ; that is, we have & & <f> = - 27r. In m other case, (that is, if the ' chord P'P cither does not meet the axis of .-u <w meets the positive part of the axis of to,) 6 Q' and </> are each in absolut' less than IT, and we have #' (= 0, So that w '' ' - 1? - 16 ON THE LOGARITHMS OP IMAGINARY QUANTITIES. and similarly , lg P' = log r' -f- i&\ and ll> 15J llf->-M<f. J^ o A' 7^ Hence .0 that, by what p.eede, logP-^P', if the choi , d * to P, cuts the negative pa,t of the axis of Hpwards , is .^ if chord cut. the ^ w part of the ax , rf m m every other case it is = loo- L & p ' or from telow to above ti o f PM P C m & m ab Ve * Wow the logarithm chan g e S from +1 to ' "1 o f ". rf ^ " le imf P art '"' Consider the integral path of the variable ,; iu order to fL , ' ^ ^ '* may de P end on tlu! ' ta Wrfh now ^ p ,, ( _ we lmye _ -p; a,d it is easy to see tllat , the path of the righfc Une x fo P coorfinatea whereof are -l, v . 0> to the point |) . We have thus -'J-* Ji ' the path in each case being a right line as abov e. The and aa * passes from 1 to ~ 386] ON THE LOGARITHMS OF IMAGINARY QUANTITIES. W value of the right-hand side is thus =log~. As regards the left-hand side, the indefinite integral is in like manner = \ogz; hut here, if the chord P'P cuts the negative part of the axis of to, there is a discontinuity in the value of log 2, viz., if the chord P'P, considered as drawn from P' to P, cuts the negative part of the axis of as upwards, there is an abrupt change in the value of log z from - iir to + iV ; and similarly, if the chord cut the negative part of the axis of at downwards, there is an abrupt change from + iir to ITT; in the former case, by taking the definite integral to be log P- log P', we take its value too large by 2zV, in the latter case we take ib too small by 2wr; that is, the true value of the definite integral is in the former ctiso = logP - logP'-2wr, in the latter case it is = log P - log P' + 2m, But if the chord PP' does not cut the negative part of the axis of ?, then there is not any discontinuity, and the true value of the definite integral is log P - log P', We. have tints in the three oases respectively P jp which agrees wilih the previous results. It may bo remarked, that it is merely in consequence of the particular definition adopted that there is in the value of log P a discontinuity at the passage over the negative part of the axis of to ; with a different definition, of the logarithm, there would bo a discontinuity at the passage over some other line from the origin ; but a discontinuity somewhere there must be. For if, as above, the chord P'P meet ^the negative part of the axis of IB, then forming a closed quadrilateral by joining by right p p MUCH the points 1 to P, P to P, P' to p r and -p to 1 ; the only side meeting the fdz negative part of the axis of a> is the side P'P ; the integral J , taken through the closed circuit in question, or say the integral +v\ dz h has, by what precedes, a value in consequence of the discontinuity in passing from P' 'to P; viz., this is =-2t7T or == 2wr, according as the chord P'P, considered as drawn from P' to P, cuts the negative part of the axis of at upwards or downwards ; but) this value - 2wr or + 2wr must be altogether independent of the definition of the logarithm ; whereas if, by any alteration in the definition, the discontinuity could bo avoided, the value of the integral, instead of being as above, would be = 0. The foregoing value - 2wr or +2tV is in fact that of the integral taken along (in the one 0. VI. 3 18 ON THE LOGARITHMS OF IMAGINARY QUANTITIES. [386 or the other direction) any closed curve surrounding the point 2 = for which the function - under the integral sign becomes infinite : but in obtaining the value as z above, no use is made of the principles relating to the integration of functions which thus become infinite. The equation log P = log r + id gives PIU ~ gm log P J.TO gi'iiiO or say (ea 4- iy} = r m e !me , where, m being any real quantity whatever, r m denotes the positive real value of r m , We have thus a definition of the value of (0 + fy)* and the value so defined may be called the selected value. And similarly, for an imaginary exponent m^p+qi, we have r p which is the selected value of ( It may be remarked, in illustration of the advantage (or rather the necessity) of having a selected value, that in an integral fa, taken between given limits along a given path it is necessary that we know, for the real or imaginary value of z correapondmg to each point of the .path, the value of the function Z, and consequently! Z is a function involving log, or.*, the indeterminate,! ess which present itsol m these symbols (considered as belonging to a single value of ,) i S) P SO to indefinitely multiplied, and fa is really an unmeaning combination of symbols, by selecting as above or otherwise, a unique value of log* or & we rondo,, function to be integrated a determinate function of the variable. ' 387. NOTICES OF COMMUNICATIONS TO THE LONDON MATHE- MATICAL SOCIETY. [From the Proceedings of the London Mathematical Society, vol. n. (1866 1869), pp. 67, 2526, 29, 6163, 103104, 123125.] December 13, 1S66. pp. 67. PROF, CAYLEY exhibited arid explained some geometrical drawings, Thinking that the information might be . convenient for persons wishing to make similar drawings, he noticed that the paper used was a tinted drawing paper, made in continuous lengths up to 24 yards, and of the breadth of about 56 inches 0); the half-breadth being therefore sufficient for ordinary figures, and the paper being of a good quality and taking colour very readily. Among the drawings was one of the conies through four points forming a convex quadrangle. The plane is here divided into regions by the lines joining each of the -six pairs of points, and by the two parabolas through the four points; and the regions being distinguished by different colours, the general form of the conies of the system is very clearly seen. (Prof. Cayley remarked that it would be interesting to make the figures of other systems of conies satisfying four conditions; and iii particular for the remaining elementary systems of conies, where the conies pass through a number 3, 2, 1 or of points and tor ' n "* "" A " f K """- -0 XOTICES OF COMMUNICATIONS TO THE [387 algebraical sum of the distances of a point thereof from three given foci is (this V,;H ^lecU-fl for facility of construction, by the intersections of circles and confocal <'uni<-s). The ijiuirtic consists of two equal and symmetrically situated pear-shaped curves, exterior to each other, and including the one of them two of the three given f'R:i, the other of them the third given focus, and a fourth focus lying in a circle with the given foci: by inversion in regard to a circle having its centre at a focus the two pear-shaped curves became respectively the exterior and the interior ovals of :i Cartesian. There was also a figure of the two circular cubics, having for foci four iriveu points on a circle; and a figure (coloured in regions) in preparation for the '"iistniction of the analogous sextic curve derived from four given points not in a circle, March 28, 1867. pp. 2526, _ Professor Cayley mentioned a theorem included in Prof. Sylvester's theory of Imvation of the points of a cubic curve. Writing down the series of numbers 1 2, 4, o 7, 8, 10 11, 13, 14, 16, 17, &c,, viz., all the numbers not divisible by 3 then (repetitions of the same number being permissible) taking any two numbers of he H*IC.,, we have m the series a third number, which is the sum or else the ifiom.ce of the two number (for example, 2, 2 give their sum 4, but 2, 7 give their ftore.ce o) and we have thus a series of triads, in each of wnich one number elLruct I ,e!t tf loirT ***** *' **"* * ****** m & CuWc rhCH ^ ^l?^ 1 ?^^ ^^l^" ' . H >. the third point g of f anf 7 th tc'ond nf ?? **!, *""' ^ fil ' 8t f them ' Wo have here L theo rem J't! t W o'l ^ *"", " " "f ****** of 4. "f i Similarly, 10, 11, 13 ,, each J Ten /I L m t - 7 ,", a ' S It. 10, 17, 19, each of them by t l u , e e nslSnl } i ". tamaaA ^ two tractions increasing by nnity L , ZB of ' 1 n; &e numhor of that the.0 constactions, 2, I ^^ SLJ?* ""^ ^ ^ theorem is - o point. Prof. Cayley mentioneTthat on , 7' b ' 5 al " OM md *e p. 29. O to on a straight line. ' ^ f * heae P mte >=eing flxed , whi|e 387] LONDON MATHEMATICAL SOCIETY. 21 March 26, 1868. pp. 61 G3. Prof. Cayley made some remarks on a mode of generation of a sibi-reciprocal surface, that is, a surface the reciprocal of which is of the same order and has the same singularities as the original surface. If a surface "be considered as the envelope of a plane varying according to given conditions, this is a mode of generation which is essentially not sibi-reciprocal ; the reciprocal surface is given as the locus of a point varying according to the reciprocal conditions. But if a surface be considered as the envelope of a qitadrio surface varying according to given conditions, then the reciprocal surface is given as the envelope of a quadric surface varying according to the reciprocal conditions; and if the conditions be sibi-reciprocal, it follows that the surface is a sibi-reciprocal surface. For instance, considering the surface which is the envelope of a quadric surface touching each of 8 given lines; the reciprocal surface is here the envelope of a quadric surface touching each of 8 given lines; that is, the surface is sibi-reciprocal. So again, when a quadric surface is subjected to the condition that 4 given points shall be in regard thereto a conjugate system, this is equivalent to the condition that 4 given planes shall be in regard thereto a conjugate system or the condition is sibi- reciprocal; analytically the quadric surface oaf + by* + cz* + dw- = is a quadric surface subjected to a sibi-reciprocal system of six conditions. Impose on the quadric surface two more sibi-reciprocal conditions, for instance, that it shall pass through a given point and touch a given plane, the envelope of the quadric will be a sibi-reciprocal surface. It was noticed that in this case the envelope was a surface of the order (= class) 12, and having (besides other singularities) the singularities of a conical point with a tangent cone of the class 3, and of a curve of plane contact of the order 3. In the foregoing instances the number of conditions imposed upon the quadric surface is 8; but it may be 7, or even a smaller number,. An instance was given of the case of 7 . conditions, via., the quadric surface is taken to be o# a -}- 6?/ a + c# 3 _+ dw* = (G conditions) with a relation of the form Abo + Boa, + Gab + ffacl + Gbd -I- Red = between the coefficients (1 condition) ; this last condition is at once seen to be sibi- reciprocal; and the envelope is consequently a sibi-reciprocal surface viz., it is a surface of the order (= class) 4, with 16 conical points and 16 conies of plane contact. It is the surface called by Prof. Cayley the " tetrahedroid," (see his paper "Sur la surface des ondes," IAQU/V. torn. xi. (1846), pp. 291 296 [47]), being in fact a homo- graphic transformation of Fresnel's Wave Surface. [Prof. Cayley adds an observation which has since occurred to him. If the quadric surface am? -I- faf + ex* + dm* = 0, be subjected to touch a given line, this imposes on the coefficients a, I, G, d, a relation of the above form, viz., the relation is A z lo + B*ca, + &al) + Pad + G*bd + H*od = ; where A, B, G, F, G, H are the "six coordinates" of the given line, and satisfy therefore the relation AJF + BG+ <7I/= 0, It is easy to see that there are 8 lines for which the squared coordinates have the same values A*, B*, 0\ F\ G 2 , f/ a ; these 8 lines are symmetrically situate in regard to the tetrahedron of coordinates, and 22 NOTKWS Ol' 1 rOMMinVK'ATIONH TO TIM; I moreover they lio in a hyporbolmd. Tim i|uiulrin mirliuv, instnid ..!' Imin^ ilrliinul above, may, it in cloav, bo dnfinod by thn i!i|iiiviilniil. u lil.iuu>i n|' fniii'liimf ( . m ')i ,,f , i" 8 given linos: that is, wo havo Um oiivrlnpi> nf a ipmdrin Miirliuv hnu-liiiiir ,,(,[, ii( 8 given lines; thosu HHUH nob bring urliit.niry MUCH, bill liciii^ a K.Vdfrm il' u \vi'v liiM-ri'il form. By what procedoH, tho nnvnL.pn i>t n .jiiai'Uc Hiiriinv. li. 'II|I|H-IU;I, linwovcr I'l'^r in virtue of tho rohition AF + JH}* (!Jf ?>(), Uim \ i,u tnu^.r M piupn-'^uuili,' nm-fi,!!^ but that it rosolvon itsoU' into MK- abdvc-iiioiil.i d liyp.'-rbi.lni.l mKru i.win- ' TrY is, i-catoi'ing tho oi-iginal A, it, &,!., in p| llt v n f /| '/.' J . ,<T M ll ivrln|.. ' !' i')", Xic -I- Jka + (Jab + 2i\id -\-Qbtl -\- Jfal - 0, (which in in ijPi'irnil u h'tiahrdmitl), in wi,".*" ^1, A (7, /'', ff, // am tin) Hi]imnl (!oonliiml,oH uf u Im,- (nr, wltiif i^ il,r -^i,,,, u^)"." passing through tho givon lino and tlirnngh the HyimMulnVMlly''^!., 'u,' L'-v'n nllMrii!',!!' J-' 1 la, IHli.S. pp. ion, nil, Frofiw Onyloy g,tvo an uooouni, h, i| lr M-.-tin,/ ul' ,i M,,,,,,,. liv ,.. ((i , H(( ,. Comus rmnnhohor ( on,ploxo ndo. Vonill^nH-MimiMK ,|,., Ktilrr',,!,,,!, H,,.,,J v u % ^ ....... -''-" ) - number of spaces, un,l .orUiu Huppl.n.nnlnry mnu.riH,, A' " J 1 . H tho bottom bo also We,,,, away, ,,,, H i . t ' ' ^ ' , 7 ,"T"i " '' "" '' ........ " supplomonhuy (1 nni.lilii!H ,., i,, ,,, ,,,,, ,T, .',, ' .' "'" , ..... ' ..... "' " f "' The ohiof diniculty ,! i ..... ,, r ,,, ''., "'.' """ lv "' ' -*'l---(./..-*")-.. 8l Wlomo,,tavy ...Jatta ^ J ,?, ..... '' '"" '" ' ........ "" ...... ""' ...... '' ...... December 10. 1808, ini l' 1P- l OlMorratioiw by I'rofoas , - I 1. 2, 3, au,l any ,, tl lr(!u i jT '," , '"' ""* >' liui " ..... > io C T OS ' caoh ^ * Hi \ ;. T , ,; v ;r"' r-- *- *. o, 8, wo havu a tmnfo, vt io n ,, . ,, . i " y ' ,:' ''' " "' ,! hl ' "'"'I' 1 " ! system " '' ' *"-A, .- 1, ; ^, l,, u ,| illK U| u t , lll|V( , nM , Si o > points 4, s, 0, rat i n >0 8implo ^ r "' 2 , "f," ' ^'" Ol " ro 'l 1 ""' 1 ^ having 387] LONDON MATHEMATICAL SOCIETY. 23 Analytically, Cremona's transformation is obtained by assuming the reciprocals of "ai 2/a 2* fa ke proportional to linear functions of the reciprocals of as lt y lt z l (of course, this being so, the reciprocals of *' lp ft, ?i will be proportional to linear functions of tho reciprocals of flj a , 7/ 3) z.j). Solving this under the theory as above explained, write M a , /> , c \ i i_ r T a' 3 u6i i/i Z\ 1 } = . L J, /_ . = 1 : + + i if Honco n Q-JB^O, &c., arc qnavtioH, or generally aQ.12, + ^R l P l + -yP, Qt = is a quartic, having throo double points (y^O, ^ = 0), (^ = 0, ^ = 0), (^ = 0, ^ = 0), and having beaides the throo points which are tho remaining points of intersection of the comes (&-0, ^-0), ( Ji, = 0, P, = 0), (Pj = 0, Q! = 0) respectively ; viz., these last are the points "1 'I 'I o :L . . : - = ei hf : fy - id : dh (/e> &c. &o. Tho double and simple points are fixed points (that is, independent of a, /3, 7). and the fonnulm coino under Cremona 1 * theory. It is, however, necessary to show that if the point* < 5', 0' are in a line, the points 1', 2', 3' are also in a line. This may be rtono as follows : Lot thoro bo throo pianos A, B, 0, and let the points of the first two correspond hv ordinary triangular inversion in respect of the triangle , on the_plane A, and _/3, oil the piano S. Lot also the pianos S, C correspond by ordinary triangular inversion in respoot of the triangle ft on tho plane B, and 7 , on the plane lh " "^ Hpondonoo between A and is tho one considered, the points 188 to ming the r,,.^',:-;".- r ^-^ r^-j-s, -i~ A^nlnrmim nrnnP.vtioa musl) apparently belong to uiemonuB uw interesting part of the theory. i's "TJeber- t ip" Grelle, torn. but in the present Paper wwnwty proportional to 388. NOTE ON THE COMPOSITION OF INFINITESIMAL ROTATIONS. [From the Q mrtef ly J omd of Pw . e and ^ vm . pp. 710,] n the last Smith , -^ COS ion to Denote M infinitesinml *X rigidly ^ in the point 388] NOTE ON THE COMPOSITION OP INFINITESIMAL ROTATIONS. 25 where I, m, n, p, q> r are constants depending on the infinitesimal motion of the solid body. Hence, first, for a system of rotations &>! about the line (a lt b lt c,, f lt </,, h^, 0)3 ,, (tt a , y 3 , C a , J 2l (/ 2 , fly), the displacements of the point (#, y, z}, are Bo) = . ?/Sca> s2ow 4- Sy 2cw , + zZaw 4- and when the rotations are in equilibrium, the displacements (80, Sy, 8,2) of any point (a), ?/, z) whatever must each of them vanish ; that is, we must have 2wa = 0, S<u& = 0, SMC = 0, Sa>/ = 0, "Za>(f = 0, Sw/i 0, which are therefore the conditions for the equilibrium of the rotations w lf a) 2 , &c. Secondly, for a system of forces P l along the line (a,, &,, c,, / 1( (/j, /M, * 3 )i Jl V*3) ^3) C a , _/ a , (/ 2 , ft fl ), &c. the condition of equilibrium as given by the principle of virtual velocities is SP (al + Imi + on+fy + gq + kr) == ; or, what is the same thing, we must have which are therefore the conditions for the equilibrium of the forces P u P 3 , &c. Comparing the two results we see that the conditions for the equilibrium of the rotations WD w a , &c. are the same as those for the equilibrium of the forces P l} P a , &e. ; and since, for rotations and forces respectively, we pass at once from the theory of equilibrium to that of composition; the rules of composition are the same in each case. Demonstration of Lemma, 1, Assuming for a moment that the axis of rotation passes through the origin, then for the point P, coordinates (as, y, z], the square of the perpendicular distance from the axis is = ( , y cos -y + z cos /3) 9 + ( x cos 7 . z cos ) a + ( to cos /S + y cos a . ) a , C. VI. 4 2ti .VOTE OX THE COMPOSITION OF INMNIT.ESLVIA1, IIO'I'A'J'IONH. aid tho expressions which enter into this formula donoto (is fnllmvH ; via if l.lih Hll r] ^ l-mt * at right angles to the plane through P ftnd t h axm nf mlul,i, m Mil ""f n* : Iie rt Ular diStanCG f ' < Mirmtw oi Q refonwl to P as origin are a cos 7 , _ g cos ffj - A'cos/9+7/cosa oach ' .V-*, *-) in place of ( an H coordinate., (, J, c , f . n : '' ^' allt " 4l < cto . of l emma 2. gts ~- ,"' "'" '''"' ........ ' ""'i'-' to t ], ..inilind.],,,^ ( ,,| u , n,l, ...... . l,v , I G f M . Motion of the 389. ON A LOCUS DERIVED FROM TWO CONIOS. [Prom tho Quarterly Journal of Pure and Applied Mathematics, vol. vm. (1867), pp. 778*.] REQUIRED tho locus of a point which is such that the pencil formed by the tangents through it to two given conies has a given auharmonic ratio, Suppose, for a moment, that the equation of tho tangents to the first conic is (HJ ay) (as by) = 0, and that of tho tangents to the second conic is (co cy) (at dy) 0, and write G (a- so that write also /,--* /- - **~ A* lv *~A' then tho anharmonic ratio of the pencil will have a given vali (&-/<;,)(& -70 = 0; that is, if or, what is the same thing, if that is, if - (B-0}* =0, ^ 8 ON A LOCUS DERIVED PROM TWO CONICS. [389 where are each of them symmetrical in regard to a, b, and in regard to G, d, respectively. Let the equations of the two conies be U =(a, b, c,f, (J , A>, y, 2) a = 0, V (*',&', ef i f,g',kJa ll y l z^ = Q i and let fa, ft 7 ) be the coordinates of the variable point. Putting as usual (A, B> G, F, G, tf) = (i c -/ 2 , ca-f, i-/ t , g} t -af, kf~bg t fg-ch), K = abo - af* - hf - c /i the equation of the tangents to the first conic is (A,B, 0, F, G, where X^W-fy, r^ and therefore 5l ,b sti t u ti, lg for 5 the vall>e _i (tf tangents, an equation of the form aZ + 2hZ7+h7*-n w l,i n i i ^ , to b e (Z-7)(X~ & 7) = 0; that is, we ht e ^ " efa ' ^ taken 1 : ft+& : 06= a ; -2h : b; an<l, in like , Mme , if the accented letters refer to the second code 1 : c -f d : cd = a' : - 2h' : b'. Substituting for a, h, b their values, and for a' h' b' th . ana 101 a, h, b the corresponding values, wo find ' ' a + b : ab i c 0/aa We then have ...Ja, A 7 ), 389] ON A LOCUS DERIVED PROM TWO CON1GS. 29 and similarly ( -d)' = -4W. ..., /3, 7)'- We have, moreover, (a + 6) (c + d) - 2 (06 + cd) = 4 (//7 3 - ^7 - Gf3>y - 2 (7V - 2 # and substituting the foregoing values, we find or putting for shortness = (EG 1 + '0 - 2J!f, ., ., Gil 1 + (?'// - ^J" - ^'^, ., . Ja, ^3, 7 ) a . the equation of the locus is where (a, /3, 7) arc current coordinates. The locus ia thus a quartio curve havi quadruple contact with each of the conies Z7=0, U' Q; viz. it touches them at th points of intersection with the conic = 0, which is the locus of the point such ti the. four tangents form a harmonic pencil. The equation may bo written somewhat more elegantly under the form 30 ON A LOCUS DERIVED PROM TWO (.IONICS, so that A - 2k ,.F=* mr (BO -^ ,..) = -(w)' -jiff, H/J -/-, and substituting these values the oqimtion in which, if A, B, denote a, p, 7 . /?, 7 i fl, j0, fy a > ft> 7 I , m, 11 f , m', n' i , W, ?l Xi tf'i '' P, 9. r p'. g', ?' f, wi', n' ^. 2, ' }' J , i P> 7 a a /3 > 7 a_{ i /3, 7 a, ft 7 _. i, m, ?i /' / / f , m, n ^ fli, w J?>> f/. '' P. ff ' X. ?', *' r, W', ft' ^'. '/, '' , 7 respectively, (A + B + 0=0} is, in fact, tho (2fi + l)M 9 or, what is the same thing, that i s ', m', *r' , 7 jugate axes, the we have K^abo, K'^ O. ON A LOCUS DERIVED FROM TWO CONICS. I suppose in particular that the two conies are a? 4- inif 1 = 0, fcho equation of the quurtic is 1 ma? + - 1 _ a m ? - m - 1 = ,, -N ,, . . or putting A, = jf/o/nrv5 tnis 1S njiS _L 1 To fix the ideas, suppose that m is positive and > 1, so that each of the conies in nu (illipso, tho major semi-axis being =1, and the minor semi-axis being = ,, -; . 3?or any roul valuo of Js the coefficient X. is positive, and it may accordingly be assumed thai) X in poHitivo, Wo have ..^..."t.. 1 ,-> < 1, or the radius of the circle is intermediate between m (m -h I) in liho aomi-axoH of tho ellipses, hence the points of contact on each ellipse are real points. Writing for shortness ^ - > ??i a + m tho equation in (a? + iwf - 1) (ma? + f - 1) - X (& + if- a) 9 = 0. For thu points on the axis of eo t we have (f-l)(flMJ a -l)-M< 1 -a) B = 0, that i ( m - \) & + {- (1 + m) + 2X] a; 3 + (1 - Xa fl ) == 0, mid bhoucu or, substituting for a its value, this ia Eemarking tlmt tho values are 32 ON A LOCUS DERIVED FROM TWO CONICS. [389 1 (m + l) a and considering successive values of X; first the value \ = ,, = - ~, wo have + ) m/ m 1 m or observing that this is ^ = 0, or \ w m The next critical value is X-m. The curve here is (a? + mf - 1) (met? + f-i\- m (& + ,* - a )a = Q that is ' ' m (tf + # 4 ) + (l + m s ) a,' 2 ?/ 3 - (m + 1) (a 3 + f) + 1 that is v ,/ y * f m - I) 3 .ty + ( 2)w - m - 1) (^ + /) + 1 _ , ?ia a = o, or, substituting for a its value, _ > l-mtfl--&??li a8 _( m -l)'(m rt ,. . m( a the equation is . m(m+iyt-' or, as this may also be written, 1 m H in or, what is the same thing, in 389' ON A LOCUS DERIVED FROM TWO OONICS. 33 which has a pair of imaginary asymptotes parallel to the axis of so, and a like pair parallel to the axis of y, or what is the same thing, the curve has two isolated points at infinity, ono on each axis, linos TKo next critical value in X = J(m-t-l) 3 ; the curve hero reduces itself to the four , (m ' m 0; and it in to be observed that when X exceeds this value, or say \ >(* + 1) 8 , the cui'vo has no von! point on either axis; but when X=oa, the curve reduces itself to (? -I- y 3 -)"<), i.e. bo liho eirclo ro 3 + f - a. = twice repeated, having in this special ciwo real pointw OH the two axes, Ik i iww eauy to trace the curve for the different values of X, The curve lies in ovory eano within the unshaded regions of the figure (except in the limiting coses aftor-muutiomsd); and it also touches the two ellipses and the four linos at the eight points /i), at which points it also cuts the circle ; but it does not cut or touch the four HnoH, tho two ellipses, or tho circle, except at the points k Considering X as varying by HUCCUBnive stops from to oo; A, = 0, tho curve is tho two ellipses, \, ^^ M.!?JljA Vj the curve consists of two ovals, an exterior sinuous oval lying in the \ W four vegionH a and tho four regions 6 ; and an interior oval lying in the region e. C. VI. 5 q * ON A LOCUS DERIVED FROM TWO CONICS. Tggc, 1) 3 - there ,s still a s i oua oval M above> aove> ^ .^^ \ w . dwindled to a conjugate point at the centre. ; X "" ; TO ; there is no interior oval, but only a J fc , the onive becomes the four lines. k>i(m-+l) 2 , the curve lies wholly in tho f,,. eonasting thereof of f olll . detaehed siuL ov a l s A TT - ^ ^ *" ^ !( + !). e^h oval approaches mm nea r' tnf ' ^ Y , lateS ' eSS from the and inflate l ine . porti l w hieh temd " ' "^ W * ini1 f ed ^ the eide And a X depart/ f fom the 1 tit T + , Z"" * V WhMl th OTal be approaches more nearly to the circuil^ h PP '' 0a heS * "' each si cont ^ 6eparates the 4 e crcu h contains the dnnon. oval. ^ 6eparates the 4 ^giona 4 e , w hich Finally, x = 0, the curve is t ], e ci role Wee repeated. 390. THEOREM DELATING TO THE FOUR CONICS WHICH TOUCH THE SAME TWO LINES AND PASS THROUGH THE SAME FOUll POINTS. [From the Quarterly Journal of Pure and Applied Mathematics, vol. vm. (1867), pp. 162167.] THE aides of the triangle formed by the given points moot; one of the given linos in tliroo points, nay P t Q, 11; and on this same line we have four points of contact, Bay A lt A tl A a , Ai\ auy two paii^i say A lt A t \ A B , A 4t form with a properly selected pair, sn,y Q, R, out of the above-mentioned three points, an involution j and we have thua the three involutions (A lt A; A aj Ai\ Q, .fi), (A,, A s \ A it A,- } H, P), (A lt A,; A 2> AS\ P, Q). To prove thia, lot ffi = 0, i/^O ^ G tno equations of the given lines, and bake for the equations of the sides of the triangle formed Toy tho given points b & + a y ~ab =0, If oi + ' y - a' &' = 0, the equation of any one of the four conies may bo written Lab LW . L"of'b" \j - ab b'os + a'y - a'V b"ai + a!'y - and if this touches the axis of ic, say at the point co= a, then we must have _^L JL' .. V/ = -/f^") a , tv-acK-aiv a" () )(;- a') (a; - a") ' 52 Uti THEOREM RELATING TO THE FOUft CONICS WHICH TOUCH THIS >r, aiming as we may do, #=-('-<)("-) (-'), this gives L a =( ft -op (a 1 -a"), MUW manne, if the conic touch the axis of y, 8il y at the poi.,; ifi =(b -^(fi'-fi", ' and thence p, ming - P = a & (a' - ft ") (y j/^ f =a'b' (a," - a )(b"~l)), wo have ^-V(a-0( ~V), and thenco ; * ; wo have in like mamel . - _ "' ' p , Putting equation is '" ' -^ 390] SAME TWO LINES AND PASS THROUGH THE SAMS TOUH POINTS. 37 and "by attributing the signs + and - to the radicals, we have, corresponding to the four conies, the equations (a - ,) V (X) + (a 1 - ffl ) V (A") + (ft" - a.) V (X"} = 0, - (a - ,) V (X) + (a' - a a ) V (A") + (a" - a 3 ) V (X") = 0, (a - .) V TO - (a' - a,) V (T) + ( a " - ,) V ( JT") = 0, (a - Oi) V (X) + <' - ,) V (*') - (" - <0 ^ (A) = 0, where a n 3( a 3) a., aro the values of a for the four conies respectively, Eliminating a" we obtain the system of three equations (2a - , - a,) V (X) + (a, - ,) V (A") + (a, - ,) V (A r ") 0, foi - ) V (A') + (2a' - , - ,) V (A r/ ) + (a 3 - ,) ^ (.Y") - 0, (a, + a - a, - a.,) ^ (A r ) + (a, -}- a 3 - a, - a,,} V (A r/ ) + (a, + 4 - a, - a 3 ) V (^") = 0, and then oliminating the radicals we have , a - a, + 4 ~ a - ==0, which is in fact 1, . +'j act,' - 0, as may bo verified by actual expansion; the transformation of the determinant is a peculiar one. The foregoing result was originally obtained as follows, viz, writing for a moment a V (X) + (if V (*"') + " V (X") = , tho four equations aro - a, <I> = 0, - <E> = 2 these give (a, 2 (a - B 38 THEOREM RELATING TO THE I'OITR CONICS WHICH TOUCH THE From the last equation we have (a, - ,) fc = 2 (0 - a V (X) - a' V (Z')J - 2a 4 {> - V (X) = 2(i-^)*-2(a- that is or substituting for J(X), *J(X') their values in terms of <]>, we find which way be written that is T or ft Of., that is or finally which is a known form of the relation I ~1 _ ft . V~ aa -> o + ft', acs' = 0, gives the involution of the quantities a, a'; ,, 4 We have in like manner 1 t . ,, . . -0, and ''olutions of the systems < a "j a . a [390 , w and < a; 1( 3 ; . a4 39 390] SAME TWO LINES AND PASS THROUGH THE SAME FOUR POINTS. It may be remarked that the equation of the conic passing through the three points and touching the axis of m in the point ; = ia (a - a)* (of - a") 6 , (ft' - ? (a" - a) V (a" - ) 3 (a - o') b" _ Q " + ' '-'' ^ ' '-"' and when this meets the axis of y we have >0. y-b " r y-V ff-6 Hence, if this touches the axis of y in the point y = {3, the left-hand side must be - (a ~ a) 2 (ft' - a") + -, (ft' - a) 3 (a" - a) + j (a" - - ) 3 (a - a') (y - /3) s , ft a -) and equating the coefficients of - a , we have i/ ( a _ a ). (a' - a ") + ^ (a' - ) 3 C^" - a) + j/ (" - a ) a ( a ~ a ') P (a _ )- (a' - O + ^ (a' - )* (" - a) + ^ (a" - ) ( - a j ft _a or what is the same thing, fa - ,f (rf - a") + ( - ) (" - a) + s ft t( ' " = 2/3 (ft - a) 3 (' - a") + ~ (' - ) 9 (" - ) which gives /3 in terms of a, that is ft, ft, A, A in terms of ,, a a , ,, 4 respectively. Cambridge, 30 November, 1863. 391. SOLUTION OF A PBOBLEM OF ELIMINATION. [From the Quarterly Journal of Pure and Applied Mathematics, vol. viu. (1867), pp. 183185.] IT is required to eliminate en, y from the equations a > b , c , d , e a' , b' , G , d', e' a", b", G " , d", e" This system may be written = 0. if for shortness Or putting we have 2/ 4 = SXe; - ft + X a j &c. - -A "* fV 391] SOLUTION 01? A PROBLEM OF ELIMINATION. 41 or, what is the same thing, X (6 + Ao) + X' (6' + &0 + A." (&" + Aw") = 3 X (o + to*) + V (c' + fctf) + V (c" 4- Ad") - 3 X (d 4- Are) + V (d f + fre 7 ) + X" (<?" + to") = ; and representing the columns a b> a' b f , a" I", b c, b' c', b" c", c d, c' d f , c" d", d e, d 1 e', d" e", ty 1, 2, 3, 4, 5, 6, each equation is of the type X(l + fc2) + V (3 + fc4) + V (6 + fc6) = 0. Multiplying the several equations by the minors of 135, each with its proper sign, and adding, the terms independent of k disappear, the equation divides by k t and we find X 2136 4- V 4135 + X" 6135 - ; operating in a similar manner with the minors of 246, the terms in k disappear, and we find X 1246 + V 3246 4- X" 5246 = ; again, operating with the minors of (146 + 236 + 245 + &24C), we find X {1236 + 1245 + k (2146 + 1246)} + X' {3146 4- 3245 4- k (4,236 4- 324G)} + X" {5146 4- 5236 4- k (6245 4- 5246)} = 0, where the terms in Is disappear, and this is X (1236 4- 1245) 4- V (3146 4- 3245) 4- X" (5146 4- 5236) = 0. We have thus three linear equations, which written in a slightly different foi X 1235 4-X' 3451 4- ^" ' X (1236 4- 1245) 4- X' (3452 4- 3461) X 1246 4-V 3462 and thence eliminating X, X', X", we have 1235, 1236 + 1245, 3451, 8462 + 3481, 5613, 5614 + 6623, C. VI. 42 .SOLUTION OP A PROBLEM OP ELIMINATION. which is the required result. It may be remarked that the second and third are obtained from the first by operating on it with A, A 3 , if A ~ 2^ + 4<8 S -f ( say the result is 1235 (1, A, 3451 5613 = 0. In like manner for the system if the columns are then the result is a? , afy, ay, ohj\ ay*, f a , b , o t d , e , f ft' > &' , c' , d' , e' , f a", 6", c" , d", e" , /" ft'", &'", c'", d"', e"' , f" a b, a'V, a"b", a,'" V", b G, V c' } 6"o", b'"o'", c tZ, c' d', c" d", c'" d'", d e, d'e', d"e", d'" e'", */> e'f, e"f"> e'" f", = 1,2, 3,4, 5, G, 7,8; (1, A, JA where 12357 34571 56713 78135 =0, 43 392. ON THE CONIOS WHICH PASS THROUGH TWO GIVEN POINTS AND TOUCH TWO GIVEN LINES. [From the Quarterly Journal of Pure and Applied Mathematics, vol. vm. (1867), L pp. 211219.] LET ro = v-0 be the equations of the given lines; s = the equation of the line ioining the given points. We may, to nx the ideas, imagine the implicit constants so determined that + tf-M-0 shall bo the equation of the line infinity. Take ff-my-O.fl-ny-O as the equations of the lines which _ by their inter- section with = determine the given points. The equation of the come is y (m) + \i (n)J V (wj) = a + y V () + 7* or, what is the same thing, (co - my) ((o - ny} + 2 (w + y V (*)) y* + y V = > so that there are two distinct series of comes according as ^ (vm) is taken with the positive or the negative sign. The equation of the chord of contact is & + y V (w-) + y* 0, which meets ,-0 in the point { ^y^nn)^ , = 0} that is in one of the ^centres of the involution formed by the lines (, = 0, y-0). (--flW-0. -^-0). It to be observed that the conic is only real when im -i* positive, that is (the lines ^ and points being each real) the two points must be situate in the same region or in opposite redon3 of the four regions formed by the two lines: there are however other real cases- e. K . if the lines a-0, y-0 are real, but the quantities m 7, are conjugate imageries; included- in this we have the circles which touch two real lines. b ^ ON THE CONICS WHICH PASS THUOUGH TWO GIVEN POINTS [3Q2 f V (m) + V ()} V W = + y V (mn) - 7 (a + y), which gives two coincident points, that is the conic is a parabola, if that is (1 " 7) {A/ (wm) * ?) - * (V W + V ()}-, or 7 9 - 7 {1 + V 0)) == i {V (m) - ^ (n)) 7 = HI + V (m) V {(1 + m) (1 + n )J], where it is to be noticed that y == i [1 + V (m) + V {(1 + m) (1 + 71)}] is a positive quantity greater than V(Mi), say 7 =^ 7 = HI + V (mn) - V {(1 + m) (1 + )}] is a negative quantity, say T = -g, ff being positive. The order of the lines is as shown m fig. i, 8ee plafce facing p _ 7 = - ro to Y = - 3 , curve is ellipse; 7as _ fff parabo]a 7- - ff to p, curve is hyperbola; 7 ^ pambola Pj) 7=p to 7 = OT, ellipse. Resuming the equation the coefficients are ~ ' *+ TV 0, (. & ' c. / ^ /O - [1, wi, 7 9 , 7V(m), 7, and thence the inverse coefficients are (4, 5, a, j, t?, jj) = or, omitting a factor, the inverse coefficients are W. a, a, j- ffl a,. [ 0) a, Considering the line the coordinates of the pole of this line ar are 392] AND TOUCH TWO GIVEN LINES. 45 or (what is the same thing) introducing the arbitrary coefficient k, we have kx + 7/i - v v* (inn) 0, ky + 7\ v = 0, kz~\*/(iwi) - /*- g" y(m) - V ()1 3 v = the first two equations give k ; 7 : -l-i^A W (*)) : v \V V(w) ffl ) :*#-/*& that is /t-\V(m)l ~"iy V (*"-) -0) , or, substituting this value of 7 in the third equation, lA-W(m)l* , w (ffw)} + ^-W ^ 0> Xw-/*y l ^ (G-y*j(mi that is (m) - V ()}' -I- ( ~ 2/ V ( + * { - ^ V (*)! y {/i - X V (wwi)} = 0, which is the equation of the curve, the locus of the polo of the line X0 + /*y+M*=0 in regard to the conic (<a - my) (a ~ny) + 2{ + y*/ (ma,)} i* + 7^ a = - In particular, if A, = /i = y = 1, then for the coordinates of the centre of the conic, we have to -. y : 2 = -7 + V(<) = -7 + 1 = V (> + * + jj^ W ( m ) ~ V ()) 8 i and for the locus of the centre, (ffl-^i{V(m)-V())' + (-y>l-^ so that the locus is a conic, and it is obvious that this conic is a hyperbola. Putting for greater simplicity {o-y =A, co - y V (ron) = Y> z Z, the equation of the curve of centres is Z B . HV (m) - V ()]' + XY i 1 + V (*)] + YZ {1 - V (n)} = 0, or, writing this under the form Y[X {1+ V (mft)} + J? {1 - V (>))] + i IV (*) - V (n)) a X* = 0, 46 OX THE COMICS WHICH PASS THROUGH TWO GIVEN POINTS [Q the equation is YQ + X* = Q, where - V ( these values give {1 - V (m)] * = [\l (m) - V ()} Q + 2 (1 + V (wi)) ^, or, what is the same thing, (1 - V (mi)} w= - V ( mn ) X + Y, [I ~ V (mn)} z = 2 (1 + V (mn)J Z + { ^ ( w ) - V ()}' whence also ' {1 + V (')} JT + 27+ (^(m) - V ( or the equation of the Hue infinity is {1 + V(0) Z+ 2F+ {V(m) - V()}' Q = 0, bo apKed to e asymptotes 7Q + X = 0. In fact we have identically asymptotes of the oouio\B=oH, n w ( . Ue n " ty> PUte iu "vidonoo the fa' a Wntnl S *' /* 7^ in the pl ao o of ^ , and . y. that is , ' * * -(u 392] AND TOUCH TWO GIVEN LINES. that is - 4/3y&' (a s 47 - (a 3 + 4/37/c') ( 2/Vc'a - a?;*) 2 - 40y F 1 (OHO 4- or, what is the same thing, _ 43# (> + W) (&V + 2/2) = {20tf oj + 2/3 a /,/y - (2/37/0' + a 3 ) ^} a - (a 2 + WO (2^'<w - ^) 3 - *^^ C which, when tw + /5i/ + 7^ = is the equation of the HUG infinity, puts in evidence the asymptotes of the conic #" + # 0. Now writing X, 7, Q in the place of en, y, z\ tt = l, and == (1 + V (")). = 2 > y={V(ft)- V()l a we navo - 16 [{1 + V (w)) a + 8 (V (0 - V ()i a ] (7Q + ^) = [4 (i+ v ()) ^ + sy - (4 iv w - v ()) + (i + v (flw - [(1 + V (m)|" + 8 [V () - V ()] s ] [4^ - (1 + V <*)} Q] a - 16 [{1 + V <ro)l ^ + 27 + [V (0 - V ()} 3 Q] 3 . and the asymptotes are = + V(l + V (WMI)}" + 8 IV C" 1 ) - V ()} a [4A r - {1 -1- V (mn)} Q}. At the centre 4 {1 + V ()} A r + 8 F - [4 (V (w) ~ V ()} 2 + {1 + V <)} 9 ] Q = 0, 4Z-{l + V(m)}Q ! =0. but the first equation is {1 + V (wn)) [4X - Q {I + V (*n){] + 8 F - 4 (V (0 - V (n)) 9 Q = 0, so that we have 4A' - {1 + V (flw)J Q, 2 7 = {V (0 - V ())' Q. the first of these is 2 {V (TO) - V ())' ( - ff) - {1 + V (ift)l* (* - y) - (1 - ) * = > and the two together give 2X {V (m) - V ()!' - {1 + V (W3i)] F = 0, so that we have 2 {V (m) - V (*)!" ( - ^ ) - [i + V ()$ {^ - ff V ()) - o, to determine the coordinates of the centre. 48 OX THE CONICS WHICH PASS THROUGH TWO GIVEN POINTS The equation of the chord of contact is } -I- 70 = 0, which for 7=1 is parallel to y = and for y = V(0 is parallel to <y=0. Jlut coordinates of the centre are a : y : z = - 7 + V (mn) ; - 7 + 1 : ^ ( wm ) + l + _. (^ ( m ) _ ^ ( n )}^ which for 7 = 1 give V(w) H- 1 + i{V (*) - V())'i 2 + 2 V * : 2-i-m and for 7 = V (ni) give a = 0, if : *-l-V(w) : V y (tttu the cent,, !, J- + y V (m) + 7'^ = 0, where, writing for *, y , , the coordinates of the ^^ w(j + 2 v () + 4 |y (m) + 1 + {V (m) _ ^ (B)} ,J = Oj (mn) -7(1 + that is or, what is the same thing, 7-7 = and consequently 7' = 7 only for 7==0 , TJ. the fixed point Lx /^ \ ft con ' es P on{lll ig positions of the chnvrl nf o rtll *. rt . n tL _ ^u pome (a- + i/V(m) = 0, *=,( -i * ^-- - cl ot con(; acli thmuwh see fig. 2 in the 392] AND TOUCH TWO GIVEN LINES, 49 Chord of Contact. Centre, 01\. P a , at infinity on hyperbola, OL(z = Q). L, (z = Q, a,'-i/ = 0). OH. 0, the lino joining this with being always behind 00. OP i. P., at infinity on hyperbola, X {-' + y V (mi') = 0] . - ' X (m = 0, y = 0). OG (parallel to y = 0). G (on line # = 0). 077 (purallol to ^ = 0) and so back to H (on line ffi=0) and so on to I have treated separately the case */(mn) = I. Consider tho conies which touch the lines y a 1 = O f y-M = and pass through the points Tho aquation is of tho form f - a? + Je(t -*? = $> and to determine ft, AVG havo c a 1 o" 1 + /c (1 )' J = 0, and therefore k ^ --- r a . (.* ~ a ) Tho (M|uixtion thiiB bocinne-s (1 - ) 3 (f - ffiS ) + c a ( - ) = > thiiti in (1 - a) 3 y a + (c a - (1 - a) 3 } ffi 9 - 2c 3 aa; + c a a 9 = 0, or IIH this may bo written {r&fi 1 a /i3/yS C\ _ \a '-^^-f^-o- Honoe tho nature of the conic depends on the sign of c G -(l-a) a , viz. if this b& poflilivo, or between the limits l + o, lc, the curve is an ellipse, (B-coordinate of centre -; c (i a) which is positive, o (1 - a) a'-semi-axis = y-semi-axis Tho coordinate of centre for a = 1 + c is = + oo (the curve being in this case a pai P!> and for 1 - o it is also = + oo (the curve being in this case a parabola P 3 ). coordinate has a minimum value corresponding to a=V (1 -o)> viz - fehis is ~i I 1 + C. VI. 50 ON THE CONICS WHICH PASS THROUGH TWO GIVEN POINTS, &0. | ,'JU'J Hence as () passes from 1 + c to V(l~ c3 ). the coordiimfce of tho eontn; pus^-n from x to its minimum value {1 +*/(! c a )j; iu the passage wo hnvo a=| rrivJiiff the coordinate =1, the conic being in this case a pair of coincident linos (; 1)' J =-. ; O. And as (2) passes from the foregoing value Vfl-c 2 ) to 1 - o, tho (inordinate of (In 1 centre passes from the minimum value ^{1 + ^(1 -c a )} to GO. The curve is a hyperbola if a lies without the limits 1 + c, I - c, .'(.'-coordinate of centre - -ZJ^. ( which has the sign of -a, 0-semi-axis = -~~ ..-L...T-- / y-semi-axis = semi-aperture of asymptotes = ' [ " A/ - (i -); ^l^r 1C ^.= 0. (Parabola), but inereases as l- illoroiu(oa coordinate of centre is = -l-c, coordinate of centre s u ' " >, 0, tho hyperbola being in this case the pair of lines *- -*,-' a re ..- .- -c')}, and then as a !>/ ,7 a mnximum l"itiv viiliu. , i. . "'ii'H us a passes irom //i _ ^ <. .. centre diminishes from Hl-V(l-<fll n T ~ C I cooiflinatp i>l negative, the Hnes y ^^ = n ^f.^,1/,.,, , i8 to bo Barker! that a 1 51' 393. ON THE CONICS WHICH TOUCH THREE G-IVEN LINES AND PASS THROUGH A GIVEN POINT. [From the Quarterly Journal of Pure and Applied Mathematics, vol. VIIL (1807), pp. 220222.] CONSIDER the triangles which touch three given lines; the three lines form a triangle, and the lines joining the angles of the triangle with the points of contact of the opposite sides respectively meet in a point S: conversely given the three lines and the point S t then joining this point with the angles of the triangle the joining lines meet the opposite sides respectively in throe points which are the points of contact with the throe given lines respectively of a conic ; snch conic is determinate and unique. Suppose now that the conic passes through a given point; the point 8 is no longer arbitrary, but it must lie on a certain curve ; and this curve being known, then taking upon it any point whatever for the point S, and constructing as before the conic which corresponds to such point, the conic in question will pass through the given point, and will thus be a conic touching the three given lines and passing through the given point. And the series of such conks corresponds of course to the points on the curve. 52 OX THE CONICS WHICH TOUCH THEEE GIVEN LINES, &O. [""SO 3 the condition in order that the conic may pass through the given point is a -J- & + G == O> and we thus find for the curve, which is the locus of the point 8, the equation {(*) or, what is the same tiling, the rationalised form of which is = 0. This is a quartic curve with three cusps, viz. each angle of the triangle in a and by considering for example the cusp (?/ = 0, # = 0) and writing the equation uiitlor the form o? (y - zf 2a- (yz* + y z z) + if& = 0, we see that the tangent at the cusp in question is the lino s/-,3=0; that is, fclio tangents ^at the three cusps are the Hues joining these points respectively with tins given point (1, 1, 1). Each cuspidal tangent meets the curve in the cusp counting 1 IXH three points and in a fourth point of intersection, the coordinates whereof in tho Oiuscs of the ^ tangent y-* = 0, are at once found to be a : y : z^ 1 : 4 : 4, or say thia IH the point (1, 4, 4); the point on the tangent x-to = Q is of course (4, 1, 4), and thai* on the tangent *-y0 is (4, 4, 1). To find the tangents at these points reHpoctivoly, I remark that the general equation of the tangent is that is X . Y . Z or for the point (1, 4, 4) the equation of the tangent is 8Z+r+^0, or 8 + y + ^0; that is the tangent passes through the point ,- = 0, or+y+^-O, boi the pomt of mtersection of the line ,- with the line . + + ;. ^hioh IB t e ne ,- wt the line . + y+ . 0| hioh IB tli a 4 Sr/fT^'JV' 1} " r ^' d fc the U ^ * o tl haimoni ' it ' ^ 1' 'n } T^^ ^ to)I h th ^ ints ot the harmonic lino a + y + g = Q mt h the three given lines respectively, of * " "' r without much difficulty. ' w Sles P 10 *^ may be effected Plate I. $5 face p. 52. 394. ON A. LOCUS IN RELATION TO THE TRIANGLE. [From the Quarterly Journal of Pure and Applied Mathematics, vol. viir, (1867), pp. 204277.] IF from any point of a circle circumscribed about a triangle perpendiculars are kit Tall upon the aides, tho feet of tho perpendiculars lie in a line; or, what is the Htimo thing, tho IOCUH of a point, such that the perpendiculars let fall therefrom upon tho Hides of a given trianglo havo their feet in a. line, is the circle circumscribed about iiho triangle. In this well known theorem we may of course replace the circular points at infinity by any two points whatever; or the Absolute being a point-pair, and the terms perpendicular and circle being understood accordingly, we have the more general theorem expressed in tho same words, But it is loss easy to sec what the corresponding theorem is, when instead of being a point-pair, the Absolute is a proper conic; and the discussion of the question affords some interesting results. Take (so = 0, / = 0, s = 0) for the equations of the sides of the triangle, and let tho equation of tho Absolute be (a, &, o, /, g, hfco, y, *0 2 = 0, then any two linos which are harmonics in regard to this conic (or, what is the same thing, which are such that the one of them passes through the pole of the other) are said to be perpendicular to each other, and the question is: Find the looiiB of a point, such that the perpendiculars let fall therefrom on the sides of the trianglo have their feet in a line. Suppoain* as usual, that the inverse coefficients are (A, B, 0,1, &, 5), K IB the- discriminant, the coordinates of the poles sides respee 54 ON A LOCUS IN RELATION TO THE TBIANGLK [394 (A, H, G), (H, B, F), (G, F, 0}. Hence considering a point P, the coordinates of which are (x, y t z), and taking (X t Y, Z) for current coordinates, the equation of the perpen- dicular from P on the side X = is X, 7, Z = 0, A, H, G and writing in this equation A'=0, wo find ( , AyL for the coordinates of the foot of the' perpendicular. For the other perpendicular respectively, the coordinates are and (Bx-Hy, , Bz-Fy}, (Cx-Gs, Oy-Fz , o ), and hence the condition in order that the three feet may lie in a lino is or, what is the same thing, , Ay-Ess, Az~ Csa-Gz, Cy-Fz, = 0; that is ( ^ - OH) & + 5 f the,ofore a cubic. and /f' for the discriminant or as th l8 may also be written , the equation i s ^ 394] ON A LOCUS IN RELATION TO THE TRIANGLE. 55 that is r 2 f A -nn -nTT\ A B C'l /A B G \fa y z\ vffrftffi (ABO - FGII) - -rv a - M ~ rrT a *F+ ( Jf lit + 7v ^ + W ^) ( B? + O? + /// ) and the cubic will therefore break up into a Hue and conic if only 2 ABO TFG-'H.' ( ~~ ' ~ j^a ~ (yi ~ p3 ~ ' and it is easy to see that conversely this is the necessary and sufficient condition in order that the cubic may so break up. The condition is ' n = zFQ'ii 1 (ABO -Fan) - ACPIP - BHW - ow a = o, we have A A' + ]&' + GO' = 3vl56' - AF* - 5G a - C// a , = 7f + 2 (ABO- and thence a j'<?7/' (^^' + J5J3' that is tt = -A G'H' (Q'ff - A'F) so that the condition il is satisfied if If = 0, that is if the equation (A,B, 0,F, ff.SJfc fl.0^0, which is tho line-equation of the Absolute breaks up into factors; that is,, if the Absolute be a point-pair. In the case in question we may write (A, B, G, ff, G, that is - . , (A, B, 0, F, G, J/) = (2aa', 2/3/3', 277', 0</ + 0<y> whence also, putting for shortness, (/ - /3V> 7' - y' a > a $' " a '^^ = ( x> ^ v ^' we have (A 1 , B', 0', F', G', #') = -(^> /*"' ^ and also JTf = 0, 2 (X^a - FGH) = AA' + BB'+ CO', = - 2 50 ON A LOCUS IN RELATION TO THE TRIANGLK The original cubic equation is J? + 77V) ay& + eufayz (py + vz} + pfipza (\cv -\- vg) 4- 77'^ (Xft. 1 + mid this in fact is (aofayz + pfi'ttxte + w'vwy) (X# + p,y -I- MS) = 0, The equation \x-\- py +vz = ft is that of the lino through tho two pointn which constitute the Absolute; the other factor gives s = 0, which is the equation of a conic through the angles of the trianglo (to = 0, y = (), z =s ()), and which also passes through the two points of the Absolute; in fact, writing' (, /3, 7) for (a;, y, z) the equation becomes a/3y ('X + ftp + y v ) = 0, and so alw> writing (', /3', -y') for (.-B, y t 2) it becomes a'/9V (oX + ^ + 7^) = 0, which relations nro identically maiw fusel by the values of (X, /*, v). Hence we see that the Absolute being a point-pair, tlm locus is the conic passing through the angles of the triangle, and tho two points of the Absolute; that is, it is the circle passing through the angles of tho triimglo. But assuming that 1C is not =0, or that the Absolute in a proper conic, thu equation fi = will be satisfied if AFG'ff + BQH'F + CHFG' + PG'H' - 0, we have F, ff,H' = Kf,Kg, Kh respectively, or omitting the factor K\ the equation becomes AF<th + BGkf+GHfg + Kfgh = Q, which is /VA 3 - bctfh* - coA 8 / 8 4- a6/ a (/ a + 2oio/^A = 0, or, as it may also be written, tho l r e "--^-^i anbstitutiug also for ^, ^ the values AT/ % /a, the equation of the cubic curve is 2 (a&c -/jA) X yz + Ay, (hy + and the transformed form is we have so that the foregoing condition _ a/* 394] ON A LOCUS IN" RELATION TO THE TRIANGLE. 0* *?/ P being satisfied, the cubic breaks up into the line -7.+ - + 7 = 0, and the conic / 9 h ' A G It is to be remarked that in general a triangle and bhe reciprocal triangle are in perspective; that is, the lines joining corresponding angles meet in a point, and the points of intersections of opposite aides lie in a line ; this is the case therefore with bhe triangle (# = 0, y = Q, 2 = 0), and the reciprocal triangle (uai -i- hy -f gz = 0, ha> + 6y +fz = 0, gio +fy + cz = 0) ; and it is easy to see that the line through the points of intersection of corresponding sides is in fact the above mentioned line ?4-^ + ?-0. It is to be noticed also that f y h the coordinates of the point of intersection of the lines joining the corresponding angles are {F, G, H). The conic 9 is of course a conic passing through the angles of the triangle {& = (), y 0, 2 = 0); it is not, what it might have boon expected to be, a conic having double contact with the Absolute (a, b, o, /, y, Ajw, y, zf. I return to the condition =0 ' abo c(/' fl by 1 * c/t 3 fyli this can be shown to be the condition in order that the sides of the triangle ( 0, y = 0, z = 0), and the sides of the reciprocal triangle (eta; + % + 0* - 0, /we + fy +/* = 0, r/ft> +/?/ -H c# = 0) touch one and the same, conic; in fact, using line coordinates, the coordinates of the first three sides are (1, 0, 0), (0, 1, 0), (0, 0, 1) respectively, and those of the second three sides are (a, h, g\ (h, b, /), ({/, f> o) respectively ; the equation of a conic touching the first three lines is ^ + + -0. and hence making the conic touch the second three sides, we have three linear equations from which eliminating L, M, N, we find which is the equation in question. 0. VI. 58 ON A LOCUS IN RELATION TO THE TRIANGLE. = 0, We know that if the sides of two triangles touch one and the same conic fclmir angles must lie in and on the same conic. The coordinates of the angles are (1 ' O ()) (0, 1, 0), (0, 0, ]) and (A, ff, G), (H, B, F], (ff, ff t G) respectively, and the anglon 'will be situate in a conic if only 111 A' I-P 111 H' B' F 1 1 1 G' F' G an equation which must be equivalent to the last preceding one; this is easily vorilii-il In fact, writing for shortness V = we have 1 1 1 1 I I a' A' ff ' A' H' G- I 1 1 I I I h> b' 7 H' B' F 1 1 i I 1 I ff* /' G' ff' G S. and the second factor is But - aGH (AF + Iff) + A.FKBG + AFgOff, = AF(a6ffI so that the second factor is which is " ^ (A* - + ~ --- L _ JL 1 , 2 o . " 394] ON A LOCUS IN RELATION TO THE TRIANGLE. so that we have identically - ABCffWH* Q = K-alcf^h- V , and the conditions V =0, D=0 are consequently equivalent. The condition L L_JL_1 + A =0 " abc a/ 3 If/" ch* fgh ' is the condition in order that the function may break up into linear factors; the function in question is / bo ca ab (<^. c '7 7- j whichiH i A B , 0\ so that the condition is, that the conic (a, o, c, /, </, /iji. ft; 1/> z) "T \^j:J g h / (which is a certain conic passing through the intonation, of the Absolute A n Is ~* i it t . - (, fc o t /, g t Kt*. y, *-0, and of the locus conic ^+--+^-0) 1 be a pair of lines. Writing the equation ef the eenic in question under the form 6c the inverse eoefficients A' t B', G', F, ff, ff of this conic, are Ale Boa Gab _^jr - a G "" i, B" - r< - tr-F ' G : S. Hence, if in regard w i.uia u ----- -- so that wo have M .(r "' ^ = ), and join the corresponding form the reciprocal ef the tmngte (.-Oj 0,j J, J ^ ^ ^ angles of ^e t,o t *. ^-^^^ fa recipvocal in reg ,d It is to be noticed that the conic 60 N A LOCQS IN RELATION TO THE TRIANGLE. [394 contains the angles of the reciprocal triangle, and is thus in fact the conic in winch are situate the angles of the two triangles. For the coordinates of one of the aiie-lon of the reciprocal triangle are (A, H, (?); we should therefore have which is or attending only to the second factor and writing GH=Kf+AW ) the condition is Kfgh + APgh + SGhf+ GSfg = 0, or substituting for If, A, B, G t F, a, H their values and reducing, this is -~__4.__ aio a/" Ziy" c/i' fgh~ t t if the trian f b i suoh that the re > tuaugle he m a oomo (or, what j. the same thing, if the sides touch a conic) then the cubic locus breaks up into the line - + 8+^-0, which is the linn inTtt otic* "' 3 f intei ' Se ti0n f the ^% -des of the two triangles, a,ul S - which is the conie through the angles of the two triangles. its alrie^rrfTh MiaeS ; g! T , a C0ni (the Abs lute ) to constraot ' S in a cic in re Md to the conic, Ho 394] ON A LOCUS IN RELATION TO THE TRIANGLE. Gl I suppose that two of the angles of the triangle are given, and I enquire into the locus of the remaining angle. To fix the ideas, let A, B, G he the angles of the triangle, A' t B', C' those of the reciprocal triangle; and let the angles A and E be given, We have to find the locus of the point G': I observe however, that the lines AA', BB', GG' meet in a point 0, and I conduct the investigation in such manner as to obtain simultaneously the loci of the two points and 0. The lines C'B', G'A' are the polars of A, B respectively, let their equations he 3 = 0, and i/=0, and let the equation of the line AB be 2 = 0; this being so, the equation of the given conic will be of the form (a, b, c, 0, 0, fc I take (a, (3, 7) for the coordinates of and (a?, y, 2) for those of G'; the coordinates of either of these points being of course deducible from those of the other. Observing that the inverse coefficients are (6c, co, ab h 2 , 0, 0, oA), we find coordinates of A are ( b, - h, 0), B (-h, a, 0). The points A' and B' are then given as the intersections of AO with 6"^'(y=0) and of BO with C'S'Cw^ )') we find coordinates of A' are (Aa + && , fcy), 5' ( , act + Aft A?), Moreover, coordinates of 0' are (0, 0, 1), C^ { w it %\ <J W !/ z )' The six looints A, B, G, A', ff, G' are to lie in a conic; the equations of the MA M AT* AY 4- 67-0 aX + hY=Q, Z=Q, and hence the equation of a lines G.O., dx*, -Aj> ^ ie /tA-rw _ ' conic passing through the ,points G', A, B is L M -ft^Q ST+Af + WC + 6F ^ Hence, making the oonio pass through the remaining points A', * 0. we find 62 ON A LOCUS IN RELATION TO THE TRIANGLE. [304 and eliminating the L, M, N, we find 1 a 1 h ' A + 6/9 1 1 A b ' ct 4- A/9 1 1 Ay aa + h y* hie + by ' z = 0, or developing and reducing, this is h cwB A Aa + 6y 1 a + A/9 _ 1 Aa + & _ As? + 6 ~ ' We have still to find the relation between (a, ft 7 ) and (a;, y, *); this is obtained by the consideration that the line A'ff, through the two points A', 7/ tho coordinntOH of which are known in terms of (a, ft 7 ), is the polar of the point 0, tho coordinates of which are fa y, z). The equation of A'B' is thus obtained in the two forms and + (Aft! + and comparing these, we have : y : z = a. or what is the same thing chy (ate + %) (Jig + by) eliminate where _ /m& ^ A a or, completing the elimination, wtioh is a quartic curve having a node at each of the oi points 394] ON A LOCUS IN RELATION TO THE TRIANGLE. 63 that is, at each of the points B t A, 6'. The right-hand side of the foregoing equation, is = - (ab - 7i 3 ) (ha, aft, so that the equation may also be written (am + hyf (hx + %) a + eA-V f oo; a + i/ + -f- xy) = 0, \ "'I Secondly, to eliminate the (as, y, z) t we have ob-h*Y 2 1 w _ h ah<t + l3 b C ' where 7_ _ chf or, completing the elimination, (a& - A 3 ) <% 3 = (A6, - 06, Aa$aa + A/3, Aa + - (06 - /i a ) A f ao? + 6^8" + ^ *P \ ii that is fa, 6, o,0, 0,^/3,7)^0. \ ft A / Writing (a 1 , y, *) In place of (a, /9, 7), the locus of the point is the conic Ub,o t 0, 0, f]Uy, *Wo, \ /t A / which is a conic intersecting the Absolute (a, 6, o t 0, 0, AJ, T/, ^) 3 = 0, ali its intersections with the lines ffi=0, i/ = 0, that is the lines C'-B' and (TA 1 . In regard to this new conic, the coordinates of the pole of C'B'(v = Q) are at once found to be (~h, a, 0), that is, the pole of G'B' is B ; and similarly the coordi- nates of the pole of Cf'A'fo-0) are (b, -A, 0), that is, the pole of ^ IB A. Wo may consequently construct the conic the locus of 0, via given the Absolute and the points A and B, we have O'A' the polar of , meeting the Absolute m two points L, th\ and O'ff the polar of 4 meeting the Absolute m the pomts (b, and b,)- the lines O'A' and 0'*' meet in G'. This being so, the required come puses through the points ou .. fti. &, the tangents at these points being Ah. **,& Bb> JP^^ eight conditions, five of which would bo sufficient to determine the conic It * to -ta remarked that the lines 0'S t O'A! (which in regard to the Absolute are h epotai- of A, B respectively) are in regard to the required come the polars of B, A respectively. The conie the locus of being known, the point may be taken at any point of thie conic, and then we have A' as the intersection of O'A w.th AO B as h, interaeetion of O'S? with BQ> and finally, G as the pole of the line AB in legaid 64 ON A LOCUS IN RELATION TO THE TBI ANGLE. [[394 to the Absolute, the point so obtained being a point on the line G'O, To each position of on the conic locus, there corresponds of course a position of G\ the locus of is, as has been shown, a quartic curve having a node at each of the points G' t A, 7?. The foregoing conclusions apply of course to spherical figures ; we see therefore that on the sphere the locus of a point such that the perpendiculars let fall on the aides of a given spherical triangle have their feet in a line (great circle), is a spherical cubic. If, however, the spherical triangle is such that the angles thereof and the poles of the sides (or, what is the same thing, the angles of the polar triangle) lie on a spherical conic ; then the cubic locus breaks up into a line (great circle), which is in fact the circle having for its pole the point of intersection of the perpendiculars from the angles of tho triangle on the opposite sides respectively, and into the before-mentioned spherical conk. Assuming that the angles A and B are given, the above-mentioned construction, by means of the point 0, is applicable to the determination of tho locus of the remaining angle G, in order that the spherical triangle ABO may bo such that the angles and the poles of the sides lie on the same spherical conic, but this requires somo further developments. The lines Q'E>, G'A' which are the polars of the given angles ^ J1 respectively, are the cyclic arcs of the conic the locus of 0, or say for shortness tho come 0- and moreover these same lines O'ff, G'A' are in regard to the conio O, the polars of the angles S, A respectively. If instead of the conic we consider tho polar conic it follows that A, B are the foci, and G'A', O'ff the oompondimr du-ectrices of tiie come 0'. The distance of the directrix G'A' from the centre of tho conic measuring such distance along the transverse axis is clearly - 00" - distance of ' ' hn boft'tfn T Vei ;r T 1S =9 ; that i8 ' the C0nic & is ft co described 0* l! ' , " aUSV r ^ (01> SUm r dUteMM of *" fci diBtomooH) -90. Considering any tangent whatever of this conic, the pole of the tangent is n ^ ^ -^ f interSeCti011 " *' P^onS- lo ta tiofl - ' P^on from the angles of the spherical triangle on the opposite sides; hence to (viz. i , ~ " "" v jjuino. luiiujyitj ne in a snhor mly to construct as before the conic 0' with th f A H " SArrjs t j^^^-i'itrr*!* ^ f the conic 0'. It i s moreovpr nW ^\ opposite sides respectively, is a taragoiit - 10 uiuieuvei Cieai, tnat ffivon n fnio-nn-ln ^ nn i ,1 i question, if with the foci AS d niangie^^^a having the property 7 in like manner with the foci A *P nn^r^ 88 aX1S ==9 W desci ' ibe a conic, ctnd ?, 0' and the same transverse axis\ve 1 h T tr ^ SVOrse axis ' aud with th foci onics will have a common tan^nt the Tol! wL P Tu^ f fln that th threb action of the perpendiculars fmm +i, i * wneieoi will be the point of infcer- B spectively, P IpentllCUlaiS flom the "^ of the triangle AEG on the opposite siclo* 395. INVESTIGATIONS IN CONNEXION WITH CASEY'S EQUATION. [From fcho Quarterly Journal of Pure and Applied Mathematics, vol. vm. (1867), pp. 384341.] IN 1,1 paper road April 9, I860, and recently published in the Proceedings of the Royal Irish Aoad&niy, Mr Casey has given in a very elegant form the equation of a pair of oh'oloH touching each of throe given circles, viz. if U = Q, F = 0, TP = be bho equations of the three given circles respoctively, and if considering the common (iangonlM of (F = 0, lf=0), of (1F=0, U=Q), and of (ZJ^O, 7=0) respectively, these oommoH tangents being Huoh that tho centres of similitude through which they roHpuotivoly posH Ho iu a lino (vis;, the tangents are a.11 three direct, or one is direct and the other two arc inverse), then if /, g, h are the lengths of tlio tangents in intention, bho equation belongs to ti pair of circles, each of thorn touching the three given circles, (There mil, it IH olenr, four combinations of tangents, and the theorem gives therefore " aquations of four pairs of circles, that is of the eight circles which touch the givun circles.) Generally, if Z7-0, F=0, W-Q are the equations of any three curves of the same order n t and if/, g, h arc arbitrary coefficients, then the equation is bhal; of a curve of the order 2n, touching each of the curves 7=0, F=0, F=0, 7i q timos, viz. it touches = 0, at its n* intei'soctions with gV W=0, 0, VI. 66 INVESTIGATIONS IN CONNEXION WITH CASEY'S EQUATION. If however the curves U=Q, V=Q, W=0 have n common intersection, then tho curve in question has a node at this point, and besides touches each of the throe curves in n"-l points; and similarly, if the curves U = Q, F=0, W=0 have k common intersections, then the curve in question has a node at each of these points, and bosidos touches each of the three curves in ?i 3 k points. In particular, if tf~0, V=0, W = are conies having two common intorseotioiiH, then the curve is a quartic having a node at each of the common intersections, ami besides touching each of the given conies in two points; whence, if tho coefficients f, ij, h (that is, their ratios) are so determined that the quartic may have tnvo mtmt nodes, then the quartic, having in all four nodes, will break up into a pair of conicw, each passing through the common intersections, and the pair touching each of thn given conies in two points; that is, the component conies will each of thorn touch each of the given conies once. Taking the circular points at infinity for the coinmon intersections, the conies will be circles, and we thus see that Casey's theorem i in effect a determination of the coefficients / g, k, in such wise that the curve (which when U=Q, 7=0, F = are circles, is by what precedes a bicirculnr queu-tin) shall have two more nodes, and so break up into a pair of circles. The question arises, given ZT0, F-0, F = 0, curves of the same order n, it in required to determine the ratios / : g : h in such wise that the curve may have two nodes; or we may simply inquire as to the number of the sots of values of (/:</: A), which give a binodal curve, V (fU) -\- V (gV) + V (h'W) = 0. ^ I had heard of Mr Casey's theorem from Dr Salmon, and communicated it toffotlwi- v u o !lT" lg considerati M to Prof - Cremona, who, in a letter dated Bolocrim. March 3, I860, sent me an elegant solution of the question as to the number of tho binodal curves. This solution is in effect as follows : ^n Um *' F== ' lf = of the samo ***** ; oonaidor the pom ( g, A) and corresponding thereto the curve fU+gV+hW-V. As long as tho rl^l' f\? 7> ^ curve /^+^+^-0,will not have any node, an ,l in order that this curve may have a node, it is necessary that the point. (/ V A) shall he on a cortam curve S; this being so, the node will lie on a curvo > tho Jacobs o the curves U, 7, W ; and the curves J and S W ill correnp d to urve * , , a llode at sorae one P^nt on the cuiye J. and convei-sely, m order that the curve fU+ ff V+hW=0 may bo a curvn havmg a node at a given point on the curve J t it JLJ that the ^n (/> 7 I) shall be at some one point of the mirvn 5 1 TK^ -/ i T l V **' ' = a binodal curve having a 395] INVESTIGATIONS IN CONNEXION WITH CASEY'S EQUATION. 07 each of the corresponding points on J ; and each cusp of S corresponds to two coincident points of /, viz. the point (/ g, h) being at a cusp of 2, the curve fU + yV+hW = Q is a cuspidal curve having a cusp at the corresponding point of J. The number of tho biiiodal curves fU+gV+hW=Q is thus equal to the number of the nodes of 2, and tho number of the cuspidal curves fU + gV +h 17 = is equal to the number uf bho cuHpa of 2. The curve 2 is easily shown to be a curve of the order 3(-l) 3 nncl class 3n(-l); and qua curve which corresponds point to point with J, it is a owrvo having the same deficiency as J t that is a deficiency = i(3ft-4)(3- r >); we liavo tlionco tho PHtokeriau numbers of the curve 2, viz. : Order is = 3(tt-l) 9 , Class = 3n(n-l), Cusps 12(tt-l)( n-2), Nodes = a(n-l)(w-2)(3'-Sn-ll), Inflexions = 3(n-l)(4n-5), Double tangents = f (-!)( n-2)(3' + 3-8). ! also ( W , it ir lg thonin n + l for ) p , and -th Ping now to the pvoposod question of tho same ordor i and we may confer the curve VCW + VW + V W = <>. of the orfer 2, having rf contacts wit] na the point (/, g, K) i o ir o ' ' Hch ; 3 a cuvvo g iven curves ff, F, If. As leng g ^ ^ orfer ^ he point (f. . shall lie on a 68 INVESTIGATIONS IN CONNEXION WITH CASEY'S EQUATION. [395 point of /. The number of the binodal curves fi is consequently equal to that of the nodes of A, and the number of the cuspidal curves fl is equal to that of the cusps of A ; we have consequently to find the Pliickerian numbers of fcho curve A ; and this Prof. Cremona accomplishes by bringing it into connexion with the foregoing curve S, and making the determination depend upon that of the number of the conies which satisfy certain conditions of contact in regard to the curve 2. Consider, as corresponding to any given point (/, g, h) whatever, tho coni r + + 2~ which paases blll ' ou gh tlll ' ee ftxecl points, the angles of tho triangle * J J A > = 0, y = Q, z = Q. For points (/ g, h) which lie in an arbitrary Hue Af+J3g+ Oh = 0, the corresponding conies pass through the fourth fixed point & : y : z = A : Ji : (J. Assume for the moment that to the points (f t g, A) which lie on the foregoing ourvu A, correspond conies which touch the foregoing curve Then 1". to the points of intersection of the curve A with an arbitrary line, correspond the conies winch piu-w through four arbitrary points and touch the curve S; or the order of the curve A is equal to the number of the conies which can be drawn through four arbitrary points to touch the curve 2; viz. if m be the order, n the class of 2, tho number of these conies is =2i + , or substituting for m, n the values 3(n-l) 3 and 3(w-l) respectively, the number of these conies, that is the order of A, is = 8 (w-l)(3- 2). 2. To the nodes of A correspond the conies which pass through three arbitrary points and have two contacts with 2, viz. if m be the order, n the class, and # tho mimbor of cusps of S, then the number of these conies is =*(2wt + )'- 2m -5n-fye t or substituting for m, n their values as above, and for K its value = 12 (71- t)(w- 2) the number of these conies, that is, the number of the nodes of A, is found to bu = f (n - l)(27?i 9 - 63tt + 23n + 16). 3". To the cusps of A correspond the conies which pass through throe arbitrary points, and_lmve with 2 a contact of the second order; the numbei of these (m, as above) a -B + ^ or substituting for and K their values as above, the number of these comes that is the number of the cusps of A, is - 8 ( - 1) (7 B - 8) Wo have thence all the Pluckerian numbers of the curve A, viz. these arc Order 3(rc-l)( 8-2) f Class = -" Nodes - f (n - 1) (2to - 63 ? i 3 + 22u + 16), Cusps = 8(w-l)( 7w-8X Double tangents = | ( n - 1) (18ft. _ 36?i a + lQn + Inflexions =12(n-l)( n _2) : d f vmr s r^rr^ ^i numbera include the result that the number of the binodaf curie! s 16), 395] INVESTIGATIONS IN CONNEXION WITH CASEY S EQUATION. 69 The proof depended on the assumption, that to the points (f, g 3 h) which He on / ) the curve A, correspond the conies - + - + - = which touch the curve S ; this so y z M, Cremona proves in a very simple manner : the points of / correspond each to each with the points of S, or if we please they correspond each to each with the tangents of S. To the Cm (n - 1) intersections of J with any curve a (viz. V (fU) + V" (gV) + V (hW} =0) j* i correspond the Gn(n 1) common tangents of S and the conic *~ + -+- = ; if fl- has x y z a node, two of the Gn (n 1) intersections coincide, and the corresponding two tangents will also coincide, that is Jl having a node (or the point (/, g, h) being on the curvo A), the conic touches the curve 3J. But it is not uninteresting to give an independent analytical proof, Write for shortness dU = Ada) + Bdy + Cdz, dV^A'dto + B'dy +G'dz, and lob (to, y, z) bo the coordinates of a point on J, (X, F, %} those of the corre- sponding point on 2, (/, ff> It) those of the corresponding point on A, Write also for shortness tiG'-B'G, QA'-C'A, Aff-A'B^P ; Q : R t thon wo have AX + BY +CZ -0, A'X+B'Y giving giving A , B , A 1 , G' = u ) A' +ff +G' =0. ^1" B +B" +0" =0, = 0, which is in fact the equation of the curve /; and moreover A", J3", Of- , Y : Z~P : Q : jK, to determine the point (A r , 7, ) on S; and h U ' vr or what is the same thing, f : ff : h = P*U : Q*V : JZ'Tf, to determine the point (/> </> A) on A - Treating now (/ (/, h) as constants, and (A r , F, ) as current coordinates, the conic ^ + 4 + |-0, will touch the curve S at the point (P, Q, B), if only the X y & 70 INVESTIGATIONS IN CONNEXION WITH CASEY'S EQUATION, equation of the conic is satisfied by these values and by the consecutive values P + dP, Q + dQ, R + dR; or what is the smne thing, if we have L + + A =0 P * Q * R ' that is :4:4= Q(iB- .BdQ : B&P - PdR : PdQ - QdP. If the functions on the right-hand side are as U : V : W, then these equations give f'.g:h = P*Ui V : R*W t that is (/, g, h) will be a point on the curve A. It is therefore only necessary to show .that in virtue of the equation /=0 of the curve J, and of the derived equation d/=0, we have QdR-MQ : MP-PdR : PdQ-QdP=U : 7 : F. Take for instance the equation V(QdR - EdQ) - U(WP - PdE) = 0, that is dR (UP+VQ+ WR) - J ( raP + 7cZQ -I- WAR) - 0, and this, and the other two equations will be satisfied if only UP 4- VQ + WM = 0, WdR=Q', we have, neglecting a numerical factor, W=Ck +&$+&'*, whence, attending to the values of P, Q t R, we have hence also so that if only :11 U, iNvrMTi.m-HiNM IN n.NNKxmx WITH CAH.UY'H EQUATION- UIN . ~ j lltll | ..ulnlifuiiiiK I'"' /'> V. ''.'"'. ./I', r/H' ihoir vain,*, fch loft-hand side is -_ri whii'li ^ "'. h-ii" "^ I" 11 ' m 'l l " ls| -"' iu-(i III-OVIM!, mid C / A\ -' ~' (lll ||,,, rnrv,- A. V/ ' ^ A) 1S a l Kjl!lt; |l. IM >., In* .M.M.--.I. ilmi Hi.. ln ,nrvr, >;. A arn K() .> mo tricnlly connected through ,!, I),,-,,,, m -1iih,iiy II.-IHI'. in lnll..wi.: vi/.. hikni); HH nxim Iho sides of the i; (|ltll ,i l, v (If" ')'>'-' I.MIM!''. tlin, ^iuiiiii; Jh.ui any pnim, (/ ) t \ of A '" Iil1 " fl'WI ^-l-^+fe^O, and finally tho jnvr.," .'--u.r ''. I ^1 ' M. v,hi.<l, l.y whal- pi.Tnl,^ i; t )nliuH 2 in the point corro- (iiKni'lm:! I" 'I*' 1 fi''""'!"! I"""' ,/', '/. /O "' A: mid i-imvomoly afcarting with an assume. I . :,,! ,,,, >" v,.' !'A(,-- (It.- '-iii t - ' I ' i ; I' wliinh unrtHiw tiliTOugh the angles of the I .> If O Mi.uu;!- .ul 1 .4".^ -i X " '1" ir'.iim-'.l ji..iul,; tlm IIIVDI-HO line / -h flfy + / = ; the luiiuini<- p-iit' ('- ) - ] ( ) "' ' !l ^ J) "'""i illld lilmll >' lll(! > vurHO point (/. ff, A), which will 1*' H "" 't" 1 ''tni..' e \. *h" j*"iM* t'Mii.'i.jimuliiii.f in llu'r iiMHUiucd point on the curve S. 72 [396 396. 03ST A CERTAIN ENVELOPE DEPENDING ON A TRIANGLE INSCRIBED IN A CIRCLE, [From the Quarterly Journal of Pure and Applied Mathematics, vol. IX. (ISCfH), pp. 3141 and 175176.] CONSIDERING a triangle and tho ciroumHcribod circle, and from any point of tho circle drawing perpendiculars to tho sides of tho triunglo; tho foot of tho throo perpendiculars lie on a lino; and (regarding tho point an a variable point on tho circle) the envelope of the line is a curve of the third olusn, having tho lino infinity for a double tangent, and being therefore a curve of tho fourth order with throo miHpH, see Steiner's paper "Ueber erne boaondoro Curve drittor ICloaao und vierton GrndiiH," Grelle, t. Lin. (1857), pp. 231237, which contains a series of very beautiful geometrical properties. Mr Greet', in a paper in the last volume of tho Journal, hot) expressed tho equation of the line in a very elegant form, vk if a, ft 7 are tho perpendicular distances of the point from the sides of tho triangle; A, tt, tho angles of tho triangle; (X, ^ *)(-, ^r, ^~); and (X, F, Z) certain cuiTont oooixUimtoH. viz. these are the perpendicular distances from the sides, multiplied by sin A tan .4, , sin G tan G respectively; then the equation of tho line is where the parameters X, /*, v are connected by the equation \ tan A + ^ tan S + v tan G - 0, or say by the equation 39(5] ON A CERTAIN ENVELOPE DEPENDING ON A TRIANGLE &0. 73 Wo havo a cubic equation in (X, /*, v) with coefficients which are linear functions of (X t Y, %)> and the required equation is that obtained by equating to zero the reciprocal of this cubic function, the facients of the reciprocal being the (a, b, c) of the linear relation ; the reciprocant is of the degree G in (a, b, c) and of the degree 4 in tho coefficients of the cubic f auction, that is in (X, 7, Z), But I remark that tho equation in (A,, p, v), regarding these quantities as coordinates, is that of a cubic curve having a node at tho point \~fj.~v t or say the point (1, 1, 1); the corresponding value of Xa + /tb + vc is = a 4- b + c, and the reciprocant consequently contains the factoi 1 (a + b -I- c) 9 , or dividing this out, the equation is only of the degree 4' in (a, b, c). The equation of the curve thus is ----- - j 1 - - ruoip. [X\ (\ - /t) (\ - P) + Yp (p - v) (/* - X) + Zv (v -\)(v- /*)] = 0, ~" being of the degree 4 in (a, b, c), and also of the degree 4 in (X t Y, Z\ that is, treating (X, Y, %} an current coordinates, the envelope is as above stated a curve of tho fourth order. A symmetrical method for finding the reciprocant of a cubic function was given by HUHBU, HOU my paper " On Homogeneous Functions of the Third Order with Three Varmblofl,'" dumb, and DitU, Math, Jour., vol. i. (1840), pp. 97104, [35]; the developed oxpreHnion there given for the reciprocant is however erroneous ; the correct value ia given in my "Third Memoir on Qualities," Phil. Trans., vol. CXLVI. (1856), see the Tublu 07, p. (H4, [144] and we have only in the table to substitute for (, ij t } the quantitiuH (a, b, o), and for (a, ft, c, /, g t h, i, j, Is, I) the coefficients of the cubic function of (X, p t v) t vk multiplying by 6 in order to avoid fractions, these are ( a, b t G, f> (J, It, , j, 1& > l ) {OZ, G7, 0^, -27, -2, -2J, -2^, -2Z, -27, X+Y + Z) rospcctivoly. The Hubstitution might be performed as follows, viz. for the coefficient of u', wo havo 76 ON A CERTAIN ENVELOPE DEPENDING ON A. where the reciprocal in question may be calculated from the before mentioned tulili 67, viz, multiplying by 3 in order to avoid fractions, the coefficients of the table arc (, b, a, f t (j t h, i, j, k, I ) = (0, 0, 0, o, a, I, b, o, a, -a-b c) respectively, and for the facients (, ?j, f) of the table we have to write (to, ;/, The expression of the reciprocant is == 6W 4- cVty + aW + &o., and dividing by (aH-jr + *)* we have the equation of the envelope in the form 6 W + C 9 ay + aWz 1 + &c. = 0, which must of course be identical with the former result boyst (cy l)zf + cazta (as co))- 4- abxy (bat ay) 8 + &c. 0. Instead of discussing the curve of the third class it will be convenient to mite (0, y, z) in place of (, y t f), and discuss tlui (nu-vf of the third order, or cubic curve U = a-ic (y - zf + ly(z~ (u) a + os (to - yf = 0, which is of course a curve having a node at the point ( = y z) t or nay at tlt^ point. (1, 1> 1), and having therefore three inflexions lying in a line. The equation <* the tangents at the node is found to be a (y - *) fl + b (s - ic) s + G(&-yf = 0, that JSj at the node the second derived functions of V are proportional to (&-fo, o + tt, a + l, a, -b, -c). The equation of the Hessian may be found directly, or by means of tho No. 61, in my memoir above referred to. It is as follows: (6 + c){a(b + (o +a){6 (c + ( + &) (c (a + 6) + 2a&) - (So? 2cct + a s 6) + Qabc] aye = 0. ^IIM.I.I: i\.i iiiui.it IN A rim-i.i-:. 77 > '>* '.' ' ' : 5 Ar, ?!(. ! . ||,| in ',,- I-, -i u)i(siiit I)HI iiiiiiH'iirul I'uHur 'I 1 , // ' 'M / '^ '!./'> ; ! /I. '//' /, .'. A'. AM -. v, 'f tl. / /M-'l" 1 ! I/). / -Mi^ 7 I *V). '-- M!*I-' t -Ui, Ml"- |""llt (1, 1. H. Hi'' lllV.illll 'Hi' 'i""Ji'l ilvlivril rilll''!i"UH inV - c!i-"!" "*' ill- 1 niiii-' i il. in i-jis 78 ON A CERTAIN ENVELOPE DEPENDING ON A [396 I write X = y ~ z, Y z cc, Z = $ ij, so that we have identically and that the equation of the tangents at the node is Z + &P + c^ = 0. I write also for shortness F'='2F + I+L, I'=F+ the equation of the cubic is then U = asoX* + &7/F- + GzZ\ = 0, and that of the Hessian is HU = X* (Lfo + F'y + I'*) + F' (J' K + L y + G'z) + Z* (H'x + 7r'y + L Now observing that we have lf i - 3 (G +J /v+scM=/v- we find + SJtf Z7 - Z a (/T - which shows that the function I2U + 3MU is a cubic function of y-z t .%-!&> -y, decomposable therefore into three linear factors; and the equation HU+3MU = Q, is consequently that of the three lines drawn from the node to the three inflexions of the cubic (or the Hessian). We know also that the Hessian of the three lines is the pair of tangents at the node^), viz. that regarding any. one of the variables X, Y, Z as a linear function of the third of them (in virtue of the equation X+ then that the cubic function of X, Y, Z has aA r2 + &7 3 + cZ' 1 for its Hessian, 1 Tailing as the canonical form of n uodal oubio I/=K 8 -f-j/ B +6toya = 0, then we have HU=x 3 -\-y a -Zlxyz = Q ; 3 =0 ia the equation of the lines from the node to the inflexions, and the Hessian of the binary oubio is^icy, where ,ri/ = is the aquation ol the tangents at the node. We obtain as the only linear functions of U, HU wMoh are ilecGinpontible, a?+# 3 and xyz, the equation .1^2=0 gives #i/ = which belongs to the tangents at the node or else s=Q, which is the equation of the line through the three inflexions : this lino ia BO obtained n little further on in the toxt. 396] TRIANGLE INSCRIBED IN A CIRCLE, 79 It in interesting to verify this ; I write Z - - X - 7, the cubic function then assumes the- form whoro (a, /3, 7, &} have the values presently given. The Hessian is (27 - 20 s ) A 3 + (S - 7) . 2AF + (2/38 - 2f ) F', or writing 2ZF = # 3 - X 3 - Y\ this is = (2a<y - 2/3 fl - S + fly) A' 3 + (2/3S - We find after some easy reductions, ct= K'+ F f , =-3 (a + c) (no (& +c ), (6 +c ), --^S- tt'+ J' r = -3 (& + o)(6o + Jlf), and hence aS - /9y = - 81 (a + c) (6 + c) {(c + Jlf ) (60 + JW) - a& (a + c) (6 + c)}, where the expression in ( } is = (06 + 2ao + 60) (a& + ao + 26o) - a& (a& + ac + 6c + c a ), ^ c (a6 (3ft + 36) + 0(6 + 2a) (ft + 26) - ft6 (a + 6 + c)}, = 2c ((t + 6) (&o + ca + a6), and thoreforo a g _ ^ 7 = 1 6 2 (6 + c) (c + a) (a + 6) Mo ; the othor coefficients may be similarly calculated, and omitting the merely numerical factor, we have Hessian =(6 which is right. I write next> 80 ON A CERTAIN ENVELOPE DEPENDING ON A or writing a-zY, y = z-\- X, this is [396 + #> [K'X-3'Y }, we may determine , so that the cubic function of X, Y, 2 contains the factor aX- + bY'-+cZ~; writing Z = -X-Y, then Contains the factor Quotient is X s ( K'+ F' JP7 - H' + 2ff' ) (a -+ o) X s 3(6c+Jlf)F. We have seen that whence the quotient is, as above stated, = - 3 (ac -H j]/) X + 3 (6c + JJf ) 7. Comparing the coefficients of X*Y, we have <& = -(- H' + W + F' + /') + 3 ( + ) (6c + jtf) _ 6c (ao + M}> (a + c) (6 + o) + 3 (a + C ) ( ffl & + ac + 26c) - 6c (a6 + Sao -I- bo), that is * = 12Hf; and the same value would have been obtained by comparino- the coefficients of ,YP. Hence SU-WU divides by a^ + iP + c^ the quotient being ,. , . -- which is or, finally it is and we thus have = - 3 {(bo + M) + (ca + j) y + (06 + jlf ) #} , so that the three inflexions are the intersections of the cubic curve by the line (ftc + JO* + (oa It may be noticed, that if we write bate + oay + ohz ~ - Mu, by+ cz= v TRIANGLE INSCRIBED IN A CIRCLE. 81 then os, y, z will be as : (c - a) {(23f ca) 6u} and substituting these values in the equation aai (y zf + ly(z a;) 3 + cz (to - y)* = of the cubic, we have a cubic equation for the ratio (u : v); and thence the values (a 1 , y, z) for the coordinates of the inflexions. It may be added, that wo have = - 3 (oX* + & y + c# a ) {(bo + Jtf)co + (ca + M)y + (ab + M) z\ which is the equation of the cubic expressed in the canonical form. Pp. 175 179. Effecting the process indicated p. 73, but writing for greater con- venience (to, y, z) in place of (X, Y, Z\ so that the substitution to be made is ( a , I , G , / , (j , h , i , j , k } I ) = (6*, Qy, Qg t -2# -20, -20, -2s, -2a, -2 respectively (where I have corrected a misprint in the formula as originally given) I find the equation of the envelope to be tyz (y z)* a 4 -I- fans (z ~ wf b :| yf c' 1 b 3 c c 3 a -1- 4?/^ (?/ a + z* 4- 3fl>y - 2y* + 6) a a b bc a ca s 4- 4no (" 4- a 3 + 3ay - 2m; + 5y^) ab a + y(y*- Vttfg - 2i/ 3 ffl -f i/s a + 3SaJi/2 + ^ + 12s a a; + ^ ca + e (z* - 22 a a; - 2^y + zffl 9 4- 38wy^ + zf + 12^^ 4- 12#2/ a ) a a b 3 4- 2y^ (llaj a 4- 7/ a + - %* 4- 24#2/ + 24^aj) a 2 bc 4- %m (lly a + 2 3 4- & - 2#w 4 24^ 4- 2%) b 9 ca 4- 2y (ll^ a + a 4 2/ a ~ 2y + 24w 4- 24y#) c s ab = 0. C. VI. 82 ON A CERTAIN ENVELOPE DEPENDING ON A TRIANGLE &C. [396 The function on the left-hand side is the quotient by -48(a + b + c) a of the sextie function, Table 67 of my third Memoir on Qualities, [144] ; the foregoing quotient was calculated without using the coefficient of the term in a a b u c 3 (grfZ 2 ) of the table, but by way of verification, I calculated from the table the term in question, and found it to be ( + y + zY - 2 (a + y + zj (ya 4- w + ay) and this should consequently be equal to the coefficient of a 3 b a c a in the product of (a-f b + c) a into the foregoing quartic function of (a, b, c) that is, it should be + y (f + z (z* - 2s 9 * - 2z*y + zaP + SSsoyz + zf + + tyz (1 lffi a + f + z* - Zyz + 24a,'y -f m (U f + z? + a? - 2^a; + a 9 + f - Ztcy which is accordingly found to be the case. 397. ' SPECIMEN TABLE jfcf = o6" (MOD. N) FOB ANY PEIME OR COMPOSITE MODULUS. [From the Quarterly Journal of Pure and Applied Mathematics, vol. ix. (1868), pp. 95 96 and plate.] IF N be a prime number, and a one of its primitive roots, then any number M prime to ff, or what is the same thing, any number in the series 1, 2, ... tf- 1, may be exhibited in the form jtf e a (Mod. tf) ; where a is said to be the index of M m regard to the particular root a. Jacobi's Gmon Arithmetics (Berlin, 1839), contains a aeries of tables, giving tho indices of the numbers 1, 2, 3...JT-1 for every prime number N less than 1000, and giving conversely for each such prime number the numbers M which correspond to the indices 1, 2. ,., (tf-1) (Tabute Numerorum ad Indices datos pertwentiim et Indicwn Nmiero dato correspondent^. A similar theory applies, it is well known, to the composite numbers; the only difference is, that m order to exhibit for a given composite number N the different numbers less than J? and prime to it, we require not a single root a, but two or more roots a b,... and that in terms of these we have Jf-a'P ... (Mod.JV). For each root there xs on index A (or say the Indicator of the root), such that a" el (Mod. N}, A being the least index for which this equation is satisfied; and the indices *&,.., extend ^ from 1 to A B respectively; the number of different combinations or the product AB... t being precisely equal to <j>(N), the number of integers less than N and prime to it. The least common multiple of A. B..,, is termed the Maximum Indicator and repre- senting it by /, then for any number Jtf not prime to Jf> we hav.e Jf'sl(Mod. iY), a theorem made use of by Gauchy for the solution of indeterminate equations of the first order. Thus JV=20, the roots may be taken to be 3, 11; the corresponding exponents are 4, 2 (viz. 3^1 (Mod. 20) 11- si (Mod. 80)), and the product of these is 8, the number of integers less than 20 and prime to it; the series [go to p. 86] J. X l 84 SPECIMEN TABLE M s Ct a b ft (lIOD. N) FOR ANY [397 SOS. 1 2 3 1 5 6 7 8 10 11 12 13 : 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 '29 30 HOOTS 1 2 3 2 5 3 3,5 2 3 2 5,7 C 3 2,11 3,7 10 5 10 3,11 2,13 7 10 5,7,13 2 7 2 3,13 10 7,11 IND. 1 2 2 4 2 2,2 6 4 10 2,2 12 j C 4,2 4,2 16 6 18 4,2 6,2 10 22 2,2,2 20 12 18 6,2 28 4,2 M. I. 1 2 2 4 2 G 2 G 4 10 2 12 6 4 4 16 G 18 4 6 10 22 2 20 12 18 6 28 4 1 3 2 4 2 G 4 6 4 10 4 12 G 8 8 1G G 18 8 12 10 22 8 20 12 18 12 28 8 1 2 1 3 1 ___. 1 3 2 5 1 2 1 4 6 0,0 1,0 0,1 1 2; fi 1 1 3 2 t 0,0 1,0 5 8 10 fl 1 B 0,0 1,0 2,0 0,0 1,0 1,1 10 11 4 7 1 17 5 16 2 0,0 1,0 0,0 1,0 2,0 1,1 4 2 8 20 1C IB 0,0,0 1,0,0 1 7 2 8 3 1 2 6 0,0 1,0 2,1 11 27 22 18 0,0 JL 3 7 1,1 8 4 3 3 g 7 3 0,1 1 7 8 1,1 3,0 0,1 5 9 14 2 4 12 15 3,0 3,0 1 6 21 2 0,1,0 8 6 8 1 16 3 in M 5 1,0 2 G 4 2 2,0 6 10 2,0 8 18 14 4 2,0 M 10 5 2 1 1 2,1 1 6 1 11 ll 1 11 4 0,1 3,0 13 5 6 0,1 6,0 8 1,1,0 16 5 13 1,1 2 it 0,1 12 6 15 3 14 9 i\. 13 8 3,1 3,1 12 4 13 1,1 0,1 3 12 0,0,1 19 8 0,1 2 ,1,0 14 2,1 3 11 7 14 If 16 2,1 2 7 6 13 11 3,1 IV 1C 8 14 4,0 10 4 4 1(1 17 3 8 3,1 5,1 7 17 1,0,1 13 10 15 4,1 7 1,1 18 4 15 U 1!) 2,1 4,1 6 0,1,1 8 7 12 5,0 Ifi 20 3,1 7 12 9. ft 21 5 10 12 9 10 22 11 17 14 ti 23 1,1,1 11 2 11 5,1 24 8,1 24 10 4 2i> li 10 4,0 H 26 9 lii 27 3,0 23 28 14 29 2,1 307] PBIMJfl Oil COMPLEX MODULUS. 85 81 2 !)!) Bd 3fi 80 37 88 39 40 41 42 3 44 15 46 7 48 49 60 3.7 ,ir> 3,10 a a,o ,10 'B a 2,14 11,21 5,1!) 28 j),21 2,2tt 5 B ,7,17 3 3 20 HO H,2 10, a 10 12, a i,a an 1H 12,2 1,2,2 40 6,2 42 L0,2 12,2 22 16 1,2,2 42 UO ill) H 10 10 10 la im 18 12 4 -10 12 10 12 22 40 4- 12 20 Hi '20 24 la ao 18 24 Hi 40 18 ia 20 24 22 1C 10 42 20 ia ID ;u 21) "a'fi " i 41 an -j 0,0 1,0 LA. 2,1 s.o 0,0 1,0 8,0 0,1_ 2,1 n,o 0,1 1 5 0,0 1,0 11,0 2,0 0, 1 ' 8,0 10, 0,0 1,0 11 Id 22 1 1 4 0,0 1,0 2,0 0,0 0,0,0 1,0,0 20 13 12 22 0,0 1,0 ay 17 '\C> K 0,0 1,0 8,0 0,0 1,0 2,0 16 1 30 18 14 17 2 38 14 16 1 0,0 ,0,0 2(5 1 10 29 1 1 2 a 4 5 11 a a,i '.) 28 88 82 12 "(V no lit If 8r> '2 1,1 1,0 8,0,1 2,0,0 1 89 88 80 8 14 7 11) 14 2 9,1 2,0 1,1 8,0 19 10 ,1,0 27 30 12 13 10 11 33 ao Ifi 2 7 8 !) 10 a 7 '28 lit it 7,0 1.1 0,1 0, 1 11,1 7 '1 El 9,0 1,1 8,1 _. B,l 11, 10, 1 11,1) 7,0 a,i 5,1 d.O 12 17 JL 10 1-1 8 7,0 0,1 0,1,0 1,1,1 11 27 81 25 87 6,1 0,1 (i 11 40 4 22 2,1 il.O 1,1 1,1 2,1 9 14 17 27 82 8 22 86 ,1,0 ,0,1 8 17 11 12 13 14 16 id 17 18 19 20 IH i H SB i-i "17 11 m 10 10 ~"c u 2H a? 1C "in J,0 fi,0 'i;T (1,1 M 11,0 M _ 112 4,0 0,0 H, 1 i, i M M Id ia ifi 10 15,0 0,1 1,1 a,n H li 7 25 28 ' ao" 17 21 1 2 4,0 2,1 5,1 11_ L 0_ 8,0 10,1 0,0 2,1,0 24 88 10 9 81 14 21) 80 18 4 5,0 4,1 !IO Ifi 81 20 41 8,1 1,1 4,0 9,0 fl,0 7 16 28 42 20 29 81 0,0,1 1,1,1 ao 25 28 U6 80 19 14 0,0,1 1, 0, 1 1,1 2,0 24 D 20 a 10 0,1 fi,o (1,0 5,1 11,0 18 2 10 11 89 10 3-1 0,1,1 2,0,0 24 88 87 10 16 13 21 22 28 2-1 25 7,1 4,1 7,0 i) in M , 0,1 0,0 6,0 '4,1 81 ...._ ad ao Id 15 10 8 11 1,1 4,1 a, 0,0 2,0,1 17 G 11 7 28 28 10 18 19 21 8,1 87 1 25 19 8,0 4,1 0,1 8,1 10,1 4 18 38 8 43 19 3,0,0 IV 8 18 14 8 2U 27 28 2<) 30 0,0 5,0 an o H '!) 8,1 5,0 . 7,1 .L 1 35 27 10 4 10 ay 18 B7 15 la 10 0,1 5,0 8,1 Al 7,1 0,1,1 1,1,0 2,1 82 27 21) 13 12 7,0 7,1 8,0 5,0 10,0 a 20 5 12 46 20 9 2,1,0 1,1,0 7 4 41 9 4 81 82 88 84 35 8,1,1 2 82 85 4,0 28 86 26 15 38 4,0 8,1 0,1 7,0 21 8 4 24 13 21 16 1,0,1 12 82 19 84 23 7 18 8U 37 38 30 40 41 42 43 44 45 KB 11,1 8!) 8,1,1 40 21 41 8,0 18 21 0,1 8,1 7,1 12 5 26 40 87 41 - 7 a, o, i 8,1,1 16 6 8 31 12 5 _43 43 5,1 44 6,1 if> l: i 22 5 11 45 47 48 49 47 2,1,1 48 21 49 10 60 50 86 SPECIMEN TABLE M = tfW (MOB. N) &C. [397 (from p. 83] of these is in fact 1, 3, 1, 9, 11, 13, 17, 19, each of which is expressible in the required form, viz. IsSMl , S-SMl , 7 = 3M1, &c. (Mod. 20): the maximum indicator is 4; viz. I 4 si, 3* si, 7* si, &c. (Mod. 20). The table pp. 84-, 85 gives the Indices for the numbers less than N and prime to it, for all values of JV from 1 to 50; the arrangement may be seen at a glance; of the five lines which form a heading, the first contains the numbers N\ the second the root or roots belonging to each number N, the third the indicators of these roots, the fourth the maximum indicator, the fifth the number </> (N). The remaining lines contain the index or indices of each of the $N numbers M less than JV and prime to it, the number corresponding to such index or indices, being placed, outside in the same horizontal line. For example, 30 has .the roots 7, 11, indices 4, 2 respectively ; the Maximum Indicator is 4, and the number of integers less than 30 and prime to it is 8 ; taking any such number, say 17, the indices are 1, 1, that is, we have 17 = 7 1 . II 1 (Mod. 30). The foregoing corresponds to the Tabulce Indicum Numero data oorrespondentium of Jacobi ; on account of multiplicity of roots there does not appear to be any mode of forming a single table corresponding to the Tctbulcs Numworum ad Indices datos perti- nentium; and there would be no adequate advantage in forming for each number N a separate table in some such form aa Roots 3 11 No a. 1 1 11 1 3 1 1 13 2 9 2 1 19 3 7 3 1 17 which I have written down in the form of a table of single entry ; for although (whenever, as in the present case, the number of roots is only two) it might have been better exhibited . as a table of double entry, when the number of roots is three or more ifc could not of course be exhibited as a table of corresponding multiple entry. 87 398. ON A CERTAIN SEXTIG DEVELOPABLE, AND SEXTIC SURFACE CONNECTED THEREWITH. [From the Quarterly Journal of Pure and Applied Mathematics, vol. ix. (1868), pp. 129142 and 373376.] I PROPOSE to consider [first] the sextic developable derived from a quartic equation, viz. talcing this to be (a> &, o, d, e\t> 1) 4 = 0, where (a, b, o, d, e) are any linear functions of the coordinates (so, y, z, w\ the equation of the developable in question is (ae - 4>bd + 3c 3 ) 3 - 27 (ace - ad 2 - & a e + 2&ccZ - e 3 ) 3 = 0. I have already, in the paper " On a Special Sextic Developable," Quarterly Journal of Mathematics, vol. vir. (1866), pp. 105-113, [373], considered a particular case of this surface, viz. that in which o was =0, the geometrical peculiarity of which is that the cuspidal edge ia there an excubo- quartic curve (of a special form, having two stationary tangents), whereas in the general case here considered it is a sextic curve. There was analytically the convenience that the linear functions being only the four functions , b t d, e, these could be themselves taken as coordinates, whereas in the present case we have the five linear functions a, &, o, d, e. The developable 88 ON A CERTAIN SEXTIC DEVELOPABLE [398 viz these equations are really equivalent to two equations, and they represent a curve f"tho fourth order which is an excubo-qnartic. We may m fact find the equations of the nodal curve by assuming (a, b, o, d, e& 1)' to be a perfect square, say to avoid fractions that it is = 3 (d? + 2/3* + tf, then we have which equations as involving the two arbitrary parameters a : ft : 7, give two equations between (a b c d, e), and we may at once by means of them verify the above- mentioned equations of the nodal curve. It also hereby appears that the nodal curve is as stated an excubo-quartic curve; viz. we have between a t b, c, d, e a single linear relation that is a quadric relation between a, /3, 7, and this equation may be satisfied identically by taking for a, /3, 7 properly determined qvmdric functions of a variable parameter 6; whence a, b, c, d, e are proportional to quartic functions of the variable parameter 0, or the curve is an excubo-quartic. The equations of the nodal curve may be presented under a somewhat different form; viz. the cubi-covariant of (a, b, o, d, e$t, 1)*-0 being 9nc' J - QlPe 5abe + loacd - IQlfd - We -Ibbce ae *, 1) = say this function, multiplied by 6 to avoid fractions, is (a, b, c, d, e, f, g^t, I)", that is 9ac 3 ~ d = 3 (+ ad* - e = 2 (+ ade + f = 1 (+ e 9 g = 6 (+ 6e 3 - Scrfe + Zd s ), ' then the equations of the nodal curve may be written a = 0, b = 0, c = 0, d = 0, e = 0, f= = 0. AND SEXTIC SURFACE CONNECTED THEREWITH. 8& It may be mentioned that we have identically ae - 4bd + 3c a = 0, af - 3bB + 2cd = 0, ag - 9ce + 8d 3 = 0, bg - 3of + 2dc = 0, bf ~ 4ce + 3d 3 = 0, and moreover a-6bf + 15ce-10d a = -6(bf- so that fcho equation of the developable may be written in the form ag-6bf+loce-10d 9 = 0, or in the more simple form each of which puts in evidence the nodal curve on the surface. The nodal and cuspidal curves meet in the points ft b _ *L - b~~ c~d e' being, as it is easy to show, a system of four points. The four points in question form a tetrahedron, the equations of the faces of which may be taken to be # = 0, 7^ = 0, = 0, w = Q; and the equation of the surface may be expressed in this system of quadriplanar coordinates. We introduce those coordinates ab initw, by taking the quartic function of t to be (a, b, c, d, e$t, l)' = w that is, by writing ftas b - G = e = Observe that (t lt U, t B , t^ being any constant quantities, we thence have e - dS<i + o2i*> - VSt&tt = tf (a -i)(o 0. VI. (7-0(7 -0(7 -0(7 -ftt) -<,)( *0(8-<8)(8-*4). 12 90 ON A CERTAIN SEXTIO DEVELOPABLE [398 and thence in particular e - a + Ga - ay + a 7 = 0. viz. this is the linear relation which subsists identically between (a, 6, c, d, e), the five linear functions of the coordinates (a 1 , y, e t w). Starting from the above values of (a, I, c, d, e), we find without difficulty J = (a - /3) s 09 - 7)* (7 - afmjz + (a - ) 3 (/3 - S) a (S - a ) a e& + (a - 7 ) 2 (7 - S) 2 (S - a) 9 aww + (/3 - 7 ) a (7 - S) 2 (S but we thus see the convenience of introducing constant multipliers into the expressions of the four coordinates respectively, viz. writing IB (/3-yS) 9 a/, where for shortness or what is the same tiling, taking the quartic to be (ft, &, G, d, e$t, l)^^ we find J^ (X> V') 3 (a; I - (XV V) [V (a)W 4- 2/'2') + X (T/'W' + /(B 1 ) + v (s'w 1 + a>y )}, where for shortness or writing we have and the equation of the developable is thus {V <>W + yV) + ft' (y'w 1 +*'') + / ((c'w' + y)} 3 - 27X>V (a/y'y -I- y W + ^Vw' + co'y'wj = 0. Observe that J=0 is a cubic surface passing through each edge of the tetra- hedron, and having at each summit a conical point; J=0 is a quadric surface passing through each summit of the tetrahedron, and at each of these points the tangent 398] AND SEXTIC SURFACE CONNECTED THEREWITH. 91 plane of the quadric surface touches the tangent cone of the cubic surface: to show this it is only necessary to observe that at the point (of = 0, y' = 0, z' = 0) the tangent cono in y'z' 4- dot H- as'y 1 ~ 0, and the tangent plane is XV 4- p'y' -I- vY = 0, and that these touch in virtue of the above-mentioned relation V (V) + V (X) + V (v') = 0- I* follows that ou the curve of intersection, or cuspidal edge of the developable, each of the summits is a cuspidal or stationary point, that is, the cuspidal curve has four stationary points; this agrees with the character of the curve as given, Salmon "On the Classification of Curves of Double Curvature," Comb, and Dull. Math. Jour. vol. v. (1850), p. 39, via. the character is there given a = 6, m = 6, n = 4, r = 6, g = 3, h = 6, a = 0, /3 = 4, as = 4, y = 6, (/3 4, that is, there are ns stated 4t stationary points). To find the equations of the nodal curve, instead of transforming the equations as given m terms of (a, 6, c, d, e\ it is better to deduce these from the equation of the surface ; viz. if there is a nodal curve, we must have : BJ : : 18 J Writing these under the form parameter C 1 ), we have XV + / \V + 0, &c., where ^ is regarded as an arbitrary -h -I- ^' i' + ff '^O = 0, w 1 ) = o, which equations (eliminating ff) mnst be equivalent to two equations only. I remark that the first three equations may be regarded as a set of linear equations in 1, w', V, 8'w' >, and determining from them the ratios of these quantities, we have, suppose, , , n 1 : _ 10 : 6 : - ffw = A : Jl : JJ) whore A' *V, *W, a/ G = The value of ff i in fact =-^, that is, instead of the fool ', \ve have really four determinate equations. 92 ON A CERTAIN SEXTIC DEVELOPABLE [398 "We have thence AD BG = and (substituting In the fourth equation) A (\'a + fjfy' + vs?) -f- C (y'z + #V -f aty') = ; each of these equations must contain the equation of the cone having (#' = 0, y 1 = 0, = 0) for its vertex, and passing through the nodal curve. The two equations are of the orders 6 and 4 respectively; aud as the curve is a quartic curve passing through the vertex iu question, the equation of the cone is of the order 3. I have not effected the reduction of the sextic equation, but for the quartic equation, substituting for A, G their values, this is - (XV + fify' + Jaf ) [XV (y - z'} + /*y (2' - of} + v'z* (z - x 1 )} + (^ + M + ay ) [xv w - *') + /*y (*' - ao + M W - y'}1 + fo*V (y 1 - z'} + vW (z 1 - x'} + XV (a/ - y 1 }} (of + tf + /) = 0, which is easily reduced to XV - a' + 'z r + ^ + --e' + y'^' (- ^ 3 + y'z -1- ^ + a;y ) (x 1 - 2/') + ^V [(of + y' + O (2/Y -I- sV 4- ,y) -I- + v'\' [(a f + y' + z f ) (y'z* + sfot 4- 'i/') + + XV [(aC + y + 2') (2/V + *V + /2/') + x'y'z 1 } (of - y 1 } = ; and I have found that this is transformable into 2 [si v (V) + y' v M + *' vcoi x [> v vc^'x^y - w'^o ^ v(/*o(^ ^ v ) *y vc^xxv - ^o V (/)} (V (') - V (V)} (V (V) - V (/*'))] = 0, viz, the two functions are equivalent in virtue of the relation V(X') 4- or, what is the same thing, they only differ by a function (a/, ?/', .s') 4 into the evanescent factor X /a + /jf 3 + 1/ 3 2/*V 2y'X' 2XV- The function in { } equated to zero is therefore the equation of the cubic cone. I do not atop to give the steps of the investigation in the above form, as the investigation may be very much simplified as follows: by linear combinations of the four- equations in co', y' t sf, w', &', we deduce (- X' - jtt' + y') (^ + w' - x 1 - 1/') + W (fffij - *W) = 0, ( V + pf + v 1 } (d + y j + z' + <of) + W (tfd + atof + x'y' + ts'w' + y'w' + z'w') = 0. Hence writing X = X' X v', at = w' + of y' ^, fj, X' + /i' i/, y = w' a;' + 1/' /, v = X' /*' + j/, 2 = w' a;' T/' + ^', w = / + / + y 1 + /, 398] AND SEXTIC SURFACE CONNECTED THEREWITH, 93 wo find X/i = -X' 2 -/ 3 + y' 4-2X>', and thence pv 4- yX + X/i = - (X' a + ^' 3 + i/ 2 - 2/*V - 2j/X' - 2X>')> = > that is 1 4. 1 4. 1 -0 ,- H --- h - = U, X /i i/ the relation which connects the new constants X, p, v* Moreover yz~MW = 4i (tfz 1 afwf), zx yw = 4 (tf a/ - y'w'}, xy zw k (afy 1 - z'w'), 3w a - fl) a - f - & - 8 (i/d + t/a/ + w'y' + a/v/ + y'w' + z'w'}> and writing for greater convenience 6 -j,, the equatioiis are transformed into ccw - yz , yw m, Ovz =zw ay, where . X p v iz. those equations, eliminating fl, give the equations of the nodal curve. From the first three equations eliminating 6, we deduce -/*)=== or, as these equations may be written, which equations, from the mode in which they are obtained, are it is clear to two equafcionB only. Using the fourth equation, and ehmmatmg * by therein for ^ ^ fo their values from the first three equations, we find 94 ON A CERTAIN SEXTIC DEVELOPABLE [398 that is 8^ + tf + ^ + rf2wf2* + - + J V so y z or, what is the same thing, sty* (3-H/ 5 4- x~ + f 4- &) - 2w fy"^ + sV + tff) = 0, we have to show that this is in facb included in the former system, for then the four equations with 6 eliminated will it is clear give two equations only. Observe that the former system may be written (p - v} ifz* -F (v - X) sPa? + (X - j^) off = 0, fir ( ti^fi ___ i and that we have thus to show that substituting for w the value 10 = fj, in the equation anje (3w a + 3 + f 4- s a ) - 27y (i/V + sV -I- a; a i/ a ) = 0, the result is (}j. - v) fz* + (v - x) 0V + (x - it) a?y = o. The substitution in question gives 3.' W -^)' (a , ) _ y (p vfyz J ^ J ' (/i y w ^ x thafc is 3a: a (^tj/ Q - vsff + 0* - v) a i/ 2 s a (a? + y 8 + ^ a ) - 2 (/i - v) (^f - v&) (f# -\- 0V + ttPif) = Q, which is in fact -_ pty (y* _ ^ ^ _ gfy + 2/iWC 9 (^ _ z *y _ uV (y - &) (af> - if) = 0, ' that is, throwing out the factor y' i zP t it is ? (f - z*) - vW (a; 3 - f) = 0. But in virtue of the equation ~H --- &-- =0, we have 1 \ j, v and the required property thus holds good, 398] AND SEXTIO SURFACE CONNECTED THEREWITH. We thus see that the equations of the nodal curve are the nodal curve is thus the partial intersection of the two cubic scrolla (skew surfaces) (ft - v) wyz = to dnf - v&), (v - X) wgai = y (v& - ^ a ), viz taking A B t ' 0, D to be the summits of the tetrahedron th 3 faces whereof at* taking A Qf ^ has ^ fop ^ nQdal C f rgt d\,cU BA CD fo, gene,ato,; the aaooud has JU) to .nodal taeto. ^ for a shmle directrix, ^D, 0J) for generators; the surfaces intersect in the line AM tL; S lint S) twice/and the le OD; the order of the residual curve, o, nodal curve of the developable, is thus 9 -(2 +2 + 1), =4 as it should bo. I remark that the equation (^-y)^ a +^-^)^ a -1-(^-^)^ 3 = ' is the equation of the cone having its vertex at the point A (0-0, 2/ = 0, *0), and . ier' of linos tomgh two points which p through a give, pomt not on the curve is in fact = 3. It remains to introduce the coordinates <. y, ., ) * the equation of the developable. "We have 1 ' - 2 and thence / / / \ a / rt , V = (w - ') - (y - giving andsimilarly 8 (^' + off) ^-tf- Moreover = (fp _ ^ _ _ + y + 96 ON A CERTAIN BEXTIC DEVELOPABLE [398 Consequently ' (yV + 2'' + afy'} - (to 4- <e + y 4- z) x { 3w 3 - 2*y (0 4- y + *) - ffi" - f - ** 4- atyY - (w + - # - *) (w - 4- y - #) (w - - y - -s), - w 9 ( 4- y + rfVi3 _L rt/3 _L_ 3 ^_ 3f4Q i nfiff __ &ff& _, ^/yi ^^ />!4l3 -4 fl^it J. i^ T ^ ~r y* y A &ui it iti^/ * Ui ij -p Putting for shortness ^^aj+2/4-xr, V = a 2 4- 1/ 2 4- s a - %yz - the two expressions are toiiS 4- m - 2p f - V or obsei'ving that - pV is we have 4-tti 11 . 4- cs 3 4- y s 4- a 3 4- that is '^' + wV' 4- 4- - 2w (p a + V ) 4- 4- Z/ a 4- ^ 3 ) (a? -\-f 4- Moreover 8 {V (a/w 1 4- 2/V) 4- /*' (tfof 4- //} 4- / (z'w 1 4- *V') 4- 1/ (w 3 - ffi a - y a 4- s 3 ), - - (X 4- /* 4- y) w 3 4- ?w a 4- /i2/ a + 398] AND SEXTIC SURFACE CONNECTED THEREWITH. and we have /* = 2 (7 - S) (a - 8) (a - v-2< whence X^-y = SX'/^V- Hence finally X, ^, v denoting as just mentioned, and therefore satisfying _ 4. i: 4. L = o, the equation of the developable is \pv {w 3 - w (a? + ;?/ 2 -I- ^ a ) + Iwyz]* -f 108 {(X + /i + v) w- - Xte 3 - pif - vsP}* = (say this is \jj,vT*-\- 108$ a = 0), and this surface (which has obviously the cuspidal curve- jS' = 0, T 0) has nlso fcho nodal curve aP (/My 8 - P) w a (vz* ~ - ^^ X I will show <)> posteriori that this is actually a nodal curve on the surface. Intro- ducing an arbitrary parameter $, the equations of the curve may be written itt supra, = anu yz, 6vz zw cey, 2(9 (X -I- /* + v) w = Sw 3 -a?-f- z and wo have thonco, OB before, Hence (X + /i -I- v) w (a? + if + z*} + (X + ft + v) -vf Xffl 3 iw/* Honce writing = 0. = > & . 2 (X + p + v) w - (3w a - 3 - f - &} 0, 0. VI. 13 ng ON A CERTAIN SEXTIC DEVELOPABLE [398 substituting for its value = , and attending to the significations of S and r l\ we have which are in fact the conditions to be satisfied in order that the point (as, y, z, w} may belong to a nodal curve of the surface Vi'2*+ 10SS*= 0. It is to be noticed that the coordinates of the before mentioned four points of intersection of the cuspidal and the nodal curves (beinff as already mentioned stationary points on the cuspidal curve) may be written SB, y t #, w = (l, 1, 1, 1), (1, - 1, 1, 1), (-1,1, - 1,1), (-1.-L 1. !) We have 'thus far considered the developable, or torse, the equation of which is [X' (x'w 1 + yV) + p' (y'w 1 + z'x 1 ) + v' (z'w + 'y')Y where VM + V(/*0 + V("') = 0; 01-, what is the same thing, writing a, 1, o, in place of V (V), V (/*0. V CO respectively, the torse where ft + 6 + c = 0. Inverting this by the equations of, y', z', w'-^, ~, -, -, we obtain a s surface {a 9 (ano + yz) + 6 1 (yw + B) + o 3 C^w + 0i/)! 3 - 27 a 6V a-^w (ffl + y + z -V iu? = 0, where o + 6 + o = 0; which surface I propose [secondly] to consider in the present; paper. The surface has evidently the singular tangent planes # = 0,^ = 0, 2 = 0, w == 0, each osculating the surface in a conic, that ia, meeting it in the conic taken fchrice, viz,, AJ = 0, in a conic on the quadrio cone cfyz + iPyw + rftew = 0, = 0, = 0, w = 0, a> + & 3 2ffi + o^y = ; 398] AND BEXTIC SURFACE CONNECTED THEREWITH. 99' and it has also a cuspidal conic, the intersection of the plane K + y + g + w*=Q with the qiwdric surface ct 3 (xw + yz) + & a (yw 4- sue) + c 3 (ziu + 2/2) = ; it may be observed that the four conies of osculation are also sections of this surface. The surface has also a nodal curve, the equations of which might he obtained by inversion of those of the nodal curve of the sextic torse above referred to; but I prefer to obtain them independently, in a synthetical manner, as follows: Take a, /3, y arbitrary, and write _ A = (b - c) a. + bfi - cy , F = by - c/3, .B (c a) /3 + cy a j G = cet ay, - G ~ (b - c) 7 + an - 6/3, ' H = a(3 - la, M = (6 - c) + (c - ) jS + (a - &) y, Q ==a 9 (&-o)a + & a (<>-3) + tfl ( a - & )r : then it is to be shown, that not only the equation of the surface is satisfied, but that also each of the derived equations is satisfied, by the values a : y : * : w^aAGHQ : WHFQ : cGFGQ : abofGHM; oticli of the quantities A, B, G, M. Q is linearly expressible in terms of ff, G, H, which arc themselves connected by the equation aF + bG+oH=Qi the foregoing values of ar, y> 2, w aro consequently proportional to quartic functions of a single variable parameter, say F+G; and there is thus an excubo-quartic nodal curve, To establish the foregoing result, we have + cF =0, aG + bff + cG - 0, aA + bB + cG = 0, aAGH +bttBF aff^BT +6J0 1 aM.ff.ff + 6"^J3^ + fl"C^ff = - abc 1*3 ( a vhioh are all of them identical equations; but as to some of them the verification is rather complex. ^ _ 2 100 ON A CERTAIN 8EXTIO SUIIPAOIC &0. Hence we have at -|- y -i- g = Q (aA Gil + blllU' 1 ~|- o(M.l) Jind thonco f [Q (a. -\- Moreover and ( + y + ^ + w) B - 27 (ak)' 1 (AMMWMQy (). Again -I- c'^) w a H. - .|- 7 ' .|. J + 6 a w + c^;i/ = ttbolWHQ* (<tIl(W+ WAG -I- o/t.71/7) . %Q (a -(- /3 -I- 7)' J , and thence (t a (,'tfiy 4- y#) -I- // J (yw -I- sis) -I- t:' J (;/ and the two oquntions inavkod (#) verily bho oi|iiabion ol' tho To verify the derived oquiifcionH, writo fin 1 it, iiiitiuouli 1* *= tt* (i/a *\- into) *\- b* (sta -[ yw) + c a (a 1 )/ + am), HO (ihali the equation of tho Hiirfuco in 7 );1 - ffltPbWGi/sw (u> -h y -I- -I- w) 11 ^ 0, and the derived equation with roHpoot to w in JL^L h .. 2 7 J <?/ ; a -|- y -I- e -I- w ' or substituting for P and flj + y + + w thoii- valucH, this IH and similarly for y, e, find w. In parliieular, iioiiHidoring tho dorivod equation in nwpuob to iw, this is a*tv + 6 a . and wo have as before which is thus verified; tho verification of tho durivod oquationn for y, ^ w can bo effected, but not quite so easily. Tho existence of the oxoubo-quai-bio nodal ourvo is thus established. 399, ON THE CUBICAL DIVERGENT PARABOLAS. [From tho Quarterly Journal of Pure and Applied Mathematics, vol. ix. (1868), pp. 185189.] NEWTON ' reckons five forma, viz. these are tho simple, the complex, the orwiodal, the aonodal, and the ouspidal, but as noticed by Murdoch, the simplex has three different forma, the ampuUate, tho neutral, and the oampaniform. We have thus the 8 forma at once distinguishable by the eye. Pllloker has in all 18 species, the division into species being established or completed geometrically by reference to the asymptotic cuspidal curve (or asymptotic semi-cubical parabola), and analytically as follows, via. writing the equation in the form ^> = <D8-3oflj + 2d, the different species are .simplex, f - a? - 3c + 2^1 ^ ampullate, i/ 8 = a 3 3a 2rfj" ' campnuiform, f = a? + 2d, neutral, 3 a, 18 2c^, campaniform, f = ?" + Scffl -f 2rf, )( I/a = ? + 3caj, / == ) 3 4- 3oaj - 2cZ, complex, i/ 2 = a 8 - 3ca) + 2rf| g * acnodal, i/ 3 = of 1 - 3cj - 2c V (c), crunodal, y = " - Sea; + 2c V (c), cuspidal, 2/ 3 = fl' 3 ; 102 ON THE CUBICAL DIVERGENT PARABOLAS. [39& but of the simplex species, there are five which are to the eye campaniform, and the three complex speeies have with each other a close resemblance in form. I remark as regards the simplex forms, that the tangents at the two inflexions meet in a point R on the axis, and that the ampullate, the neutral, and campaniform forms are distinguished from each other according to the position of R, viz. for the ampnllate form, R lies within the curve, for the campaniform form R lies without the curve, and for the neutral form, R is at infinity. It is to be observed, as regards the complex forms, that here R always lies without the curve, between the infinite branch and the oval. The further division of the simplex and complex forms so as to obtain the 7 + 3 species of Pliicker, may be effected by considering in conjunction with the point R a certain other point 7 on the axis ; it is to be remarked that excluding the inflexion at infinity the cubical divergent parabola has in all eight inflexions, two real and six imaginary, viz. the inflexions lie by pairs on four ordinates, or if ID be the abscissa corresponding to an inflexion, as is determined by a quartic equation; this equation has always two real and two imaginary roots, each of the imaginary roots gives a pair of imaginary inflexions ; one of the real roots gives a positive value for i/ 3 and therefore two real inflexions, the tangents at these meet in the above-mentioned point R on the axis; the other real root gives a negative value for y a and therefore two imaginary inflexions, but the tangents at these meet in a real point on the axis, and this I call the point /, It is clear that for each of the four pairs of inflexions the tangents at the two inflexions meet at a point on the axis, so that if X be the abscissa of such point, then X is determined by a quartic equation; two of tho roots of this equation are imaginary, the other two roots are real, and correspond to the points R and / respectively. . The equation of the curve being as above y" = '-3aB + 2cZ J then the coordinate m belonging to a pair of inflexions is found by the equation ar-6ca.' a -l-8cfa;-3c a = 0, or what is the same thing, (1, 0, -c, 2d, -3c>S>, 1) 4 = 0, (the invariant I is =0, and hence the discriminant, = 2V/ 3 , is negative, or tho roots are two real, two imaginary, as already mentioned) : the corresponding value of X is easily found to be y _ x 3 + 3ctc - 4d 3(a a -c) ' and we thence obtain 3oZ 4 - 4fLY 9 - 6c a Z a + WodX - (c 3 + 4>$) - 0, or what is the same thing, (80, -d, -c*, Sod, - <? - ^Z, 1)* = 0, ON Tllli (HWKJAL DIVERGENT PARABOLAS. 103 lor Uin (.!(|ua(;ii)n in A r ; the qiiadrinvariant I is =0, and hence the discriminant, - 27./' J , in imgtitivu ; that w, the roots are two real, two imaginary, as already iimiitioiiod. Miu Himplux forms, first, if o = 0, thou for the ourvo f = ? + %d t il iippimi'H that 74 lion nl; infinity, I within. tho ourvo; and for the curve f ;" - 2d, that A' lia withuiili tlui ouvvu, / at inlinity, .It Curtilmr appcavH Uvuli \vlusn (Z = 0, or Ixir liho ourvti, ft, / lio nipudiHtiiuttr frotn Uio vwtox, R without, / within tho ourvo, ]|niutt! in tlui nui'vn 7/aa W + atvr + 2rf, wiiMjn, when (/, ~-~ 0, Uio pointM jf(, I aro oquidiHtant from tho vortx, and for c = 0, the point. H IH at inlinity, it in nany to iufur by wmliinuity that tho points 7J, / lie R l', / within tin? imvvn, / liuin^ noiiror to tint vortox. Anil miniltirly in \>\w c.uvvo l?/ a =; l -|-Hft';-2(i F , that Mm pninlH Jt, I lit: ./i without, ./ within tho (iiirvo, R boing naaroi- to tho vortox. Ajriiiti, in ihn I'tu-vi' ;/ J ^rt;' l --;itoH-2^ fiiutn, in thii uiirvn f -* u? -\- l\w + $d, R IH without, / within tho ourvo, and an o ln'iioini'H rc(), 74 IIUBMIIH oil' Ui iniiiiity, it unpoai-H that a having ohnnguil its sign, or lor Llm (iiirvo now in ipiOHtiou, H liitvin^ pnmud thwnigli inlinity, will bo tdtuato within Lhu mirvti; that in, ,/i', / lit! uiuih of thoni within tho curvo. And Himilavly for Llin mirvo yj=/M-3c,-2d, it appi'in-H that H, I lio wich without thu ourvo. co, Ibmlly, lor tlio Hiinplox i'oniiH, wo havo tho 7 apcoioa of Pltickor f M flu - 3ou -|- 2rf, c 3 x ainpulhito, ,/i, f within the curve; 2/i ?- Sow -2(2, c u ix (iainpaniform, 74, / without the curve; simplex neutral, J within tho ourvo, JK at infinity; 104 ON THE CUBICAL DIVERGENT PARABOLAS. f = a*-%d t simplex campaniform quasi-neutral, R without the curve, / at infinity ; y s = fl! 3 +3cflj + 2d, simplex campaniform, R without and further from, J within and nearer to the curve ; simplex campaniform equidistant, viz. R and I are equidistant from the curve, R without and I within ; f = a + Sea- - 2d, simplex campaniform, R without and nearer to, J within and further from tho curve. Passing to the complex forms, suppose for a moment that a is the diameter of the oval and J3 the distance of the oval from the vertex of the infinite branch ; the equation of the curve then is f = as (as - a) (<o - a - /3), or changing the origin so as to make the term in & 1 to vanish, this is y- (as or, what is the same thing, 3 ) a - fa ( - ) (2a + jS) (a + 2/9), or comparing this with y 3 = ar 1 - 3c + 2d, fZ is =+, or , as </3, o = y9, >/3, or say as the oval is smaller, mean, or larger ; viz. the magnitude of the oval is estimated by the relation which the diameter thereof bears to the distance of the oval from the infinite branch. In the case d = 0, or for the curve y* = x*~ Sea; it appears (as for the corresponding simplex form ?/ 3 = $ + Sea 1 ) that the points R, I arc equidistant from the point 0, which is in the present case the middle vertex, or vertex of the oval which vertex is nearest to the infinite branch. As the oval diminishes, so that the curve becomes ultimately acnodal, 7 remaining within the oval ultimately coincides with the acnocle ; and as the oval increases so that the curve becomes ultimately cruuodal, R remaining between the oval and the infinite branch; ultimately coincides with the crunode; and it hence easily appears by continuity that for a smaller oval I is nearer to, R further from the middle vertex ; while for a larger oval, J is further from, R nearer to the middle vertex. Hence for -the complex forms the species are smaller oval, I nearer to, R further from the middle vertex; ^ of-BotSt mean oval, R and J equidistant from the middle vertex ; ?/ 3 = a; 3 -3ftB-2(Z, larger oval, J further from, R nearer to the middle vertex: and the division into species is thus completed. Gwribridge, June 16, 1865, 400. ON THE CUBIC CURVES INSCRIBED IN A GIVEN PENCIL OF SIX LINES. [From tho Quarterly Journal of Pure and Applied Mathematics, vol. ix. (1868), pp. 210221.] WE have to consider a pencil of six lines, that is, six lines meeting in a point, and a cubic curve touching each of the six lines. As a cubic curve may be made to satisfy nine conditions, tho cubic curve will involve three arbitrary parameters ; but if wo have any particular curve touching the six lines, then transforming the whole figure homologoitHly, tho centre of the pencil being the pole and any line whatever the axis of hornology, tho pencil of lines remains unaltered, and the new curve touches tho six linos of the pencil ; the transformation introduces three arbitrary constants, and fcho general solution is thus given as such homologous transformation of a particular solution, To show tho same thing analytically, take (<e = 0, y = 0, 2 = 0) for the axes of coordinates, tho lines &=(), i/=0 being any two lines through the centre of tho pencil, HO that tho equation of the pencil is (*$>, #) a =0, then if <b(x, v. *)=*0 is the cquati 10 (J ON THE CUBIC CURVES INSCRIBED IN A [400 in fact, comparing the two sides of this equation, we have each of the seven coefficients of the sextic equal to a function of the seven quantities V( C )> W (c), &V(c)j &V(c), j, J> ft so tliat conversely, these seven quantities are determinate (not however rationally) in terms of the coefficients of the given sextic. And when the sextic is expressed in the foregoing form, then it will presently be shown thafc we have (a, h, k, 6]fa y} 3 +3z(j, i,/3fc 2/) a + ^ 3 = 0, or, what is the same thing, (a, 6, o,f t 0, A, 0,j, fcjflf, 2/,) 3 = 0, as the equation of a cubic curve touching the six given lines ; and by what precedes, it appears that this may be taken to be the equation of the general cubic curve which touches the six given lines. On account of the arbitrary constant c, it is sufficient to replace s by oa: + j3y + .&, or, what is the same thing, to consider = as the equation of an arbitrary line, but without introducing therein an arbitrary multiplier. To sustain the foregoing result, consider the cubic (a, &, c, /, g, h, i, j, /c, Zjffi, y, 2) 3 = 0, then in general if 4 = (,$, y, zf, = ($>> y, *)'(, /?, 7), C^^&a, y, 2) (a, /3, ry) Q , -^ = (>>$. a > &> 7) 3 i * ne equation of the pencil of tangents drawn from the point (, /3, 7) to the curve is A*D* - GABOD + 4vi0 3 + ^"D - 35 3 a a = 0, but wi'iting for shortness (,$<*, y,gy = (A' t ff t (?, J)'J1, #)', so that A' = (a, A, fc, 6 Jflj, 2/) a , ^ O'.t/J*,ff)'i 0" ((/, i$ffi, ?/) , D'= c then for the tangents from the point (# = 0, #=0), writing (a, /3, *y)(0, 0, 1), we ha-ve A = (A' > S, G'.D'^l, zf, B - (ff, G', jD'Jl, *), and thence the equation of the pencil of tangents is AW - QA'S'G'D' + *A'C' & + 4>B' a D' - 3J3' 3 a /a = 0. Hence for the curve (a, I, o,f, 0, k, 0,j, k, Z$tf, 2/) 3 = 0, 400] GIVEN PENCIL OF SIX LINES. 107 we have g*=Q, i' = 0, and therefore C" = 0; the equation of the pencil of tangents is J l' a D' 3 +4#' 3 D'=0 ) or throwing out the constant factor D', and then replacing A 1 , B\ D' by their values, the equation of the pencil of tangents is o((a, h, k, frjffi, yyp + qO*, 1,/Jia, 2/) 3 ] 3 = 0, which is the before-mentioned result. The coefficients V(c), &V(0), &V(o). ^V(c), j, I, f, or (as we may call them) the coefficients of the cubic curve, are, it has been seen, functions of the coefficients of the given sextic (*$#, y) 8 ; hence the invariants S and T of the cubic curve are also functions of the coefficients of the sextic, and it is easy to see that they are in fact invariants (not however rational invariants) of the sextic. To verify this, it is only necessary to show that the invariants S and T are functions of the invariants of the functions A/( C ) (> /*> k, &$X Z/) a and (j, I, f$x, y)"; for if this "be so, they will be invariants of the function !>(, h, k, l>lx, ?/) s ? + 4[(j, l,f$x, yf}\ that is of tho soxtic. Wo have in fact the general theorem, that if P t Q, R,... be any qualities in (a;, y, ,,.), and <(P, Q, It,...) a function of these quantics, homogeneous in regard to (iv, y, ...), then any function of the coefficients of $, which is an invariant of the quantica P, Q, .R, ... is also an invariant of 0. Considering for greater convenience the function (a, h, k, &$a, y) s in place of V (o) . (ft, h, k, b^ic, y)\ the invariants of the two functions (a, h, k, &][#, y) 3 and (j, I, f$fo, y)' J are as follows: D = a 9 b - Gabhk + 4a/c 3 + 4&A 3 - 3W, V- fj~l\ @ = j (bh - k") + I (lik - ab) +f (ah - h*\ R = + 1 a 3 / 8 + 6 - G - 6 + 12 + l - B + 12 bhjl* - 6 6/c a 2 + Q - 18 + 9 - 8 108 ON THE CUBIC CURVES INSCKIBED IN A [400 viz, Q, V tire the discriminants of the two functions respectively, and 0, R are simultaneous invariants of the two functions, R being in fact the resultant. The corresponding invariants of the functions V(c) . (a, h t k, 6}J>, y}\ and (j, I, /Joj, y}* are obviously c 2 D, V, c@ and cR. The values of 8 and T are obtained from the Tables 62 and 63 of my "Third Memoir on Qualities," Phil. Trans, vol. CXLVI. (1856), pp. 627647, [144], by merely writing therein g=*i=Q. It appears that they are in fact functions of c a D, V, c and c-R; viz. we have 8 = V 2 +c, T = 8 V 3 + c (4R + 1 2 V 0) + cTJ. The invariants of the sextic (*$>, y)\ if for a moment the coefficients of this sextic are taken to be (a, b, c, d, e, f, r/), that is, if the sextic be represented by (a, 6, c, d, e,f, 0]j> ( T/) 9 are the quadrinvariant (=af/- G6/+ looe- 10d), Table No. 31 and Salmon's A., p. 203 ( l ), the quartin variant, No. 34, and Salmon's B., p. 203, the sextinvariant No. 35, and Salmon's C., p. 204, and the discriminant, which is a function of the tenth order ^a s g s +&c. recently calculated for the general form, Salmon, pp. 206207, say these invariants are Q,, Q,, Q g and Q 10 . These several invariants are functions of the above-mentioned expressions c a D, V, c and cR\ whence, con- versely, these quantities are functions of the four invariants Q 9 , Q 4 , Q tl Q n - and the invariants S, T of the cubic curve, being functions of c 3 D, V, c and cR, are also, as they should be, functions of the invariants Q a , Q t , Q a and Q n of the sextic pencil (*", y) 8 - To effect the calculation of Q a , Q t and Q Si I remark that inasmuch as by a linear transformation, the quadric (j, I, f$ce, 7/) 2 may be reduced to the form 2%, and that the invariants of (a, h, k, b^so t y) 3 and 2% are Q = a &s _ Qabhk + 4a/c 3 + 46A* - 3/iVc 3 , V = - Z 2 , = - I (ab - We), .K = - 8l 3 ab, hence, writing j = 0, /= 0, and writing also c=l, we may consider the sexfcic [(o-j A, /c, o]j_A', y) 3 ] 3 + 32i a a; 3 7/ 3 , that is ( 2 , ah, i(Za/c + Sh*') ] ^ (ab + 9M- + 3 Ql s ), ^(26A+3A?), i/c, 6Jw, y) fl , the invariants whereof are found to be functions of the last mentioned values of D, V, , _K; to pass to the given sextic (*#, y) , put equal to c [(ft, h, k) oQco, 2/) 3 ] 3 + 4 JY j, have only to consider D, V, 0, R as having their before-mentioned general values, pft the coefficient c by the principle of homogeneity. to Salmon's Lessons Introductory to the Modem Higher Algebra (Second Edition 1866). 1885, the values are given, pp. 260 2G6. ' 400] GIVEN PENCIL OF SIX LINES. 109 As regards the discriminant Q ln , this as already remarked, has been calculated for tho general form, but for the present purpose it is easier, by dealing directly with the form [(a, k, k, 6Ja?, 2/) B ] a + 32J B afy s , and then interpreting D, V, 0, R and restoring the coefficient o as above, to obtain the discriminant Q,o of the function [c (a, h, k, 6$X 2/) 3 ] a + 4[(j, /,/][&', i/) 3 ] 3 in the required form, as a function of c a D, V, c, cR. I find after some laborious calculations O 9 = 10 No. 31 = c 3 ( + ) 4 = 10000 No. 34= c j {- Q s = 1000000 No. 35 = c" { + 9 a 40 jR 288 V 256 V 3 99 D 3 400 RU 2304 V0D 8640 3 12800 .ft V 82944 V 2 2 -1- 4608 V 3 D 20480 + 147456 65536 V" 7992 D 3 72000 RQ* 145152 V@^D 622080 3 D 160000 ^D 691200 -RV0D 3456000 R 3 3815424 V a @ a D 36080640 V 4 635904 VD a C+ 33177600 .RV 2 3 + 4669440 ,BV B D + 217645056 V 3 3 + 23003136 V J D + + 110 ON THE CUBIC CURVES INSCRIBED IN A + (? (+ 8192000 J^V 3 11010048012V 4 509607936 V 2 1^+ 14155776 V e D + G J 62914560 12V 452984832 V 7 134217728 V [400 = multiple of discriminant . 4- 64 48 to which may be joined QJ= c 1 [ 81 D a 720 RU\ 5184 VD 1GOO R 3 -f 23040 R V + 82944 V 2 3 ^+ 4G08 V ft D + c J 20480 1V 3 \+H745G V J @ + 65536 D and : C" + C B ''- 180 1120 + 2880 VD - 8640 @ B o a f 1600 E a 1+ 10240 RV 270 D 3 720 51840 77760 400] GIVEN PENCIL OF SIX LINES. in r - 23200 ]&*n - 5356SO-R0VD 4- 432000 R& - 1762560 V-6 2 D 4- 4510080 V 1 + 17280 f - 64000 R 3 - 1382400 _R 2 V0 - 5806080 .RV 2 3 + 30720 JRV'Q + 3317760 V 3 3 - 1105920 V^D - 204800 JJ 2 V 3 -39321fiOEV*. The foregoing values of S and T give + c 3 + 24. VfciD -64.0" 16 J$ + 96 -RV - 48 V 2 4 so that Qio. cltt 3 ' and therefore [oR (V which are interesting in the theory. 112 ON THE CUBIC CUEVES INSCRIBED IN A [400 Wehave and if by means of these values we eliminate o and c a Q, we obtain Q 2 , Q 4 , & and Q, as functions of S, T, V and aft Choosing instead of Q 4 and Q, the combinations Q<?-Q t and Q a ~8Q/, and forming also the expression for the combination Qa(Qt-Qd, we have tints the system of formula? = 9 2 T - 432 S - 72T _ 72 + 86* + 144 + 128 VScR, + 272* - 4212 PViSf + 6588 T a V 3 + 252 T'cB - 7776 TO 3 - 16848 TVS + 3456 2' - 2592 - 1296 - 1824 - 448 + 544320 -461376 + 74304 + 15552 - 10368 - 10728 V 7 S - 3264 - 1536 + 81 972 612 108 - 3888 - 5184 - 11052 - 2016 288 2'V" 352 yV a 64 - 77760 V + 153792 V - 1728 + 29376 + 26996 + 576 + 1088 + 512 4:00] GIVEN PENCIL OF SIX LINES. 113 and, as mentioned above, Q 10 = c 3 -R 3 (- 2* The juat-mentiouod value of Q 10 should, I think, admit of being established a priori, and if this be HO, then the substitution of the values of 8 and T in terms of c 3 D, V, c, c.R, would be the easiest way of arriving at the before-mentioned expression of Q 10 in terms of these same quantities. The calculation by which this expression was airived at., is however not without interest, and it will be as well to indicate the mode in which it was effected. Calculation of Q w , Wo liavo to find the discriminant of o[(a, k, k t 6]J>, yf]'+Wl*atof. Consider for a moment the more general form P 2 -f 4<Q 3 , then to find the discriminant, wo have to eliminate between the equations -. dy dy these aro sabiBfiod by the system P0, (2 2 = 0, and it follows that if R be the resultant of tlio equations JP = 0, Q 0, then the discriminant in question contains the factor Ji a . For tho other factor we may reduce the system to pdP ( dt}^ _ da) dy dy Now writing Q - 2?a;y, these equations become dx , ^_ L X dot ^ dy tho resultant of which is = I 3 into resultant of the system dco dP dP _ 'm~" <s dy ' 15 0. VI, 114 ON THJ3 OUJUC OU11VJW INK01UKUI) IN A [400 but in virtue of bho second (Munition, wo Inivo . , / dP dP\ ,. (IP *-*(*<& + * dyrfo'dy' which reduces tho flrnt u(|iiiH.i<m lit) a/y'-'/VM^wy^o, '' (A'W (/!/ or omitting tho factor Sy, to v' ( "'.|-72JW <>, w / rf/ J Honco, writing P=V (o) . (, A, /:, /'$. ,'/)", and fc]wlbro ^-H V (o) . (, A, ^'H ,'y)' J , ditto y-l, tlin two (iqiiatioiiH boconio (a, A, %, ])w- tho second of whicli IH morn hiiuply wi-ibtiju Honco, i-oHtoring tha factor #', and iitwi to iivditl fractimm iuUwhioing tho i'uclov Hit*, tlio rcHultunt of Iho two oquatioiiH in 8i(tUI [HZW -l-o(((, h, Q>, !)(/*, A', ftj*. 1)}. where II donotoH tho producit of tho iucitfun comtHinuiding to Iho tliroo routH ,'c,, ?,,, - a of tlio oqiintion (, A, -ft, -tjfli, l^wO, or what is th< Raino tiling, fc fl H- Arc 1J ~Awj 6 so that tho aymmotric functions iiru In bo fontid from Tho required disovimiimnt i the foregoing , wsnultunb inultipliod by ft, or wiy by c 3 Ji a , that is tho discriminant Q lo i if for shortnoHS wo write ft(a, A, A5, I) 1 , (a, *, 400] GIVEN PENCIL OF SIX LINES. H5 and when the symmetric functions have been expressed in terms of the coefficients, the result is to be expressed as a function of Q, V, , R by means of the values c a D = ft a &- + 3 + 4&fc" - Gadhk - V = - Z 3 , c @ = - I (ab hk), cR = - 8l 3 ab. Thus, for instance, the first term of the result is = c a # 3 . 8#a which ia which is a term in the before-mentioned expression for Q w 401. A NOTATION OF THE POINTS AND LINES IN PASCAL'S THEOREM. [From the Quarterly Journal of Pure and Applied Mathematics, vol. ix. (1868), pp. 268 TAKING six points 1, 2, 3, 4, 5, 6 on a conic ; let A, B, 0, D, E, F, G, H, I, J, K, L> M, N, denote each a combination of three lines, thus 12. 3*. 56 = ^ 13 . 45 . 62 = J3 14. 66. 23 = 15.62.34 = _D 16. 23. 45 = # 13 . 46 . 25 = Q 14 . 52 . 36 =H 15 . 03 . 42 = I 16. 24.53 = J 12.3G.45 = tf 13 . 42 . 56 = L 14.53.62 = ,/Y 16.64.23 = ff 16.25.34 = then any hexagon formed with the six points may be represented by a combination of some two of the letters A, B, &c., viz, the three alternate sides are the lines repre- sented by one letter, and the other three alternate sides the lines represented by the other letter: for example, the hexagon 123456 is AE\ and so for the other hexagons. Any duad AE thus representing a hexagon may be termed a hexagonal duad ; the number of such duacls is sixty. Each Pascalian line may be denoted by the symbol of the hexagon to which it belongs; thus, the line which belongs to the hexagon is the line AE, I form the following combinations : IMO.DHJ each involving all the duacls 12, &c. except those of 123 . 456, DE&.BNO ELM, BO J HLff.CGI EH.JKN AEH.OKO AMN.ODF AQJ.ELO ABI.DKL Q'KM.BFH 124 . 356, 125 . 346, 126 . 345, 134 . 256, 135 . 246, 136 . 245, 145 . 236, 146 . 235, 156 . 234, 401] A NOTATION OF THE POINTS AND LINES IN PASCAL'S THEOREM. 117 and also the combinations : AEQMI involving all the duads 12,13, &c,, ASHJN BOFIO ODGJK DEFRL KLMNO which I call respectively the ten-partite and six-partite arrangements. It is to be remarked that (considering IMO . DHJ as standing for the six duads IM, 10, MO JJIf, DJ, HJ> and so for the others) the ten-partite arrangement contains all the tiixfcy hexagonal duads : and in like manner, (considering AEGMI as standing for tho ten duada AS, AQ, AM, AT, EG, EM, El, GM, GI, MI, and so for the others) tho aix-parliite arrangement contains all the sixty hexagonal duads. Tho GO Pascalian lines intersect by 4's in the 45 Pascalian points p, by 3's in 20 points g and in 60 points A, and by 2's in 90 points m, 360 points r, 360 points t t 360 points 2, and 9 points w. The intersections of the Pascalian lines thus are i<5 p counting as 270 20 fir 60 60 A 180 90m 90 360 r 360 360 1 360 360 z 360 90 w 90 1770 = ^60.59, and the intersections on each Pascalian line are Bp counting as 9 1 g a 2 3 A . 6 3m & 12 r 12 12 1 12 12* ,. 12 3w ^_ 59. 118 A NOTATION OF THE POINTS AND LINES IN PASCAL'S THEOREM. [401 For the ten-partite arrangement, any double triad such as ABI .DKL gives 15 intersections; 10x15 = 150; and any pair of double triads such as ABI .DKL and AEH.GKO 'gives 36 intersections; 45x36 = 1620; and these are (Off 60o 10 x ' J J 45 x 9m 90m 150 4 A 270 p 180 A 8r 300 r 8* 360 j 82 3602 .2* 90 w 1620 rfo 1770. For the six-partite arrangement any pentad such as ASHJN gives 45 intersections; G x 45 = 270 ; and any two pentads such as A.BEJN and AEQMI give 100 inter- sections ; Ifi x 100 = 1500 ; and these are 6x 15 30 A 180 h 15m A.K. 90m 270 ' Irtf 40 18 p 24 r 60 g 270 300 r 24 1 360 1 24,0 3600 w 10f> 90 w 1500 J.UV j_(_/u \j 1770. I analyse the intersections of a Pascalian line, say AE, by the remaining 59 Pascalian lines as follows: Observe that ^i? b&longs to the triad AEH, the complementary triad whereof is GKO\ it also belongs to the pentad AEIMG. We thus obtain, corresponding to AE, the arrangement ESS HA HE B N J F L $ IMG KG 401] A NOTATION OF THE POINTS AND LINES IN PASCAL^ THEOREM. 119 viz. HAENJ, is the pentad which contains HA, the arrangement of the last three letters B Y N, J thereof beiug arbitrary ; HEFLL is the pentad that contains HE, but lihe lust three letters are so arranged that the columns HBF, HNL, HJD are each of bhem a triad, IMG is then the residue of the pentad AEIMG, and KOO is the complementary triad to AEH t but the arrangement of the letters IMG, and of the letters KOO, are each of them determinate; viz. these are such that we have BFICO, NLMKO, JDGGK, each of them a pentad. And this being so we derive from the arrangement 20 All, EH\ 3m KG, KO, CO; Qh AI, AM, AG; El, BM t EG; 12* IS, IF, MN, ML, GJ, GD ; HB t HP, HN, EL, HJ, HD; Qp A'B, AN, AJ; EF, EL, ED; BF, NL, JJ); 12 r GB, GF, GJ, G3)\ OB, OF. ON, OL; K$, KL, KJ> D- t Ut FL, FJ), LD; BN } BJ, NJ; IG t 10; MX, MO; QK, GO; 3w IM, IG, MQ\ 59" via, bho line AE in question meets AH, EH each of them in a point g\ KO, KO, GO each in a point m; and so on. By constructing in the same way an arrangement for each of the lines AS, &c., we find the nature of the point of intersection of any two of tho lines AS t AE, AH, &o. ; and we may then present the results in a table (soo Plato), which shows at a glance what is the point of intersection (whether a point 0, m, li, z, p, r, t, or w) of any two of the Pascalian lines. I further remark that representing the 45 Pascalian points as follows : 12 . 84 = a 13.24=0 14.23=?^ 15.23 = s 16.23 = 7/ 12.35 = 6 13.25 = A 14 . 26 - n 15.24 = * 16. 24 = # 12.36 = c 18 . 26 = i 14 . 26 = o 15 . 26 = u 16. 25 = a 12 . 45 = d 13 . 45 = j 14.35=2 15.34 = y f fl QA ~ fi 12 . 46 = e 13. 46 = ft 14.36 = 2 15.3P 12.56=/ 13.56 = 2 14 . 56 = r 15 '- 23 . 45 = e 25 . 34 = X 34 . 56 = p 23 . 46 = C 25.36 = /i 35.46 = o- 23.56 = 7? 25 . 46 = v 36.45 = r 24 . 35 = 6 26 . 34 = 24 . 36 = i 26 . 35 to 24 . 56 = K 26 . 45 = TT 120 A NOTATION OT? THE POINTS AND LINES IN PASCAI/S THEOREM. the sixty hexagons and their Pascalian lines then are AE 123456 12.45 23.56 34.61 AH 125634 12. 63 25 .34 56. 41 cXr EH 145236 14. 23 45 .36 52. 61 mra GK 123654 12. 65 23 .54 36. 41 feq GO 143256 14. 25 43 .56 32. 61 npy KO 125436 12. 43 25 .36 54. 61 ft/iS AM 126534 12. 53 26 .34 65. 41 bffr AG 125643 12. 64 25 .43 56. 31 G\l AI 124365 12. 36 24 .65 43. 51 CKV EG 132546 13. 54 32 .46 25. 61 tt DF 126435 12. 43 26 ,35 64. 51 a^ FL 124653 12. 65 24 .53 46. 31 f&k DL 134265 13. 26 34 .65 42. 51 ipt BN 132645 13. 64 32 .45 26. 51 keu BJ 135426 13. 42 35 .26 54. 61 ff"* jar 153246 15. 24 53 .46 32. 61 fay GK 125463 12. 46 25 ,63 54. 31 <w KM 126354 12. 35 26 .54 63. 41 faq 10 152436 15. 43 52 .36 24. 61 Vfti MO 143526 14. 52 43 .26 35. 61 fr EM 145326 14. 32 45 .26 53. 61 mTTf) El 154236 15, ,23 54 .36 42. 61 STZ AN 123465 12, ,46 23 .65 34, ,51 eyv AJ 124356 12, 35 24 .56 43. 61 fa/3 AB 126543 12. 54 26 .43 65. 31 d& DE 154326 15, 32 54 .26 43. 61 S7T/3 EL 132456 13, 45 32 .56 24. 61 fa* EF 123546 12. 54 23 .46 35. 61 dfr CD 143265 14. 26 43 .65 32. 51 ops GF 123564 12. .56 23 .64 35. 41 .ftp 401] A NOTATION OF THE POINTS AND LINES IN PASCAL'S THEOREM. 12 L GG 132564 13. 56 32. 64 25 .41 l& GI 142365 14. 36 42. 65 23 .51 qxs Mtf 146235 14. 23 46. 35 62 .51 mtru GJ 135246 13. 24 35. 46 52 .61 ffa-a SI 136245 13. 24 36. 45 62 .51 ffru DG 134625 13. 62 34. 25 46 .51 flue LM 135624 13. 62 3o. 24 56 .41 Mr FI 124635 12. 63 24. 35 46 .51 0&03 EH 136254 13. 25 36. 54 62 .41 hro m 125364 12. 36 25. 64 53 .41 GVp FO 125346 12. 34 25. 46 53 .61 avy LO 134256 13. 26 34. 56 42 .61 hpz DK 126345 12. 34 26. 45 63 .51 aw KL 124563 12. 60 24. 63 45 .31 fv SO 134526 13. 52 34. 26 45 .61 AfS NO 152346 15. 34 52. 46 23 .61 wy KG 132654 13. 65 32. 54 26 .41 leo OJ 142356 14. 35 42. 66 23 .61 P*y JK 124536 12, 53 24, 36 45 .61 1*8 KN 123645 12. 64 23. 45 36 .51 eeiv DH 143625 14. 62 43. 25 36 .51 00, HJ 142536 14. 63 42. 36 25 .61 pta HL 136524 13. 52 36. 24 65 .41 hir EN 146325 14. 32 46, 25 63 .51 BF 126453 12. 46 26. 53 64 .31 DJ 153426 15. 42 53. 26 34 .61 LN 132465 13. 46 32. 65 24 .51 GM 135264 13. 26 35. 64 52 .41 IM 142635 14. 63 42. 35 26 .51 GI 136425 13. 42 36. 25 64 .51 0. VI. 122 A NOTATION OF THE POINTS AND LINES IN PASCAL'S THEOREM. [401 Each Pascalian point belongs fco four different hexagons; viz. a to the hexagons KD, KQ, FD> FO; and so for the other points, thus: , 0) * (D 6 (A, K)(M, J) V (0, N)(J, 0) G (A, F}(H, 7) * (#, 0)(7, L) d (A > ff}(B,M!) (E t J)(G, 11} e (A, J0(ff,'tf) /3 (-4 / (a, L)(K, F) 7 ( ff (B, G)(J,V) S (5 h (B,L)(H, 0) e (S,K)(0,N) i (J),M)(G t l) r (a.^XJ 1 , G) j (E,1C)(G,L) TJ ( k (B t L)(F,N) e ( I (A, G)(S, G) t (H, K)(J,L} m (E, N)(H, -M) K (A, G}(I, J) n (0, M)(G, 0) \ (A, D)(G t H) o (B,D)(0,ff) p (G, 0)(/,^) p (Q,H)(F,.J) v (F,N)(H } 0} ff (Q t M)(I t K) f (A, 0)(B t M) r (A,L)(H t M) (B t ty(F, J) s (V,E)(D,I) TT (D, M)(E,K) t (J,L)(D,N) P (0,L)(D t 0) u (B, M)(I, N) a- (G, N)(J, M) v (4,0 )(#,/) r (B,ty(B,I) w (D,Jf)(ff, /O I have constructed on a very large scale a figure of the sixty Pascalian lines, and the forty-five Pascalian points, marking them according to the foregoing notation; but the figure is from its complexity, and the inconvenient way in which the points are either crowded together or fly off to a great distance, almost unintelligible. AB Al AO AH Al AJ JIM AN BC BF BH Bi BJ Bti BO CD Cf CC Cl CJ CK CO DE Of OC OH DJ BK DL EF A A A A A A A AN B B B B B B B C c, G C C c c D D DJ OK DL Ef EC N / EL EM Fl FH n fO Gl t> P h ff & P h P P ft ff 7l h p r t P X.%, z x- t t r T r t t 7 t % x, t r t T p z x. h z w w t r r r r t X ff Z X Z T i Z p p if z h it- t t r z p 2 t z t z p m m t t 7 if z w 7i % m m r r t p h ff p 7ip 7t,p h y r t r T r t t mm j m-Tii p t z x,p r t p t T z r p p % t r w m % t p p z r t x r t r l r r mm z t r p r z z p t r 7 T T t m t r r ? r f r r r m r t t r r t p li r z t p z I r t t w t p T % p r t 2 p % & h j? b p r p p h phffhppj hyp p_h p ft t ~*- ff J\ h p ph p x % 7i p y 7i p p h z t p h h /j It ji fft'f h p p h p h ff {, t 1> r p t z x r t / p p r i z t r f h y k p P h z h t irx. li z t h z w I w p ff y z x z g z % % hJi h t x. w h t H- ff p h t t \y z ir z hp p prrtztp&hl r r t r T 1 nnnp x t 7' r ? t- r m m h- / j> t $ r z x t r p t z, t z r p p t r A 1 z r Z t pp r t ff z. P h it ji -p x, r t x. r t t z r P % r t x, t w r t r r 7- m in x, z Z7- p t z p i f r i p z m r m t {, r r / x h ?ti t m r T r r t h P P fj 7i p n h p z P Ji x g z z x. z z p h p x x h. t w w L p P /i p z z r t Irx, h p t z w z h w t i p p i> h t n- % 7i x 7ip h p z z x- % g fyjj li. ff?> f r t x, p h p g t \Y t H- %. p t z r t ffhpft z if z t w g p 7i P T z- x> T f li ji a h t Iv % t trp 7b ff p % z ff x. x. 'h p h p T 7/1 t. rn r t r r p t t m, r m t r r ? h w rpzztptrrt T t r t r T m m z f r z p p t z t r t r 6 ZPP r z r f ji x 1 p z z j' p r t x p m r f 7- mi T r t r m, t r t m r r r r t P P t z z 7* t r t p r t x p %. h _p Ji p // ^ it- z z h j} h p h.p t r r mm p z z p t Z t p r t g ff z z x, T r t m. m h. z h x w p r z t r p z t r z z. r p t r h z w t, h r m, t r r h z w t, w t, m r r r p Z t r f, z z r r i z jy p z r ff ( z w t w h x, r y r z t r I z P t t r h y hp p h J Z z h p h j} p h ft wt ffp p h hp x T h hp ff p h w w t p Tvh ff ~\7i.p t h- zp p kp h y t Xp Ji, 7i jo h p ff tr r t fv h z, w t t ff t /i fi z if x if t r r zgzffZK&xrz %1l W X: 7l t \Y t t %, r p tzt T zpmt t Kw zwt h x,7n,r z z tpp r t r wTi, z. z T pp t r t z ff r tmt T m r T t p t T m T t m. r r n- h X-wh fhzwtx,t r t ?'7tim x. p -p %, t p p t t r t't T r ?' pp r r t p z Zp 7- f r 7n ? r t f: r r m r t 7m- r r r t tm z h t t-w% h x, lit y z &x, zzp r tp z t 7' p zzp r t r x-wt zhzh t wt 'PP r t z % ff ff x. f r ff z, ff x>x, x x># m mh tr %,'kx, t w h r tp x,x,t r r t ff r7*7i;wx,-wt x, fa Ji, r r p t %, t r p x. p t wxp p t r r t ff ' t wfat 7i,xis wfc p Kwli, t w t X, bli, z-frtp r t r p z,j? X'X,-X,71T} T t & 7- ; CK GU HJ Hi HN IM , 10 JH JN % Zffzz & z %. zr tp r t zxp tm t wht w z?i %.7'm wxh t 7i,t- wzz tzrprtpx-rx, '%-tJitwxbtz. % ww t ht h zt t x- t trp r zp ww r z t r pp x,x, r t %wt hzh-zt t pp t r w t & %,?i m r t r r wt h z w n-t r r r t r p z t it h-w t zp t r p P r t zp t Immr i- t wh z r rmmt rp r_p x, r t- %p r tTi,tz,Ji,tw%,Jtst trpzpwtwth Y t hhx. t r t r p vt x, t kz, kw t r 7i x, z t h t wft, % T z_P x, r p r t z# t t r % x, z ff ff z x, t "rwwtK&xhw ' r t t r p %,p %, t P t r tmt r r x, '. h r r m m r r x, *-%- t t iy X< hh Km, '.p 7' T t Z p Z 7} 7ft. 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T % t -x. T p r B E G H At AJ AM N BO BF BH Bt BJ BfJ BO ffj GK CO Of DF DO 0H DJ OK p x, T r -X- 7* T* t Tn Is T 7' m x, %, t p r r r tm-r X, 7* p t X r' P X. t t ff p p %, x, t x-P t t ff % X.P p t r r in t p T % r % t Imr r t %. % r r p p t t X p p ff t X X' x. t t, p x x. ff t P P r X T p t /W- 7' t & T t x. % t r r 7n,p T r P ff P P % & t x, % t x- y x. %p t t 7' r x, 7 r P t % t p ff p %. t- P t x- ff t x p t x- t p % t p p t x. i, pjy t t m t x t x- t p ff %. ff p t, h, ft, fa tih ft' w w fa fo h w w fa h w h w k w A w h 7i 7i A Ji- li ff h 7i 7i w hw h -ir~Ji h ww?i< Jt w h h~w h wh EF EG EH Et EL Fl FH 'A FO 8f BJ OK *flf HJ HL HN /M to JK KL KM KN KG LM LN LO NQ H 1 ELEMFI FH FL FO f I 8J GN OM NJ HL Hff IM iO MJNKL KMKN KO LI* LN lOMfJ MO NO 402. ON A SINGULARITY OF SURFACES. [From the Quarterly Journal of Pure and Applied Mathematics, vol. ix. (1868), pp. 332338.] A SURFACE having a nodal line has in general on this nodal line points where the two tangent planes coincide, or as I propose to term them "pinch-points." Thus, if the nodal line be the curve of complete' intersection of any two surfaces P = 0, <3 = 0, then the equation of the general surface having this curve for a nodal line is (a, 6, eJP, Q) 2 = (where a, b, c are any functions of the coordinates), and the pinch- points are given as the intersections of the nodal line P = 0, Q = with the surface = 0. Consider the case where the nodal curve is a curve of partial intersection represented by the equations P, Q> = 0, or say by the equations > = 0, P', Q', r = (viz. p, q, r denote the functions QR'-Q'R, RP'-R'P, Ptf-P'Q respectively), and consequently we have identically (P, Q>&l!t.p, ff. )-<>, (P 1 , Q', -fi'&j, * O-o, or what is the same thing, (\ p) being arbitrary, (XP + ^P', *Q + t*Q', \R + ?&$$, V, r)-. The general surface having the curve in question for its nodal line is represented by the equation ^, ' (a, 6, c,f, g t h$p t q, '-) a = G, (where (a, 6, o, /, g, Ji) are any functions of the coordinates), and it is easy to see that the condition for a pinch-point is the same as that which (considering p, q t r as coordinates and all the other quantities as constants), 'expresses that the line > ') = - 1 fi O J. U ~~""tj 124 ON A SINGULARITY OF SURFACES. [402 touches the conic (a, b } c,f, ff, /t)(p, q, r) s = 0, viz. A, B, G f F, G, H being the inverse coefficients, 4 = &c -/ 2 , &c., this condition is (A, 11, 0, F, G, or what is the same thing, the pinch-points are given as the common intersections of the nodal line p~Q, 5 = 0, r=0 with each of the three surfaces (A t S t C t F, &, J/)(P, Q,tt)* = 0, (A, B, C, F, G, S)(P t Q, R) (F, Q', R 1 } =0, (A, J9, 0,F, G, H) (P', Q', JS') a = 0, these last three equations in fact, adding only a single relation to the relations expressed by the equations 2> = Q, ,7=0, r-0. If the functions P, Q, R, P', Q', R' are linear functions of the coordinates, then the curve (p 0, 2 = 0, r = 0) is a cubic curve in space, or skew cubic ; and if moreover (a, b, c, /, g, h] are constants, then the equation (a, b } c, /, g, /i}J>, q, r) a = 0, belongs to a quartic surface having the skew cubic for a nodal line : this surface is (it may be observed) a ruled surface, or scroll. With a view to ulterior investigations, I propose to study the theory of the pinch -points in regard to this particular surface ; and to simplify as much as possible, I fix the coordinates as follows: Considering the skew cubic as given, let any point on the cubic be taken for the origin; let #=0 be the equation, of the osculating plane at 0; y = that of any other plane through the tangent line at 0; s = t that of. any other plane through 0, not passing through the tangent line ; and w = that of a fourth plane ; then the equation of the cubic will be y, z ==0, or what is the same thing, the values of p, q, T are yw Z*, zy new, and &g T/' J respectively. And conversely, the cubic being thus represented, the point (# = 0, i/ = 0, 2 = 0) may be considered as standing for any point whatever on the skew cubic; the osculating plane at this point being 'as = 0, and the tangent line being <w = 0, y = 0. For the purpose of the present investigations, we may without loss of generality write w~l; and for convenience I shall do this; the values of p, q, r thus become y 2?, yz ES, fez i/ 3 , and the equation of the surface is (ct, ft, c,/ 402] ON A SINGULARITY OF SURFACES. 125 At a pinch-point, we have (A, B t G,F t G, 3) fay, if = 0, (A t B, G,F, G,S)fay t *) (y, z, 1) =0, (A, B, O.F, G,H) &*, 1) 9 0, and hence the origin will be a pinch-point if = 0, that is, if ab-ltf Q. This however appears more readily by remarking, that the equation of the pair of tangent planes at the origin is (a, 1), e t f, (J, K$y t -as, 0) a = 0, or what is the same tiling, (a, h, &fe, -#) a = 0; the two tangent planes therefore coincide, or there is a pinch-point, if only 6-/i 3 = 0. By what precedes, it appears that if we wish to study the form of the quartio surface, 1, in the neighbourhood of an arbitrary point on the nodal line; 2, in the neighbourhood of a pinch-point; it is sufficient in the first case to consider the general Kiir face (, 1>> o,f t g, in the neighbourhood of the origin; and in the second case, to study the special surface for which ab - h* = 0, or writing for convenience a = 1, and therefore 6 = 1$, the surface (1, M c,f, g> in the neighbourhood of the origin. Considor first the surface (a, 6, c,f, ff, A plane through the origin is either a plane not passing through the tangent line (a,0 = 0), and the equation ^=0 will serve to represent any such plane; or if it pnaa through the tangent line, then it is either a non-special plane, which may _be represented by the equation y-0; or it is a special plane: viz. either the osculating plane > = of the nodal line, or else one or the other of the two tangent planes ( A IVy -aO a = of fche surface - I G0n8ider thei ' eft)re the secfclolis of the surlace by these planes * - 0, y = 0, w-0, (a, k, bty, -^) 2 = respectively. Section by the non-special plane 2 = 0, The equation is (a, i>,o,f,gJ>&y>-~>-fy^> which represents a curve having at the origin an ordinary node the equations of the two tangents being (a, A, Vfy. -tf-0, vk these are the mte^ctrons of the two tangent planes by the plane 2 = 0. 126 ON A SINGULARITY OF SURFACES. [402 Section by the non-special plane through the tangent line, viz, the plane y = 0. The equation is (a, &. G,f, g, K$r& t -a?, fl#) a =0, or what is the same thing, ba? - ZfaPs + Shies* + cV - 2i/#3 3 + ass 1 = 0. Writing as usual a* = ,4 z* -J- &c, we have and since ub fi 2 is by hypothesis not =0, ^i has two unequal values; we have at the origin two branches as -djS 3 -f- B^ + &c., ,'C = .A 2 s 9 + .5 2 2 3 + &c., having the common tangent <c=0 (viz, this is the tangent a;~0, y~ of the nodal curve), and with a two-pointic intersection of the two branches, that is, the point at the origin is an ordinary tacnode. Section "by the osculating plane so = 0. The equation is (, &, o, f, g, K$$ - z\ yg t - yj = 0. Wo may write y = z* + Az> 1 + Ssa,, we at once find ^ = 3, and then (a-, &, c f /, ;x, that ia (a, h t A has two unequal values, and the "branches through the origin are viz. the branches liave the common tangent line ?/ = (the tangent = 0, 2/ = of the nodal curve), but in the present case a three-pointie intersection. Section by one of the tangent planes (a, h, b^y, a;) 3 0. Writing y^-tno), and therefore (a, h, &$m, -1) 9 = 0, the equation is (a, &, c, /, (/j h^mx-z' 1 , -ts-mztc, aw mV) f = 0, which represents of course the projection of the section on the plane z = 0, at - 0, but which (since there ia no alteration in the singularities) may be considered as representing the section itself. Developing, the coefficient of a? is am 3 + 2Am + b, which is e= 0, and the equation becomes a + cm 4 + 2 [Am" + (6 - ff) m -/) a^ -I- ' 2m 9 (/w - c) _ + 2{om + ft) CM? + (6 + 2^) m 3 - 2/m + + 2 (1m - g) ax 8 + a ,3 4 = 0, so that the curve has at the origin a triple point, the tangent to one branch being the line = (the tangent a? == 0, y = of the nodal curve), 402] ON A SINGULARITY OF SURFACES. 127 Consider next the surface (1, A 9 , c,f, ff, A&-*, $*-*> aw-J^-0, being as already remarked, the general surface referred to a pinch-point as origin. Section by the non-special plane z 0. Tho equation is (l./^c./tf./^,-*,-- 2/7-0, whore, attending only to the terms of the ^lowest order, we find (1, h, A'Jff, -.) a = 0, that is (y - /(A') a = > showing that the origin is a cusp. Section by the non-special plane through the tangent line, viz. tho plane Tho equation is (1, h\ Otf.g.Ki-** -, awy-O, or what is the same thing, = 0) that i , . (ha + 2 a ) a - 2> 2 s + ca;V - S^a'S 3 = 0, writing to *. + ^. find at onoo , = i and then ,1' = |-|, *at tho branch* a,e to-^i^i whence we have at the origin a c U3 p of the second ordov or node cusp. Section by the osculating plane #=0. The equation is . n . (I, h* t o t f,ff,K$#-*. **>-& = *> siting y-^-W + ^ ^ easily find ?-*, and then * A ** diaannear of themselves, the terms in * give whore the terms in *, and ^ +4 diaappeai jl" + 2jy/i = 0, and the branches are via. there is a cusp of a superior order. Section by the tangent plane y~ha. Tho equation is 12 g ON A SINGULARITY OF SURFACES. [402 representing the projection on the plane of m. Developing, the equation is 2 (/- gh) a? (Me -z) + 2* 1 =0, and there is at the origin a triple point (= cusp + 2 nodes) arising from the passage of an ordinary branch through a cusp; the tangent at the cusp being it will be noticed the line = 0, that is the tangent = 0, y=Q to the nodal curve at the pinch-point. The results of the investigation may be presented in a tabular form as follows : Nature of Section. Plane of Section. Non-special . Ditto, through tangent line of nodal curve. Osculating plane of nodnl curve. Either of the two tangent planes. The single tangent plane. Origin, an ordinary point. Node, Tacnode = 2 nodes, Triple point, one branch touch- ing the tangent of nodal line. Origin, a Pinch-point. Ousp. Node-cusp, = node + cusp. Triple point, = cusp + 2 nodes ; the cuspidal branch touching the tangent of the nodal line. I have not considered the special cases where one of the two tangent planes, or (as the case may be) the single tangent plane of the surface coincides with the osculating piano of the nodal curve. 403. ON PASCAL'S THEOREM. [From the Quarterly Journal of Pure and Applied Mathematics, vol. ix. (1868), pp, 348353.] I CONSIDER the following question : to find a point such that its polar plane in regard to a given system of three planes is the same as its polar plane iu regard to another given system of three planes. The equations of any six planes whatever may be taken to be A r 0, Y=0, J?=0, U^O, F=0, F=0, where X+ Y+ Z+ U+ V+ 17=0, and so also any quantities X, Y } Z, U t V, W satisfying these relations may be regarded as the coordinates of a point in space ; we pass to the ordinary system of qnadriplauar coordinates by merely substituting for 7, W their values as linear functions of X, Y, Z } U. This being so, the equations of the given, systems of throe planes may be taken to be and if we take for the coordinates of the required point (, y t z, u, v, to), where 0+ y+ + + + w = 0j aw 4- ly + GZ +j% -f gv + hw = 0, then the equations of the two polar planes are , , a? y z u v w o. vi. 17 130 ON PASCAL'S THBOHEM. [403 respectively, and we have to find (a, y, ,, u> v> w ) t 8uch thftt these fcwo e tions raay represent the same plane, or that the two equations may in virtue of the linear relations between (X, F, Z, U> V, W) be the same equation. The ordinary process by indeterminate multipliers gives 1 - + X + (j,c = 0, *nd we have the before-mentioned linear relations between fo y . ,, u> v> w} . these last are satisfied by the values if only in fact, ^ satisfying this equation, the relation + # + # + w- is obviously satisfied; and observing that we have . ft- a-tf* " > f-6 we have J +fti-{-ffv + hw BO that the relation + ff , + *.0 is also .atisBed. Substituting the the six equatiom oontahiing 7c - x - * wm be ai1 403] ON PASCAL'S THEOBEM. 131 The coordinates of the required point thus are 1 1 1 c-0' /-(?' <j-6> li-Q where - 4- i n T a- 6 ' 6 - c - f-6 g-6 h- and, the equation in being of the fourth order, there are thus four points, say the- points Oi, S , O s , 0,,, which have each of them the property in question, It will be convenient to designate the planes X =Q, Y~Q, j?=0, U=Q, V~=Q, W = Q as the planes a, b, c, /, g, h respectively; the line of intersection of the planes X = 0, F=0 will then he the line ab, and the point of intersection of the planes X = 0, Y~0, Z=Q the point abc\ and so in other cases. I say that from any one of the points it is possible to draw a line meeting the lines af.bg. ck (1), ag . bh . of (2), ah , bf . G(j ' (3), af.bh.cg (4), aff.bf.ch (o), ak.bg .of (6), and consequently, that the finir points are the four common points of the six hyperboloicls passing through these triads of lines respectively. In fact, considering d as determined by the foregoing quartic equation, and writing for shortness (c-6)Z = 0, (h-6)W=H, so that the equations A+J?~Q, B+G = Q, (7 + JJ=0, are equivalent to two eqiiations only, and it is at once seen, that these are in fact the equations of a line through the point meeting the throe lines af, bg, oh. respectively, 1 1 The equation A + F 0, is in fact satisfied by the values X : U= - ^'; j 7., (t O J C7 and by X Q, U=0; it is consequently the equation of the plane through and line (if' ' " _i_ /3 (I Jo tl-io omicifinn fiF +.ITO rilnnw f.lirniin'Vi O n.TiH f.lid 132 ON PASCAL'S THEOREM. [403 bg; and 0+B"=0 is the equation of the plane through and the line e/t; and the three equations being equivalent to two equations only, the planes have a common line which is the line in question. The equations of the six lines thus are : (1) , (2) , (3) . (4) A+F =0, (6) A + G=Q, It is further to be noticed, that if in any one of these systems, for instance in tho system A + F Q, B + G = 0, G + ff==0, we consider as an arbitrary quantity, then the equations are those of any line whatever cutting the lines of, &</, c/t; and hence eliminating 9, we have the equation of the hyperboloid through the three lines of, fy, ch ; the equations of the six hyperboloids are thus found to be ax +fu _ by + yv _ cz + hw V / _LI W + fl 2 + W ' ass + gv _ by + hw __ 02 +/H /0 v cw + fey % +> W ( . . cfl! +/w _ byj-hw _ cz + gv ^ J "+ir ~ 'w+w ~ ~^^ ,_, ax + ffv by+fy c^ + /wy (o) = = " ' - y +u /f\ wr _ by + gv _ c^+fu w/ ~~ respectively; the equations in the same line being of course equivalent to a single equation, For each one of the six lines we have (A,B t 0)-(-.F, -ff, -H) in some order or other, and it is thus seen that the six lines lie on a cone of the second order, the equation whereof is X 2 + S 3 + ^ - .F 3 - G B - H 3 = 0, 403] ON PASCAL'S THEOREM. 133 Consider now the six planes a, b, c, f, g, h, and taking in the first instance an arbitrary point of projection, and a plane of projection which is also arbitrary the Hue of intersection ab of the planes a and b will be projected into a line ah', and the point of intersection of the planes a, b, c into a point abc; and so in other cases. \Ve have thus a plane figure, consisting of the fifteen lines ab, ac, ...gh, and of the twenty points aba, abf, ...fgh\ and which is such, that on each of the lines there lie four of the points, and through each of the points there pass three of the lines, viz. the points abc, abf, abg, abh lie on the line ab ; and the lines bo, ca t ab meet in the point abo, and so in other cases, If now the point of projection instead of being arbitrary, be one of the above-mentioned four points 0, then the projections of the lines of, by, ch meet in a point, and the like for each of the six triads of lines; that is in the plane figure we have six points 1, 2, 3, 4s, 5, 6. each of them the intersection of three lines as shown in the diagram, l = af.bg,ch, Z-ag.bh. c/, 3 = all . bf , eg, 4 = af . bh . eg, 5 = ay bf . ch, 6 = ah . bcf . of, and these six points lie in a conic, It is clear that the lines af t ag t ah; bf, bg, bk; cf, G(j, ch are the lines 14, 25, 36; 35, 16, 24; 26, 34, 15 respectively. Conversely, starting from the points 1, 2, 3, 4, 5, 6 on a conic, and denoting the lines 14, 25, 36; 35, 16, 24; 26, 34, 15 (being, it may be noticed, the sides and diagonals of the hexagon 162435) in the manner just referred to, then it is possible to complete the figure of the fifteen lines ab, ac,...gh and of the twenty points #fco, abf,...fffh, such that each line contains upon it four points, and that through each point there pass three lines, in the manner already mentioned, Of the fifteen lines, nine, viz. the lines af> ag, ah ; bf, bg, bh ; of, eg, ch are, as has been soon, lines through two of the six points 1, 2, 3, 4, 5, 6 ; the remaining lines are bo, ca, ab ; gh, hf, fg. These are Pascalian lines, bo of the hexagon 162435, ca 152634, ab 142536, gh 162436, hf 142635, bg * 162534, 134 ON PASCAL'S THEOREM. [403 which appears thus, viz. line 6c contains points bcf , Icy , Ich , = bf . cf t if/ . c(/ t l)k .ch, = 35.2G, 16.34, 24.15; s that is, l)c is the Pascalian lino of the hexagon 162435 ; and the like for the rest of the six lines. The twenty points ale, abf, . . .fyh are as follows, viz. omitting the two points abo, fgh, the remaining eighteen points are the ' Pascalian points (the intersections of pairs of lines each through two of the points 1, 2, 3, 4, 5, 6) which lie on the Pascalian lines fie, ca, ab, gh } Jif, fy respectively ; the point ale is the intersection of the Pnscalian lines be, ca, ab, and the point fgh is the intersection of the Pascalian lines f/h, hf, fg, the points in question being two of the points P (Steiner's twenty points, each the intersection of three Pascalian lines). "We thus see that \ve have two triads of hexagons such that the Pascalian HUGH of each triad meet in a point, and that the two points so obtained, together with the eighteen points on the six Pascalian lines, form a system of twenty points lying four together on fifteen lines, and which points and lines are the projections of the points and lines of intersection of six planes ; or, say simply that the figure is the projection of the figure of six planes. It is to be added, that if the planes are a, b, c> f, g, h, then the point of projection is any one of the four points which have the same polar plane in regard to the system of the planes , 6, c, and in regard to the system of the planes / g, h, The consideration of the solid figure affords a demonstration of the existence as woll of the six Pascalian lines as of the two points each the intersection of three of these lines. 404. REPRODUCTION OF EULER'S MEMOIR OF 1758 ON THE ROTATION OF A SOLID BODY. [From the Quarterly Journal of Pure and Applied Mathematics, vol. ix. (1808), pp. 361373.] EULER'S Memoir "Du mouvement de rotation des corps aolides autour d'nn axe variable," Mem. de Berlin, 1758, pp. 154193 (printed in 1765), seems to have been written subsequently to the memoir with a similar title in the Berlin Memoirs for 1760, and to the "Theoria Motua Oorporum Solidorum &c.," Rostock, 1766, and there are contained in the first-mentioned memoir some very interesting results which appear to have escaped the notice of later writers on the subject; via, Euler succeeds in integrating the equations of motion without the assistance furnished by the consideration of the invariable pla/ne. In reproducing these results I make the following alterations in Euler's notation, viz. instead of cc, y, 2 I write p> q, r\ instead of Ma*, Mb*, Mo 2 (where M is the mass) I write .A, B, 0, these quantities denoting the principal moments, and in some equations where the omission or j., ' " ' " ar ' vnallir immofnviol T nn-lfn A R H i f.lin nlnnn /. 136 REPRODUCTION OF EULEB's MEMOIR OF 1758 [404 and introducing the auxiliary quantity u such that du = pqrdt, we have $ = + r" = (+2 where 51, 3i, < are constants of integration, and thence f= V {(51 where bite integral may without loss of generality be taken from w=0; u, and consequently jp, g, r, are thus given functions of t\ and it is moreover clear that 31, 93, ( are the initial values of p", g s . T*. We have also if w be the angular velocity round the instantaneous axis Euler then assumes that the position in space of the principal axes is geometrically determined as follows, viz, (treating the axes as points on a sphere) it is assumed that the distances from a fixed point P of the sphere are respectively I, in, n, and that the inclinations of these distances to a fixed arc PQ are respectively X, p, v. We have then the geometrical relations cos 3 1 -f cos a TO + cos a n = 1 ; * f . cos I , , cos in cos ?i sin (u, y ) = -. : . cos ( u, v ) = : : . sramaintt ^ ' smmsinw* --'- - """*-,, oos( ! ,-X) = - co8 ' !cosZ Bin n Bine x ' smresin sin I sin m r sin I sin m ' whence also __ cos X cos n sin X cos I cos m Sin ill i J - ; sin I sin m = s !-?-^ CQS T1- ~ GOS ^ cos ^ c _ s w sin j sin m ' cos X cos m + sin X cos I cos n sn v = sin I sin t am X cos m cos X coa I cos ?t CQSV = sm i sin n 404] ON THE BOTATION OT A SOLID BODY. 137 The geometrical equations connecting the resolved angular velocities jy, q, r with the differentials of I, m, n, X, p, v are dl sin I =dt(q cos n - r cos m), d\ sin 3 1 = dt(q cos m + r cos n ), dm sin m = dt (r cos ( ^cos?fc), dp sin 3 m = dt (r cosn 4-jtf cos I ), rf sin n = fZ( (p cos m q cos 2 ), rfy sin 3 n eft (p cos -f 5 cos m). Multiplying the equations of motion respectively by cos I, cos m, cos 11, and adding, we obtain an equation which is reducible to the form d (Ap cos I + Bq cos m + Or cos n) 0, whence integrating Ap cos 4- Bq cos m + (7r cos n = 2), 2) being a constant of integration. One other integral equation is necessary for the determination of the angles I, in, n. The expressions for dl, dm, dn give at once p dl sin I + q dm sin m + r dn sin n = 0. Instead of the arcs I, m t n, Euler introduces a new variable , such that v = p cos I 4 q cos in + r cos n ; by means of the last preceding equation, we find civ = dp cos I + dq cos m + dr cos n, and then, substituting for dp, <fy, d? 1 , their values, (L cos 2 Jf cos m. Ncasn\ , ...... - ----- .......... ' 4* I C""> ff from which the relation between u and , is to be determined. "Wo have cos a I + cos 3 m + cos 3 n 1, jfljp cos + Sq cos wt + C 1 ?- cos n = 3), p cos 1 4- # cos m + 7' cos w v, which give cos I, coam, oosw in terras of u, v t the resulting formula* contain th& radical ) 5y 4- 6V)J J which for shortness is represented by VlOOl- We bnen havo * ~ COS t to'Ja-a j,2 i JMi 4- CMoti (ifp - Zr B ) + C. VI. cos m - = I?A*<?i* 4- M*B*i*p -^'N^G^f (MBf - LA.(f) + XBrti (g a - Jtfp 1 ) + ^Vgpg V{(-)] 18 138 . REPRODUCTION OF BULER'S MEMOIR OF 1758 [4Q4 and substituting these values in the differential equation ~ - -^ CQS ^ , AfjMam Ncosn du p q + r the equation to be integrated becomes Now substituting for p, % r their values, we h ve and writing for shortness 9[+ = K, where K = substituting these' values and observing that the radical of the formula becomes Vl(-)l = V (-ST- 2iJf J^ffitt + SiD'iJIf jYw - 3>ff + 23) JV - and the differential equation becomes +(0) which can be reduced to the form Kdv - - CTJ n r ^^T, 404] ON THE ROTATION OP A SOLID BODY. 139 Euler remarks thai) as the right-hand aide of the equation contains only the variable u, the solution will he effected if we can find a function of ,, a multiplier of the left-hand side; he had elsewhere explained the. method of" finding such multipliers, and applying it to the equation in hand, the multiplier of the left-hand side, and therefore of the equation itself, is found to be J^T^lWWGu' V wlmt 1S the aame thing Multiplying by this quantity, the right-hand side may for shortness be represented by dU, so that (H - iLMNFu} </ (ff) dit. ___ dU ~ (K-2LMNGu).J {(ZLu + SI) (2Jlf + S3) (Sflfo + )} ' and U may he considered as a given function of u, or what ia the same thing of t. As regards the left-hand side, attending to the equation K = E&- F*, the radical multiplied into V (<?) may he presented under the form and consequently the left-hand side becomes (K - ZLMNGhi) Gdv + LMNG (Gfv - 3XF) (7f - iJKffff tt) V{(<? - ^ a ) (^' - ZLMNQu) - (tfv - WJf'fl ' which putting for the moment K-*LMNG~p> Gv-$>F-q, ff-S) 3 -/ 2 , becomes _J$$lJ-ty. the integral of which is am- 1 !-; henco resfcoring. the values of p, q, /, p*j(f*p*-<?y w the integral is . sm Hence considering the constant of integration as included in U, or writing (H- ZLMNFit) V (ff) rf we have for the required integral of the differential equation sm 7(e^ whence also TW^' and ~~ ~" " =cos U, 182 140 HEPKODUCTION OF EULER's MEMOIR OF 1758 [404 so that the value of the original radical is , ,, .. ^(Q-VWKZ-ZLMNGU)} V i(- - - - coa (/. Substituting iu the expressions for the cosines of the area I, m, n, these values of t) and the radical ; the formulas after some reductions become sm + ff ~-&7(I{ - ZLMNGu) . ff ff V (K - 2IJIT jy fft*) V (ff) V ----- - -- _ ___ _ - ___ . . _.. ___ . ff ff V (tf - 2MjVff tt ) V (ff) V (JST - ZLMNGu) J where for shortness _p, (jr, r are retained in place of their values V (2Z + 9t)' V (2Jlf -|- 93), The values of ^, ?ji, being known, that of \ could be determined by tho differential equation 7-x (U(qcos in + z cos n) IvA* "-" """ - ; r^5 - -- , sin" I ' and then the values of /*, i/ would be determined without any further integration; but it is better to consider, in the place of any one of the principal axes in particular, the instantaneous axis, which is a line inclined to these at angles , (3, y, tho cosines of 33 Cf 1* which are-, i, - (if as before o> a = i > + g' + j*). Considering the instantaneous axis as a point of the sphere, let j denote the distance OP from the fixed point P, and tf> the inclination OPQ of this distance to the fixed arc PQ, We have cos j = cos a cos I -f cos cos m -f cos 7 cos } sin j cos {/> = cos a sin 2 cos X + cos /3 sin m cos /* + cos 7 sin ^ cos y, sin j sin = cos a sin I sin X + cos /3 sin m sin /4 + cos 7 sin n sin y, a { A _ JA _ COS 7 COS TO - COS ff COS 7^ V W _ _ smisinj ' V W si n ; sin j cos 0* - <) = CQ3 ^- CQSmcos -?' cos f tt - ^ - cos cos K.- cos .y cos ^ ^ W sinmsinj > cos (/^</0 -- sinmsin / - cos (v - A) = ""r-CMftcoaj cog , , , cos ff cos I - cos a cos m sinwsmj ' ^ ^ sih 7i sin j - ' 404] ON THE ROTATION OF A SOLID BODY. 141 so that X, /i, v are determined in terras of j and (ft. These expressions give dcf} = a . 3 . {cos I (qdr - rdq) + cos w, (rdp pdr) + cos n (pdq - qdp}}, 7 sm 3 which is reducible to Euler's equation p (jlf < - JV33) cos I + q (Mi - m) cos m + r (i93 - Jl/21) cos and thence, substituting for cos I, cosm, cosra their values, and observing that Af (JlfE - #33) + #2 a (OT - ZS) + C?- 3 (Z93 - JlfSI) =>-(H- BGp* (m - N$? + 04Q' LA (Jlfg-JVffl) +MB the equation becomes 7/ - 1* 4- dt = where it is to be remarked that 3 - ( ff - 5) a ) (If - ZLAfNQu) sin Now the differential d$ can be expressed as a fraction, the numerator whereof is sin U <-u) and the denominator (ff - 2) a ) ^ a + g (/f - ZLMNQu) - 22)^ V (<? - > a ) (^ - ZLMNGu} sin Z7 - (g - 2) a ) (/^ - ZlMNQu) sin 3 To simplify, write > 9 ) = A, 142 REPRODUCTION OF EULER's MEMOIR OF 1758 [4Q4 the numerator is and the denominator W + Gs- - 23X?fts sin U ~ AV siii a which, observing that /t^ff-3) 3 , is equal to (*7i - 35s sin U? + <?s a cos 3 U, and we have the integral of which i s where S is the constant of integration, or substituting fo r A| 8 their values, the equation is tan (A + g W ^V (g - 3> 3 ) ~ S> sin g V (JC cos Oy iff (K-%LMNGu)} It way be added that cosj = v = _ _ u sn and therefore r>f\<3 n *~^ LV "^ } \ ~ j&i/jri-ii Crt'jf sin (_/ UUd / " j ~~^ ti. ,^ ,, /j Euler remarks that the complexity of the solution owing to the circumstance that tae nxed point P is left arbitrary; and that the formulas may be simplified by taking this point so that tf-S-O, and he gives the far more simple formula corresponding to his assumption; this i s in fact taking the point P in the direction of the normal TO Miff inVfintt'" mlrunji ,>J J.1, _ li- n , .... J pane, and the resulting formula are identical with the ordinary for the solution of the problem. The term invariable piano is not used by or' T;M 5"? ^ OOaoatA La S Ean * rtl " Essai "" le P^leme do trois corps, Pros de I'Aoad. de Berlin, t h ix., 1772. To prove the before-mentioned equation for (fy; stai'ting from the equations COS j = COS CE COS / + COS /? COS m -h COS ry COS 71 = - , a sinj cos tj> = cos a sin i cos X + coa j3 sin w cos /* + cos 7 siu cos v sin j sin < = cos a sin i cos X + cos sin m sin ^ + sin 7 sin n sin v, we have cos j cZj cos $ - sin/ sin 404] ON TEE ROTATION OF A SOLID BODY. 143 the second term is cos \ cot I (a cos n r cos in} to cos cos fj, cot m (r cos I p cos n ) to 4- cos v cot 11 (p cos m q cos I ), to and the third term is + sin X cosec I (a cos i 4- r cos n ) a -j- ----sin ft, cosec wt (r cos n +p cos ) to 4- - sin i> cosec n (p cos 2 -f- q cos TO), to Hence the second and third terms together are pa f , cosZcos?i coswioosw . cosm, . ooa 3 ^ -^ COS X - ^-i -- COS U, - ; - + Bin X -r- -T + Sill /* -r I + <W!., co \ sin 2 sin w sin f B _pq ( cos X sin n cos (y - X) + sin X sin n sin (y - X)| ^ ~ "w |+ cos ^ sin n cos (/A v) + sin p sin ?t sin (/* - c)J 7 . f cos X cos (y X) + sin X sin (v X) =- L ^ am ft ^ , C , . w [+ cos /* cos (/t v) + si cos{X + (y -X)} *-* sin n ( cos v + cos c) + &c,, = ; Q) we have therefore cosj dj cos $ sin j sin < (2$ = - sin a da sin 2 cos X - sin /3 d{3 sin i cos /t - sin 7 dy sin n cos y, q . . r . . = d . sin 2 cos X 4 d - . sin ?n, cos w 4 ft . sin n sin w to a> w 1 = 4. _ ( s i n l C os X o5p + sin m oos /4 dg- 4- sin 71 cos y ar) _ ^ (ain2oo8\p + sinmooa/4g' + sinwcosyr) 144 REPRODUCTION OP EULER's MEMOIR OF 1758 [404 Hence therefore sin j sin Add = - cot j cos Ad ~ a ~ (sin I cos X dp + sin in cos /* dq + sin n cos v o ittu / " 7 -\ I * -t -^ t^sin ( cos A, i + sin m cos pq + siun cos y r) , I/ = - cotj cos (f> - (cos I dp + cos m dq + cos n dr) dto + co\,j cos ^ j (p cosl + q cos m + r cos n) - - (sin / cos X dp + sin m cos ^ dq + sin n cos v di dto , , + ^ (sin i cos X , ^j -f sm m cos p , q + ^ K eot J cos ^ cos ^ + sin Z cos X)p + &c.J. But we have . sm Z sm j * cos a - COt ?, sm t smj J) siu (X - 4) = cos Y cos m - cos ff cos n sin? sin j and thence cos \ = cos ((X - 0) + 0) - C03 ^ ^ cos g ~ cos ^ C08 -?') ~ "P ^ (oos 7 cos m - cos ^ cos n) sin { sin j whence also cot j cos $ cos Z + sin Z cos X = siSj f cos * cos ^ + cos ( cos - cos ; cos j) - sin A (cos 7 cos m - cos /3 cos n)}, sln7 l cos a cos ^ ~ sm (cos 7 cos m - cos cos )}, GOS ^ ~ siu ^ ( r cos m - ? cos '01- 404] ON THE ROTATION 7 OF A SOLID BODY. Hence the expression for smjsin0d< is = -- -- r . fi ?J cos <i sin <A (r cos m w cos n)} dp + ,. .1 l r + -j . Ho co8d> -sin A (? cos m - <; cos n)l + ,..] ' LU r -- __; . [a) do) coa </> sin {(?' cos m </ cos H) rfjj + ...}] . <u 2 cos (i = sin j sn Bin j r or finally 1 , 1 Slllrf) r , -. , . n -I di = ~v ^ [(' cos w - g cosii) dp + *c.J, r a L J - that is sin i sin J (w smj 1 C (r cos t gr cos ?t) w sin"j I + (p G os n - r cos 1} dq [ + (f? cos Z > cos wi) di 1 J which is the required expression for d<f). Recapitulating, A, B, G, p> q, r denote as usual, L = G , M = -, N = > du-pqrdt, r = ^ (( + SI -i- 23 + su so that C, VI. 146 REPRODUCTION OF EULEJl's MEMOIR OJT 1758 &0. [404 sm G COS ft = __ " cos sm cos ? = sin U _p cos ^ + ^ cos in + r cos n ZLMNGu)} sin [7], tan (d, + $) = le angles which _ determine the position of the body are thus expressed in )l w, which is given as a function of t hy the fort where p, q, r denote given functions of w.1 405. AN EIGHTH' MEMOIR ON QUANTICS. [From the Philosophical Transactions of the Royal Society of London, vol. GLVII. (for the year 1867). Received January 8, Bead January 17, 1867.] THE present Memoir relates mainly to the binary qnintic, continuing the inves gations in relation to this form contained in my Second, Third, and Fifth Memoirs Qualities, [141], [144], [156]; the investigations which it contains in relation to qu antic of any order are given with a view to their application to the quintic. j the invariants of a binary quintic (via. those of the degrees 4, 8, 12, and 18) i given in the Memoirs above referred to, and also the covariants up to the degree it was interesting to proceed one step further, viz. to the covariants of the degree in fact, while for the degree 5 we obtain 3 covariants and a single ayaygy, for t degree 6 wo obtain only 2 covariants, but as many as 7 syzygies; one of these however, the syaygy of the degree 6 multiplied into the quintic itself, so th excluding this derived syssygy, there remain (7 - 1 =) 6 syzygies of the degree 6. T determination, of the two covariants (Tables 83 and 84 post] and of the syzygies the degree 6, occupies the commencement of the present Memoir. [These eovariai 4-1,,, ,,:* M nn A iff nf 4-V.rt r,nnm. 1 J.9 " TnWnc nf n - : -" L 148 AN EIGHTH MEMOIR ON QUANTICS. [405 I then apply it to the quintio equation, following Professor Sylvester's track, but so as to dispense altogether with his amphigeuous surface, and making the investigation to depend solely on the discussion of the bioora curve, which is a principal section of this surface. I explain the new form which M, Hermite has given to the Tschirn- hausen transformation, leading to a transformed equation the coefficients whereof are all invariants; and, in the case of the quintic, I identify with my Tables his cubi- covariants f (<r, y) and fa(x, y). And in the two new Tables, 85 and 86, I give the leading coefficients of the other two cubicovariants fa (at, y) and fafa, y), [these are now also, identified with my Tables]. In the transformed equation the second term (or that in ^) vanishes, and the coefficient SI of & is obtained as a quadric function of four indeterminate* The discussion of this form led to criteria for the character of a quintic equation, expressed like those of Professor Sylvester in terras of invariants, but of a different and less simple form; two such sets of criteria are obtained, and the identification of these, and of a third set resulting from a separate investigation, with the criteria of Professor Sylvester, is a point made out in the present memoir. The theory is also given of the canonical form which is the mechanism by which M. Hermite's investigations were carried on. The Memoir contains other investigations and formula in relation to the binary quintic; and as part of the foregoing theory of the determination of the character of an equation, I was led to consider the question of the imaginary linear transformations which give rise to a real equation; this is discussed in the concluding articles of the memoir, arid in an Annex I have 'given a somewhat singular analytical theorem arising thereout. The paragraphs and Tables are numbered consecutively with those of my former Memoirs on Qualities. I notice that in the Second Memoir, p. 126 we should have No. 26 = (No. 19)' -128 (No. 25), vk the coefficient of the last term is 128 instead of 1152. [This correction is made in the present reprint, 141, where the equation is given in the form Q f = G* - 128Q.] Article Nos, 251 to 25*. -The Binary Qmntio, Govariants and Syzygies of the degree 6. 251. The number of asyzygetic covariants of any degree is obtained as in my Second Memoir on Qualities, Philosophical Tmnsactiom, vol. CXLVI. (1856), pp. 101126 [141], viz, by developing the function ' ' ' _ (1 -*)(!- aa) (1 - o&) (1 - a?z) (1 - ate) (1 _ aft) ' as shown p. 114, and then subtracting from each coefficient that which immediately precedes it; or, what is the same thing, by developing the function (1 -*)(!- aw) (l-^)(l-ai85)(l which would lead directly to the second of the two Tables which are there given: the Table is there calculated only up to *, but I have since continued it up to * so as to show the number of the asyzygetic covariants of every order in the variables up to the degree _ 18 in the coefficients, being the degree of the skew invariant, the highest of the irreducible invariants of the quintic. The Table is, for greater convenience, arranged in a different form, as follows : 405] AN EIGHTH MEMOIR ON QUANTICS. 149 Table No. 81. Table for the number of tho Asyzygetic Covariants of any order, la 13 u to the degree IS. ID 10 17 18 o i 1) 1 2 3 1 1 o 1 u 1 1 1 2 3 3 4 i 3 1 (} 2 1 4 g. u i it 2 3 1 1 2 3 1 5 7 S 3 4 2 1 4 3 7 t! 11 l> 4 5 1 1 2 3 5 11 S 10 13 5 6 1 1 4 a 7 7 2 11 18 6 j 1 a 4 G 8 10 13 10 7 9 10 11 12 '3 14 15 16 '7 iB '9 no 21 J3 25 26 27 28 ay 3 1 1 1 1 2 1 2 1 1 I 3 2 2 a 2 1 1 1 2 4 3 2 3 2 2 1 1 1 4 G 4 G 4 4 3 8 2 2 1 1 u a 7 G 7 5 11 G 3 3 7 7 S 8 8 7 8 U 5 G II 10 8 11 11 1) 10 8 I) U 7 U 11 11 13 is 13 12 12 11 11 11 !) 11 10 15 13 1G 14 17 1-1 10 13 14 12 13 14 10 17 10 13 18 19 17 17 15 LI 17 111 21 19 23 SI 24 21 20 22 111 10 20 24 24 26 20 27 2(1 27 25 25 18 18 24 2S 27- 32 20 33 30 29 20 23 27 2U 32 33 30 35 37 SO 87 36 18 211 23 13 J2 39 37 42 JO 4t 10 41 3 9 10 ii '3 '4 '5 16 '7 18 '9 20 11 =2 23 24 35 26 28 29 3 3' 33 34 35 30 37 38 39 40 A 1 2 1 1 1 3 2 2 1 1 5 3 3 2 7 7 5 5 3 12 10 7 7 15 13 12 10 10 18 19 10 10 13 23 20 20 17 31 27 28 24 25 35 30 20 40 42 33 30 35 32 33 34 35 36 37 38 39 40 4' 1" 13 44 i 2 1 3 2 G G 7 7 13 10 10 1-t 31 21 29 28 31 42 43 4-1 45 46 4, 1 2 * 5 10 13 17 23 31 47 4 7, 48 1 3 5 7 11) 17 21 27 48 49 49 . 8 7 14 27 50 So I S 10 18 5' 5 1 2 5 13 22 52 53 o 3 7 17 53 S3 L 6 1ft 22 54 54 1 2 7 14 55 55 1 3 10 18 56 2 5 iy 57 57 3 7 17 SB 58 i B 10 59 59 1 2 7 14 fa 60 1 3 10 61 61 2 S 13 f,* 6a 3 7 63 63 1 5 10 64 64 1 2 7 65 65 1 8 10 66 66 2 I! 67 67 3 7 68 68 1 5 69 <x> 1 2 7 70 70 1 3 7' 7 1 2 5 72 73 3 73 73 I G 74 74 1 75 75 1 3 76 76 2 77, 77 1 3 78 79 79 So 1 1 2 So 8t 81 a 83 82 83 83 i 84 84 1 i "5, Bt 87 87 8S Bt 89 i 89 90 90 150 AN EIGHTH MEMOIR ON QUANTICS. [405 [In regard to this table No. 81 it is hardly necessary to notice that for any column with an even heading the numbers of the column correspond to the even outside numbers, while for any column with an odd heading the numbers of the column correspond to the odd outside numbers. The table is in fact a table of the differences of the numbers of the o/-table, 142 ; thus in this table writing down cols. 5 and 6 and in each of them forming the differences by subtracting from each number the number immediately below it, we have cols. 5 and 6 of the table No. 81, viz, : 1 1 2 3 6 7 9 11 U 16 18 19 20 col. 5, 13-12 of /-tnblo. 1011222 2 3~3 2 1 T col. 6 of table No. 81. 1 1 2 3 5 7 10 13 16 19 23 25 29 30_38_J3 col. 6, IB of tf-tnblo. TTTl 2 2 3 2~"T 3 4 2 4 ~1 2 col. 6 of table No. 81.] 252. The interpretation up to the degree 6 is as follows; [In the following Table No. 82 as originally printed, the heading of the fourth column was "Constitution. Nos. in ( ) refer to Tables in former Memoirs except (S3) and (84) which are given post," and the covariants were referred to by their No/ accordingly.] J " UHt 405] AN EIGHTH MEMOIR ON QU ANTICS. Table No. 82. 151 Degree. Order. No. Constitution. Notation is the alphabetic notation of Id3, A the quintic itself, ZJ, G qitadricovnrifmta,. &c. ^:;r 1 viz. the abaoluta constant unity. 1 5 1 A Hf. 2 10 1 A* 6 1 C JHf. " 2 1 E N. 3 15 1 A* 11 1 AG 9 1 ]i> XT )f 7 1 AS 5 1 E N. 99 3 1 T> ff. '4 20 1 A- 1 16 1 j\?Q )9 H 1 AF H 12 2 G a A 3 J1 99 10 1 Ah 8 2 AD t EG >9 6 1 I N. Jl 4 2 //, J? N. 9> 1 G N. 5 25 1 A D 99 21 1 AG 99 19 1 A*F 99 17 2 A*JJ, AG* I) 16 2 A z E t GF ,, 13 2 A*, ABG 99 11 2 AI + J3F ' GE S. 9> 9 3 AW, All, CD 19 7 2 BM> L N, 99 6 2 AG, BD 91 3 1 K iV, 1 1 J N. 6 30 1 A 6 )9 26 1 A 4 G 9J 24 1 - A S F 99 22 2 A 3 G 2 , A*B 1) 20 2 A OF, A*J$ 18 3 W + 4C 3 + AW-AiBG =0 S. 91 16 2 A{AI+ J3F ~ GE} =0 S', 1) 14 4 -6AG&- l$F -4C a + AW =0, AW 3 S. 12 3 AL H- SDF ~2CI =0, ABE S. 9) 10 4 WG +12ABD- AW+W =0, CH S, 8 2 AK + 257 -32)22 =0 S, 6 4 A J -j- 2JS// j5 3 GG- 9J? a = S. 19 4 1 N ff. 9) 2 1 M ff. 152 AN EIGHTH MEMOIR ON QUANTICS. [405 253. For the explanation of this I remark that the Table No, 81 shows that we have for the degree and order one covariaut; this is the absolute constant unity ; for the degree 1 and order 5, 1 co variant, this is the quintic itself, A ; for degree 2 and order 10, 1 covariant ; this is the square of the quintie, A*\ for same degree and order G, 1 covariant, which had accordingly to be calculated, viz. this is the co variant C'; and similarly whenever the Table No, 81 indicates the existence of a covariaut of any degree and order, and there does not exist a product of the covariants previously calculated, having the proper degree and order, then in each such case (shown in the last preceding Table by the letter j\ r ) a new covariant had to bo calculated. Oil coming to degree o, order 11, it appears that the number of asyzygetic invariants is only = 2, whereas there exist of the right degree and order the 3 com- binations AI, BF t OE\ there is here a syzygy, or linear relation, between the combinations in question; which syzygy had to he calculated, and was found to be as shown, AI+BFCE=Q t a result given in the Second Memoir, p. 126. Any such case is indicated by the letter 8. At the place degree 6, order 16, we find a syzygy between the combinations A Z J, A*BF, AOE\ as each terra contains the factor A, this is only tho last-mentioned syzygy multiplied by A, not a new syzygy, _and I have wiitten 6" instead of S. The places degree 6, orders 18, 14, 12, 10, 8, 6 indi- cate eacli of them a syzygy, which syzygies, as being of the degree 6, were not given in the Second Memoir, and they were, first calculated for the present Memoir, It is to be noticed that in some cases the combinations which might have entered into the syzygy do not all of them do so ; thus degree 6, order 14, the syzygy is between the four combinations ACD, EF> SO 3 , A*H t and does not contain the remaining com- bination A Z S. The places degree 6, orders 4, 2, indicate each of them a new covariant, and these, as being of the degree 6, were not given in the Second Memoir, but had to be calculated for tho present Memoir. 2134. I notice the following results: Quadrinvt. QH = 3<?, Cubinvt. QH . =- H +54GQ, Disct. (o5 + #M) = (-(?, Q, -BUfa, /3) a , Jac. (3, H) = &M t Hess. 3D =JV, the last two of which indicate the formation of the covariants given in the new Tables MNo. S3 and N No. 84: viz. if to avoid fractions we take 3 times the covariant D t being a cubic (a, ...} 3 (x, 2/) 3 > then the Hessian thereof is a covariant (a, ...)"(*> 2/) s j which is given in Table, M No. 83; and in like manner if we form the Jacobian of the Tables S and H which are respectively of the forms (a, , ,) a (w, y) 3 , and (, ..) e (X $> tn ' s i 8 a covariant (a,..) 6 (#. yf, and dividing it by 6 to obtain the coefficients in their lowest terms, we have the new Table, N No. 84. I have in these, for greater distinctness, written the numerical coefficients after instead of before, the literal terms to which they belong. 405] AN EIGHTH MEMOIll ON QUANTICS. The two new Tables are : Table No. 83. M = (*$, y}\ See 143. Table No. 84. JY = (*$>, i/) J . See 143. 153- Article No. 255. Formula for the canonical form arc 5 4- by rj + cz* = 0, wAere a; 4-^4-2 = 0. 255. The quintic (, b, c, d, e, /$>, 2/) 5 may be expressed in the form r B 4 sv 5 where , w, w arc linear functions of (to, y) such thab + D + w = 0. Or, what is the same thing, the quintic may be represented hi the canonical form where 0+y+* = 0; this is = (-c, - C) - c , -o, -o, 6 - oj, Z/) B > d the different covariants and invariants of the quintio may hence be expressed in terms of these coefficients (a, b, c). For the invariants we have Q = J = 6 9 c 2 -i- c 9 ft 2 + ft a 6 3 - 2a6c (a + b 4- c), Q = 7f = 2 & a c a (60 If = / = 4ft B &V (6 - c) (c - a) (a - i). [Observe that throughout the present Memoir, the invariants, instead of being called Q Q ~U, W are called 7, J, K, L, viz. the /, J, K, L in all tliot follows denote the invariants, and not the covariante denoted by these letters in 142, 143, Moreover D is used to denote the invariant Q', which is in fact the discriminant of the qnmtic.] Hence, writing for a moment a +b + c =p t and therefore J =<f-4/jr, be + ca+ab~q K = r*q> abc =r & = r *> (a -6)9(6-o)(o-a)'=j)Y- 4 ? s -^ v + 18 ^ m " 2l7)l0 ' and thence , , (,-, 73 = iQr w - 45 3 - 4p* i 4- ISp^ 1 - 27r"), = 16r 10 (pY - 4g 3 - 42^' 4- ISpqr - 27r a }, 20 154 AN EIGHTH MEMOIR ON QUANTICS. [405 that is, which is the simplest mode of obtaining the expression for the square of the 18-thic or skew invariant / in terms of the invariants J, K, L of the degrees 4, 8, 12 respective!}'. If instead of the invariant K of the degree 8 we consider, the invariant D [~Q' -as before- mentioned] of the same degree, this is Q' = D = {Jw + cW 4 a* - 2a&c (a 4 b 4 c)) 2 - 128 a i 2 c 3 (bo + ca 4 ab), d 4 .and we have also the following covaviants ; B = ( ao, ab ~ac 6c, == bcyz 4- ca^ffl -f abxy. = ( ttc, -Sac, -Sac, a6-ftc~ic, -86c, - 36c, - bciftP + cu&t? + a6a#y. Z) = (0. ubc, aba, 0^-, yy* aba as-t/2, 4- ( oac 2 4 otc a - 5a&c) j*i/ 4 ( - 10ac a 4 lO&c 2 - Sale) afy 3 4 ( - 10c a 4 'lOfcc 3 4 2a6c)/y 4( - 5c 9 4- S6c a 4 4 (- afr - c 3 4- 6c 9 4 6\) 4 (fc - o) a 2 *- + (o - a) &y 4 (a - 6) c^ s - aJc (i/ -*)(*- ) (a; - y) (yz +MB + xy], Article No. ^Q. Expression of the 18-thio Invariant in terms of the roots, 256. It was remarked by Dr Salmon, that for a quintic (a, b, c, d, o, / $, i/) B which is linearly transformable into the form (o, 0, c, 0, e, 0$>, 2/) B , the invariant 7 is = 0. Now putting for convenience y = 1, and considering for a moment the equation then writing herein ~r for , the transformed equation is 405] where AN EIGHTH MEMOIR ON QUANTICS. 155 ny , &c.; P l-m/3 1 7 l-m/y hence m may be so determined that ' + </ may be =0; viz. this will be the case- if jfl + 7 - 2j)ijS7, or m-^J^. In order that S' + e may be =0, we must of course have w, = "t- 6 , and hence the condition that simultaneously j3'47 /= and S' + e' = is 2oe jQ I X 1 I ".J.7 ~ "Jj.5 ; that is, (& + 7) Se - 7 (S 4- e) = 0. Or putting IB -a for a and /3-, 2/37 2oe 7 , &c. for /3, 7, &c., we have the equation (a, - a) ( - /3) (0 - 7) (0 - 8) (* - e) 0, which is by the transformation as-ct into ^^y^ changed into (dj - a') (IB - 0') (as - 7') (a - S') (as - e') = (where a' = a), and the condition in order that in the new equation it may be possible to have simultaneously /S' 4- 7' - 2a' = 0, 8' + e' - 2a' = 0, is or, as this may be written, J 1, jS + 7, /?7 1, S+e, Se =0. I-Ienoo writing ts + tx' for a, the last-mentioned equation is 'the condition in order that bho equation _ / . . (? - ) ( - 0) ( ~ 7) (* - S > ( - e ) = may be transformable into J whoro jfl' + y^O, S'+e'==0, that is, into the form a, (?- ^) (? - 8*) = 0. Or replacing i/, if wo have (a, 6, o, rf, e./jfrtf, y ) fl = - then the equation in question is the condition in order that this may be transformable into the form (*', 0, c', 0, e', <%, T/); that is, in order that the 18-thic invanant I mav vaiiiah. Hence observing that there are 15 determinants of the form in question, and that any root, for instance , enters as rf in 8 of them and in the simple power a in the remaining 12, we see that the product 1, 2 , a 3 ^0 2 1, S+e, Se 156 AN EIGHTH MEMOIR OX QUANTICS. [405 contains eacli root in the power IS, and is consequently a rational and integral function of the coefficients of the degree 18, viz. save as to a numerical factor it is equal to the invariant /. And considering the equation (a, . .$#, 7/) fl = as representing a range of points, the signification of the equation 7=0 is that, the pairs (y9, <y) and (, e) being properly selected, the fifth point a is a focus or sibiconj ugate point of the involution formed by the pairs (/3, 7) and (S, e). Article Nos, 257 to 267. Theory of the determination of the Character of an Equation; A utciliars ; Facultative and Non-facultative space. 257. The equation (, b, c,.,$>, #) = is a real equation if the ratios a : b : c,.. of the coefficients are all real. In considering a given real equation, there is no loss of generality in considering the coefficients (a, b, c..) as being themselves real, or iu taking the coefficient a to be = ] ; and it is also for the most part convenient to write # = 1, and thus to consider the equation under the form (1, 6, c..,]a;, 1)" = 0. It will therefore (unless the contrary is expressed) be throughout assumed that the coefficients (including the coefficient a when it is not put =1) are all of them real; and, in speaking of any functions of the coefficients, it is assumed that these are rational and integral real functions, and that any values attributed to these functions are also real. 258. The equation (1, b t c...fa, 1)<^0, with a real roots and 2 imaginary roots, is said to have the character ar+2@i; thus a quintic equation will have the character or, & + 2t, or r + 4i, according as its roots are all real, or as it has a single pair, or two pairs, of imaginary roots. 259. Consider any m functions (A, E, ... If) of the coefficients, (m= or <n). For given values of (A, B, ... 1C), non constat that there is any corresponding equation (that is, the corresponding values of the coefficients (b, c, ...) may be of necessity imaginary), but attending only to those values of (A, B,...K) which have a corresponding equation or corresponding equations, let it be assumed that the equations which correspond to a given set of values of (A, B,,..K) have a determinate character (one and the same for all such equations): this assumption is of course a condition imposed on the form of the functions (A, B,,..K}> and any functions satisfying the condition are said to be "auxihaiu" It may be remarked that the n coefficients (6, c, ...) are themselves aimliars; in fact for given values of the coefficients there is only a single equation which equation has of course a determinate! character. To fix the ideas we may con- sider the auxiliars (A, ,..,%) as the coordinates of a point in ^dimensional space, or w.-space. Any given point in the m-space is either "facultative," that is, we have ang thereto an equation or equations (and if more than one equation then by wna* precedes these equations have all of them the same character), or else it is " non-facultative," that is, the point has no corresponding equation. 261. The entire system of facultative points forms a region or regions, and the entire system of non-facultative points a region or regions; and the m-space is thus divided into facultative and non-facultative regions. The surface which divides the 405] AN EIGHTH M13MOIR ON QUANTICS. 157 facultative and non-facultative regions may be spoken of simply as the "bounding surface, whether the same be analytically a single surface, or consist of portions of more than one surface. 2G2. Consider the discriminant D, and to fix the ideas let the sign be determined in such wise that D is + or ' according as the number of imaginary roots is = (mod. 4), or is = 2 (mod. 4) ; then expressing the equation D = in terms of the auxiliars (A, #,.,../(!"), we have a surface, say the discriminatrix, dividing the m-space into regions for which D is +, and for which D is , or, way, into positive and negative regions. 263. A given facultative or non-facultative region may be wholly positive or wholly negative, or it may be intersected "by the discriminatrix and thus divided Into positive -and negative regions. Hence taking account of the division by the discriminatrix, but attending only to the facultative regions, we have positive facultative regions and negative facultative regions. Now using the simple term region to denote indifferently a positive facultative region or a negative facultative region, it appears from the very notion of a region as above explained that we may pass from any point in a given region to any other point in the same region without traversing either the bounding surface or the discriminatrix ; and it follows that the equations which correspond to the several points of the same region have each of them one and the same character; that is, to a given region there correspond equations of a given character, 264. It is proper to remark that there may very well be two or more regions which have corresponding to them equations with the same character; any such regions may "be associated together and considered as forming a kingdom ; the number of kingdoms is of course equal to the number of characters, viz. it is =(+ 2) or ( + 1) according as n is even or odd ; and this being so, the general conclusion from the preceding considerations is that the whole of facultative space will be divided into kingdoms, such that to a given kingdom there correspond equations having a given character; and conversely, that the equations with a given character correspond to a given kingdom. Hence (the characters for the several kingdoms being ascertained) knowing in what kingdom is situate a point (A, J3, ... /), we know also the character of the corresponding equations. 265. Any conditions which determine in what kingdom is situate the point (A, B,.,.K) which belongs to a given equation (1, 6, c, .."&, l) ft =0, determine therefore the character of the equation, it is very important to notice that the form of these conditions is to a certain extent indeterminate ; for if to n. given kingdom we attach any portion or portions of non-facultative apace, then any condition or conditions which confine the point (A, B t ... K) to the resulting aggregate portion of space, in effect confine it to the kingdom in question; for of the points within the aggregate portion of space it is only those within the kingdom which have corre- sponding to them an equation, and therefore, if the coefficients (&, c,.,.) of the given equation are such as to give to the auxiliars (A, B, ... K) values which correspond to .a point situate within the above-mentioned aggregate portion of space, such point will of necessity be within the kingdom. 158 AN EIGHTH MEMOIR ON QU ANTICS. [4Q5 266. In the case where the auxiliars are the coefficients (&, c, ...), to any given values of the auxiiiars there corresponds an equation, that is, all apace is facultative space. And the division into regions or kingdoms is effected by means of the discrhninatrix, or surface D = 0, alone. Thus in the case of the quadric equation (1, cc, y$ff, l) a = the m-space is the plane. We have D = a?-y t and the discriminatrix is thus the parabola ^-^ = 0, There are two kingdoms, each consisting of a single region, viz. the positive kingdom or region (tf-y = -\-) outside the parabola, and the negative kingdom or region (a?-y = ~) inside the parabola, which have the characters 2s- and 2z, or correspond to the cases of two real roots and two imaginary roots, respectively. And the like as regards the cubic (1, a, y, z\Q, 1)' = 0; the m-space is here ordinary apace, D = ~ 40"# + 3^y + fayz - % 3 - z\ and the division into kingdoms is effected by means of the surface D = 0; but as in this case there are only the two characters Br and )+ 2i t there can be only the two kingdoms D = + and D = - having these characters 3r and r + 2i respectively, and the determination of the character of the cubic equation is thus effected without its being necessary to proceed further, or inquire as to the form or number of the regions determined by the surface =0: I believe that there are only two regions, so that in this case also each kingdom consists of a single region, But proceeding in the same manner, that IB, with the coefficients themselves as auxiliars, to the case of a quartic equation, the TO-spn.ce is here a 4 -dimensional space, so that we cannot by an actual geometrical discussion show how the 4-space is by the discriminafcrix or hypersurface D = divided into kingdoms having the characters 4?-, 2r + 2 4i respectively. The employment therefore ^ of the coefficients themselves as auxiliars, although theoretically applicable to adequation of any order whatever, can in practice be applied only to the cases for which a geometrical illustration is in fact unnecessary. 2G7. I will consider in a different manner the case of the quartic, chiefly as an instance of the actual employment of a surface in the discussion of the character of an equation; for in the case of a quintic the auxiliars are in the sequel selected in such manner that the surface breaks up into a plane and cylinder, and the discussion is in fact almost independent of the surface, being conducted by means of the curve (Professor Sylvester's Biooni) which is the intersection of the plane and cylinder. Article Nos. 268273. Application to the Quartw equation, 26S. Considering then the quartic equation (a, b, c, d, e\0, 1)< = (I retain for symmetry the coefficient a, but suppose it to be -1, or at all events positive), then if /, J signify as usual, and if for a moment S- = a?d - 3a6c + 26", X = we . have identically + *) a* 9 (6" - 405] AN EIGHTH MEMOIR ON QUANTICS. 159 (see my paper, " A discussion of the Sturmian Constants for Cubic and Quartic Equations," Quart. Math. Journ., vol. iv. (1861) pp. 7 12), [290]. And I write a = b' 2 uc, 269. I borrow from Sturm's theorem the conclusion (but nothing else than this conclusion) that (a 1 , y, z) possess the fundamental property of auxiliare (that is, that the quartic equations (if any) corresponding to a given system of values of (x } y, z) have one and the same character). The foregoing equation gives O&y s?z y 3 a square function, and therefore positive ; that is, the facultative portion of space is that for which Qn) s if a?z y s is =+. And the equation is that of the bounding surface, dividing the facultative and n on -facultative portions of space. 270. To explain the form of the surface we may imagine the plane of coy to be that of the paper, and the positive direction of the axis of s bo be in front of the paper. Taking z constant, or considering the sections by planes parallel to that of &y t z~Q, gives f (QoP - y) ~ 0, via. the section is the line y. = 0, or axis of ts twice, and the cubical parabola y~a?. * = (non.fao.) T(non,fcto.) V s /- z = +, the curve ic 3 = has two asymptotes y = ^vz, parallel to and equi- jy z distant from the axis of a, and consists of a branch included between the two parallel asymptotes, and two other portions branches outside the asymptotes, as shown in the figure (2 = +). 1S . the curve f- AN EIGHTH MEMOIR ON QUAKTICS. [ 405 has no real llsympt ot e , and consists of a single branch, resenrb.in, in its appearance the cubical parabola as shown in the Hgnvc Taking ., as constant, or considering the sections by planes parallel to that of the equation of the action is ,-, which i s a cllbical pm , bolil) facultative rerions ^4' /? 0). and if (., y } } ' ' ' Wl e "' one of the regions is in the region B, the character is 4r, A '> >, 4i, -e co r on, 7; to obtain *- + ,*- + J ff-+ includes the whole of facultative region is, (a, y f ,) being eac h positive, the character is 4rj *== ' IV I = -, 2,= - I Delude each a part and together the whole of facultative region A' t 405] AN EIGHTH MEMOIR ON QU ANTICS. 1.61 that is, z being +, but (cc, y) not each positive, the character is 4i; include each a part and together the whole 'of facultative region G, v = - , y = - ) s = , as = - , y = + does not include any facultative space, that is, z being -, the character is 2r-|-2i; and the combination of signs z = ~, ,%== , y = + is one which does not exist. The results thus agree with those furnished by Sturm's theorem ; and in particular the impossibility of z --= - , = -, y = + appears from Sturm's theorem, inasmuch as his combination would give a gain instead of a loss of changes of sign. Article Nos. 274 to ^S^Determination of the characters of the quintic equation. 274. Passing now to the case of the quintic, I write J = G, K = Q, T _ 77 jj u, I = If; vi. J is the quartinvariant, K and D are octinvariants (D the discriminant) i is 12-thic invariant, and J is the 18-thic or skew invariant. Hence also > J, V, i L-J arc invariant of the degrees 4, 8, 12 respectively; and forming the combinations 2Z L -J* ^ -D =J I assume that (* y. *} are auxiliars, reserving for the Concluding articles of the pi-oaont memoir the considerations which sustain this assumption. 275. The separation into regions is effected as follows :-Wo have identically (see ante, No. 256) or putting for K its value = T ^(e/ 3 ~D), this is + DJ (- 4J + 61 o. vi. 162 AN EIGHTH MEMOIR ON QUANTICS. [405 or writing as above _ 2" L - / 3 D *' -- T 3 ' ^75* whence also i ^ 2ni 1 + * jT, this is 2 M --* d 80 +*)-!} + 36 l+*'-J.l + *-* -or, what Is the same thing, Ta = <(&, y) suppose, 276. Hence a-lao writing s~J, we have saf> fa y)==2.2 33 ~= + , -or the equation of the bounding surface may be taken to be x$ (0, y) = 0, that is, the bounding surface is composed of the plane = 0, and the cylinder <t>(> 2/) = 0. Taking the plane of the paper for the plane ^0, the cylinder meets this plane in a curve <f>fa i/)=:0, which is Professor Sylvester's Bloom: this citrve -divides the plane into certain regions, and if we attend to the solid figure and instead of the curve consider the cylinder, then to each region of the plane there correspond in sohdo two regions, one in front of, the other behind the plane region, and of these regions m solido, one is facultative, the other is non-facultative (viz. for given values of (*, y], whatever be the sign of (, ff ), then for a certain sign of g t gffa y) will be positive or the solid region will be facultative, and for the opposite sign of ^toy) will be negative or the region will be non-facultative). It hence appears that we may attend only to the plane regions, and that (the proper sign being attributed to *, that is to J) each of these may be regarded as facultative. It is to be added that the discnnimatnx is in the present case the plane y = 0, or, if we attend only to the plane figure, it is the line # = 0; so that in the plane figure the separation into regions is effected by means of the Bicorn and the line y = 0. AN EIGHTH MEMOIR ON QUANTICS. 163 277. Be verting to the equation of the Bicorn, and considering first the form at infinity, the intersections of the curve by the line infinity are given by the equation y s (2y-a))^0, viz. there is a threefold intersection if = Q t and a simple intersection 2?/-# = 0; the equation ?/ s = indicates that the intersection in question is a point of inflexion, the tangent at the inflexion (or stationary tangent) being of course the line infinity; the visible effect is, however, only that the direction of the branch is ultimately parallel to the axis of as. The equation 2y-af = Q indicates an asymptote parallel to this line, and the equation of the asymptote is easily found to be 1y _ g; + 5 = Q. 278. The discussion of the equation would show that the curve has an ordinary cusp; and a cusp of the second kind, or node-cusp, equivalent to a cusp and node;. toho curve is therefore a unicursal curve, or the coordinates are expressible rationally in terms of a parameter </>; we in fact have whence also 279. The curve may be traced from these equations (see Plate, fig. 2, where the- bloom ia delineated along with a cubic curve afterwards referred to): as tf> extends from an indefinitely small positive value e through infinity to -I e, we have the branch of the curve, viz. (ft = e> gi voa -'V=GQ, i/ = ~oo, point at infinity, the tangent being horizontal, ft = cc, gives / = 1, ?/==+, the node-cusp, tangent parallel to axis of y, $ = _ 2, gives to = 0, y = 0, the tangent at this point being the axis of at, ( f> = ~l G t gives i = oo, a/ = +, point at infinity along the asymptote; and as c/> extends from ,v;~-l+e to ?-*, we have the lower branch, viz. 164 AN EIGHTH MEMOIR ON QUANTICS. [405 281. The form of the Bicorn, so far as it is material for the discussion, is also shown in the Plate, fig. 3, and it thereby appears that it divides the plane into three regions ; viz. these are the regions PQR and S, for each of which < (#, y) is = , and the region TU, for which </>(#, y) is = + ; that is, for PQR and 8 we must have J~ t and for TU we must have J=+. Hence in connexion with the bicorn, con- sidering the line y=Q, we have the six regions P, Q, R, S, ?', U. It has just been seen that for P, Q, R, S we have J=~, and for T, U we have /=+; and the 2"i - ,/ s D sign of ./ being given, the equations ce = j- , y = -^ , then fix for the several regions the signs of 2"Z / 3 and D t as shown in the subjoined Table ; by what precedes each of the six regions has a determinate character, which for R t S, mid U (since here D is =-) is at once seen to be 3r-j-2t, and which, as will presently appear, is ascertained to be or for P and r+U for Q and T. 282, We have thus the Table P, D= + t J=~, U, so that we have the kingdom or consisting of the single region P, the kingdom r + 4t consisting of the regions Q and T t and the kingdom 3r + 2i consisting of the regions R t 8, and U. 283. For a given equation if D is =-, the character is 3r + 2i; if D = +, J = +, the character is j- + 4i; if D = + , J=~, then, according as M-/ 3 is =+ or is --, the character is 5r or r+4&, But in the last case the distinction between bho characters or and r + 4i may be presented in a more general form, involving a parameter /*, arbitrary between certain limits. In fact drawing upwards from the origin, as in Plate, fig. 3, the lines #-2y = and tc + y = Q, and between thorn any line ^ whatever an-/(,y0, the point (at, y), assumed to lie in the region P or Q, will lie in the one or the other region according as it lies on the one. side or the other side of the line in question, viz. in the region P if te + fjaj is =-, in the region Q if a? + py is = -h But we have and J being by supposition negative, the sign of M - J* + pJJ) is opposite fco that of + ^y. The region is thus P or Q according to the sign of ^L-J^^JD' and completing the enunciation, we have, finally, the following criteria for the number of real roots of a given quintic equation, viz. If D = - , th e character is 3r + 2?, If D = +, J-+, then it is r+ 4 405] AN EIGHTH MEMOIR ON QU ANTICS. 165 .But if D -I-, J=~ } then /* being any number at pleasure between the limits +1 2, both inclusive, if 2"i - J a + fiJD = + , the character is 5<r, 2H4, The characters 5?- of the region P and r + 4i of the regions Q and T may be ascertained by moans of the equation (a, 0, c, 0, e, 0&0, 1) 5 =0, that is tlioro in always the roal root # = 0, and the equation will thus have the character or or r-M'f! according as the reduced equation a# 4 + 10c0 a + 50 = has the character 4? 1 or <l<t, ,l.li in clear that (a, e) must have the same sign, for otherwise 8* would have two real values, one positive, the other negative, and the character would be 27 - + 2i. And (it; (?) having the same sign, then the character will be 4?', if 8 3 has two real positive- values, that is, if e 5o is = , and the sign of c be opposite to that of a and e, or, what is the same thing, if ce be =~; but if these two conditions are not HtitiHfuul, Chen the values of 0* will bo imaginary, or else real and negative, and in oithor oasti the character will be 4r. 2Hr>. Now, for tho equation in question, putting in the Tables 6 = rf=/=0, we find J= 16 OG (tie + 3c 3 ), J ;I = 2 ia ce s {2 (ae - c 2 ) 4 - o 2 (oe + 3c 3 ) s j 2 ia ce 3 (ae - oc 2 ) (Stfe 8 + a c 3 e a + 8oo*fl + 5c"). Wo hiivo by supposition 73 = +, that is, ae = + ; hence J has the same sign as ce; whoncn if ./'= + , thon also 00 = +, and the character is 4i; that is the character of the rogion T is j--h4f. But if J = -, then also ce = -. But ae being = + , the sign of 2. - /" w the name as that of ce (ae - 6c 2 ), and therefore the opposite of that of tie - (V : hcncn D =s -h , J" - , the quartic equation has the character 4?' or 4i according an 2 n /y-i/ is =4- or =-. Hence the region P has the character Br and tho nsgion Q tho character r + 4i ; and the demonstration is thus completed, Article Nos. 286 to 203. HEHMITK'H new form of TSOHIRNHAUSEN'S transformation, and application thereof to the quintic. 2H(i M Hormite demonstrates the general theorem, that if f(a, y] be a given cpuintio of tho -th order, and <j>& y) any covariant thereof of the order -2, then wnwidoring the etjuation /(, 1)0, and writing 166 AN EIGHTH M.RMOIU ON QUANTI08. [405 (where //(*, 1) is the derived function of /(, 1) in regard do ;), thtm eliminating a 1 , we have an equation in z, the coefficients whoroof tiro nil of them invariants of 287. In particular for the qaintic /(,*, y) (, 6, (-', rf, e t ffcai, jry)", if ftfa y), &0, y), <M (t '> y), <M<i'', y) arc tiny four covariant cubics, writing 2 = - (vix. the numerator is a covuriimt ubio involving bho iutlutun-niiiititi! iMmllUjinntK t, j u, i), w) than, in the tiiuisformod oquatittn in z, kho cooillcioutH uvo all ul' liluini invnriants of fcho given quiutio. ContUioting tho invostigntion by IIKMUIH ol' a oorlitiiu canonical form, which will bo rofon-od bo in tin.! soquol, ho lixtw tho wgnirioation o(' IIJH four covariant cubica, these being roMpeetivoly oovarmnli culmw of tho tlogrouK ,'{, " (| 7, and 9, defined as follows; vk startiupf with tho form 3 f> - ;'A. * i 6, v , c , d , u , d d , o >' > f -8A -8(^ ^ 0, tfjfa y y, or (-JU and considering also the quadrio covw-iaiit (. A 7?*, ?/)', then ti, ^ a) ^ 3 , 0,, arc derived from the form (A, B, 0, ia, we have , .#, Cf, 1 5, 6', whero ^(flv y)J and [0,( ffi , y) ) ft ro tlio functions originally eallod by him fcfe ) and ^(w. y): those uliamatoly HO called by him are 9 <V 405] AN EIGHTH MEMOIR ON QUANTICS. 167 whore -fa (a, y) is the cubico variant (- 27.4 3 D + QABC - 25 3 , . .$, </)' of fa (x, y), = (~'3A, -5, -<7, -3]J>, y)\ ut sivprcb, The covariant fates, y) has the property that if the given quintio (,..,][#, y) 5 contains a square factor (k + mi/)' 2 , then fafa y) contains the factor Ix -\-rny: [fa (as, y)} and [fa(x, y}\ are covariants not possessing the property in question, and they were for this reason replaced by fa (as, y} and fa (as, y) which possess it, viz. fa (as, y) contains the factor la-t-my, and fa(ai, y) contains (kv + myj 1 , being thus a perfect cube when the given quintic contains a square factor. 288. The covariants fa (as, y) and fa (as, y) are included in my Tables, viz. we have fa ( a! , y } = - 87) of 142, 143, (observe that in K the first coefficient vanishes if it, 0, b 0, which is the property just referred to of fa(a), y))', the other two covariants, as being of the degree 7 and 9, are not included m my Tables, but I have calculated the leading coefficients of these covariants respectively, viz. Table No. 85 gives leading coefficient (or that of ,' 3 ) in fa(sv, y), and Tablo No. 86 gives leading coefficient (or that of a; 3 ) in fa (to, y) [and by means thereof we have the values of the covariants in question], The coefficients in question vanish for a 0, 6 = 0, that is, ^ 3 (ai, y) and fa(ss t y) then each of them contain the factor y\ if the remaining coefficients of fa(to t y) were calculated, it should then appear that for a = 0, 6 = 0, those of a*y, esf would also vanish, and thus that fa (a), y) would be a mere constant multiple of y s . Table No. Ho [= leading coefficient of 16J5J aV* + 1 fl^V 8 - 1 fl& 3 (// a + 64 &v a f ,9 f nf9 i ~\ f\ ft ft J 11 tMcdf* - 94 rtiVy - 54 frto/ - 144 cWf -32 ft tJCG / + 86 & a c a y la 48 6V +135 ttV '+16 ft Oft C/ + 106 fliWe/ + 184 6*0 V + 108 a*bd# - 96 i a ce 3 135 6 9 c^y + 288 V/' 2 + 63 ab*d s f 272 CflC J i5U rcWfi/ -188 fli a ( 2 e 2 + 243 bWe + 80 rtw' + 32 a&oy - 06 Wtfdf - 360 oVSy + 60 + 68 aficVy + 212 a6c a f/c 8 + 148 69 t 1 O OS + lOO 6 2 c a (? 2 e + 360 a d% - 36 abod^e 412 Fed 4 - 160 abd 5 + 144 6c B / + 108 rttf 1 ^ - 36 6c 4 c?e 180 a u ) o f(C fi 'iO ic 3 d 3 + 80 flC 3 (Z 3 e + 124 ftc ft 48 + 32 + 415 + 1119 + 1294 168 AN EIGHTH MEMOIR ON QUANTICS. Table No. 86 [= leading coefficient of S']. [405 atcej* + 9 cMef* - 9 1 rtW/ 3 4- 120 6V ;J - 576 i B / ;i + 192 aWf 3 + 21 tfbcdf 3 - 162 W/" - 21 fl&W- + 672 b*cef 2 - 1440 rtWe/ 3 - 7S :i 6c&y a + 99 aW/ 3 -H 480 o6V/ - 359 6"(Z 4 / 2 - 192 V/ + 48 n^e/ 2 + 309 aWcdef 2 - 2160 6 !1 cV-' + 3456 We 3 / - 1080 %&'/ + 12 aWfcfi 3 / + 1023 H6 B crfV a - 864 6V ' + 2025 n %^ - 240 W/ B + 120 abWf+ 3094 iVW/ a + 2592 a 3 c 3 / a - 81 a6w/- 1053 & 3 ce 4 ' - 3915 6W/' + 3546 aWdef + 1026 aWa* + 1314 aPWy + 528 frcd'af 4- 5280 W/ - 768 rt a ic s e/'- - 1863 aJWfl 3 - 45 ^crffl 5 - 13500 J crf a / 2 - 738 a*lc*d*f* + 253S afiV^' - 2592 bWf - 4800 aW/ - 5G4 ?l<?d*f + 2340 rtfiW/ - 9747 ftW + 7800 ft ; 'ct& 4 -3- 1066 s 6cV + 672 alMPef- 8496 6V/ S - 648 W + 756 tfbctPef + 2820 6Vf/e ;1 + 26610 iWe/ - 14040 W - 69G WV _ 7812 a&txPf 4- 8544 /> J cV + 3075 -6^y - 3024 a&ctPe - 16650 6W/' H- 9120 2 6cZV + 457 S oJ'rf'a + 720 6 3 cW + 16350 ftV<J/ a - 324 (tbo s f' + 972 WC&B - 19200 aW/ -f 3b88 6c 4 rfe/ + 24048 W + 4800 Vrf*e/ - 8748 6c 4 fi s - 4464 fiVff/ 1 + 4860 a a c a rfe :i - 4800 6c :1 (/ ;1 /' - 15984 6W/ 1 - 3240 aW/ + 4248 flfiflWe" - 30108 6 2 cW - 8100 V(/V -f- 14520 abcWe + 35088 6-c-W ;! c + 9000 tPecPc -11448 aicrf" - 8640 W - 2400 a rf- + 2592 oV - 7776 atftPf + 5184 j atMe* + 12960 ] fluWe - 14400 I fflc 3 ^ + 3840 + 7S 3258 +41253 124716 [The values thus are fa (at, y) = lGBJ-UJ}G; fa fa $ = S'.] 2S9. The equation in g is of the form + 68640 where D is the discriminant of the quintic and 91, 53, <, 2) denote rational and integral functions of the coefficients (a, 1>, c, d, a, /). And the covariants fa fa y), fa fa y\ fa fa y), fa fa y) having the values given to them above, the actual value of 21 is obtained as a quadric function of the indeterminates (t t u, v, w) t viz. this is - [D,/ 3 - QBDto - D (A - IQAty v*-] +D[~ JS a + W,uw + Q(B$~ IQADJ v^ where _D,= 25^5 + IGG, these quantities, and the quantity _A r (= Df-lQABfy afterwards spoken of, being in the notation of the present Memoir as follows; A = J (== G}> = OZ D = D A = 9 (16Z - JK\ N = 1152 (18Z 9 - JKL ~ 10). 405] AN EIGHTH MEMOIR ON QUANTICS. 169 290. If by establishing two linear relations between the coefficients (t, it, v, w) the equation Sf = can be satisfied (which in fact can be done by the solution of a quaclric equation), then these quantities can be by means of the relations in question expressed as linear functions of any two of them, say of v and w\ and then the next coefficient 33 will be a cubic function (v, wj 1 , and the equation 33 = will be satisfied by means of a cubic equation (v, wj l = 0, that is, the transformed equation in z can be by means of the solution of a quadrie and a cubic equation reduced to the trinomial form ( 3) and M. Hermito shows that the equation [ = can be satisfied as above very simply,, and that in two different ways, viz. 291. 1. 21 = if Dtf - QBDtv - ( A - 104fl ) v n * = 0, Btt* - W t uw - (9BD - IQADJ itf = 0, that ifi, N denoting as above, if 292. 2. Writing the expression for 9[ in the form A (i 3 - 7?w + Wuw - IQADuP) + ED (IQAv' 1 - Qtv -' + then 9( = 0, if i a - Du 2 + Wuw - WADi^ = 0, IQAv* - Gtv - u* + 9 Dw* = Q. These equations, writing therein - ' V2 become - f>A 7 2 + 8VOTF+ U 2 + 1QA UW+ (25^ 9 - 9) Tf = 0, the first of which is satisfied by the values T = pW~U, F=pF + ij7; and then substituting for T and V, the second equation will be also satisfied if only a = BA O. VI. , 22 AN EIGHTH MEAIOIR ON QUANTICS. [405 Article Nos. 29S to 295. HERMITE'S application of the foregoing results to the deter- mination of the Character of the qitintic equation. 293. By considerations relating to the form D (D, - WAS) iff] + D[~ Bit + W t uw + QBD M. Hermite obtains criteria for the character of the quintic equation /(JB, 1) = 0. 294. If D = - , the character is 3r + 2i, but if D = + , then expressing the fore- going form as a sum of four squares affected with positive or negative coefficients, the character will be 5r or 2 + 4i, according as the coefficients are all positive, or are two positive and two negative. Whence, if N denote as above, then for D = + , JV=-, A = + , 5 = -, character is 5?-, J> + , jp-^-, -BA 1-> anc ' r character is r + 4ij J) = + ,JY = + j and further, the combination = + , N = -, ]) l = - ) = + cannot arise (Hermite's first set of criteria), 295. Again, from the equivalent form which, if <a, of are the roots of the equation 90 a - 1040 + .D = 0, is then by similar reasoning it is concluded that JJ = + t 25 A* 9.Z) = + , A = , N~~ t character is 5r, (Hermite's second set of criteria.) 405] AN EIGHTH MEMOIR ON QUANTICS. 17 1 Article Nos. 296 to 303. Comparison with the Criteria No. 283: the Nodal Cubic. 296, Tor the discussion of Hermite's results, it is to be observed that in the notation of the present Memoir we have A = J, B -If = A = UL - JK = -fa (2" L - J 3 + JD), N = 1&&-JKL-IP * foa OS! 7" 3 I A TT ( Ta T\\ f Tn or, putting as above, " -Li J' -is i ji e i i " -" i L/ JP .. ^ ^ -=- j aiiu tiiiereiore i 4- is ~~y^~ > * 2/ fs i/ / <y j we havo A = J, = ,7 . {T/" - 3?/ a + Bay 4 9# 2 + HT/ + 100). It thus becomes necessary to consider the curve <$> (a), y) = y 3 - &f + tooy + 9* 3 + lly + lOai = 0, the equation whereof may also be written -i- 5 = (y - 1) V25~^9y. 297. This is a cubic curve, viz. ib is a divergent parabola having lino 9; + 4y + 6 0, and its ordinates parallel to the axis of a node at the point #--1, y- + l, that is, at the node curve is thus a nodal cubic; we may trace it directly fro to be noticed that qti& nodal cubic it ' is a unicursal curve therefore rationally expressible in terms of a parameter ^; we in fact have 81(0 + 1)= ^(^-8), whence also 172 AN EIGHTH MEMOIR ON QUANTICS. [405 298. We see that \]r = <x>, gives x = co , (/ co , point at infinity, the direction of the curve parallel to axis of <v. ^=9, a; 0, #=0, the origin, A/r = S, a' = 1, t/ = -fl ) the node, tangent parallel to axis of ?/. ^ -tfi =$&7> y = 2 $?~> tangent parallel to the axis of y, ^ = 4, # JgijS, y = %-, tangent parallel to axis of ss. ^p 0, (i; = 1, ^ = -4-1, the node. i/r = -16, .i; = -7Gff, y = -41f, the cusp of the bicorn. A^ = GO, i = co, i/ = GO, point at infinity, direction of curve parallel to axis of IK. 299, The Nodal Cubic is shown along with the Bicorn, Plate, fig, 2; it consists of one continuous line, passing from a point at infinity, through the cusp of the bicorn, on to the node-cusp, then forming a loop so as to return to the node-cusp, again meeting the bicorn at the origin, and finally passing off to a point at infinity, the initial and ultimate directions of the curve being parallel to the axis of at. 300. It may be remarked that, inasmuch as one of the branches of the cubic touches the bicorn at the node-cusp, the node-cusp counts as (4 + 2=) 6 intersections; the intersections of the cubic with the bicorn are therefore the cusp, the node-cusp, and the origin, counting together as (2 -f 6 -f 1 =) 9 intersections, and besides these the point at infinity on the axis of a, counting as 3 intersections. This may be verified by substituting in the equation of the cubic the bicorn 0-values of a and y. But to include all the proper factors, we must first write the equation of the cubic in the homogeneous form (9w + 8y + 52) 3 z - (y - *) a (25* - 9y) = 0, and herein substitute the values : y i * = -(0 the result is found to be 6< - 9)* - (20 + 3) a (4f + 40 a -I- 180 + 27)} = 0, that is and considering this as an equation of the order 12, the roots are = 0, 3 times, = 2 J 1 time; = -, 2 times, and = oo, 6 times, 301, The cubic curve divides the plane into 3 regions, which may be called respectively the loop, the antiloop, and the extra cubic; for a point within the loop or antiloop, ^(a t y) is = -, for a point in the extra cubic $(&, y} is ==+. If in conjunction with the cubic we consider the discriminatrix, or line y~Q, then we have in all six regions, viz. y being = +, three which may be called the loop, the triangle, 405] AN EIGHTH MEMOIR ON QUANTICS. 173 and the upper region; and y being =*-, three which may be called the right, left, and under regions respectively; the triangle and the other region form together the antiloop. 302. It is now easy to discuss Hoi-mite's two sets of criteria; the 6rsfc set becomes == -\ = - Jas + =+, t(. !/) = -> character or, , , . ,. character - + 4u ?/ = +, tf-l = +, J( + ff) = -' f(^2/) = - cannot exist. Referring to the Plate, fig. 4, which 'shows a portion of the cubic and the bicorn, then 1 the conditions ?/ = +,f (to, y} = - imply that the point (a>, y)_is within the loop or within the triangle of the cubic; the condition y-l=- brings it to be within the triangle, and for any point within the triangle we have + # = -, whence also the condition J> + ff)- + becomes J=-J hence the conditions amount to / = - (to ) within the triangle; but by the general theory (ta t y), being within the triangle, that is, in the region P or T, if /=-, will of necessity be within the region P; so that the conditions give J- , (, y) within the region P; the corresponding character being 5r, which is right. 2 y = + ty (a y) = - t the point (x> </) must be within the loop, or within the triangle; if (*, y} is within the loop, then y-l + , + yl, and the condition J(tf-l)(+y) = + becomes J--, that is, we have J = - and (, y) within the loop, that is, in the region T. And again, if (w, y] be within the triangle, then y-l=-, *, + +, and the condition J(y- 1) (0 + ff) =+ still gives /-; but /=-, and <, ) -within the triangle, that is, in the region T or P, will of necessity be in the region T;' so that in either case we have /=-, (, y) ^ the region T, which agrees with the character r + 4i. 3 +, t(*. 2/) ==+ > (" *) is in the Upper regi n| - or 2'; if (a, y} is in the region Q, then of necessity / = -, and if in the region 1, then of necessity /== + , that is, we have /=-, (a, y} in the region Q, or /=+, (aj, y) in the region P, which agrees with the character r + 4i. And it is to be observed that the portions of T under 2 and 3' respectively make up the whole 'of the region T, and that 3 relates to the whole of the region & so that the conditions allow the point (a, y} to be anywhere in Q or A which is right. ysa+i ^( Xi y ) = _, (a, y) is in the loop or the triangle, and then 2/-l = + sai i , , implies that it is in the loop, whence a> + y = + , and the condition J (* becomes / = -; we should therefore if .the combination existed have J=-, (*, y) within the loop, that is, in the region T; but this is impossible. AN EIGHTH MEMOIR ON QUANTICS. [405 303. Hermite's second set of criteria are y-+> --2/ = +> </=-, -$(%>!/} = -, character or, y = +, ^-y = +, J=-, ty(x,y) = + } y = +, 2g.~y = +, J=+ t - character r -f 4i. y = +J ^f-y=-, J 1. If y=+ t -$*(&; y} = -, then the point (a?, y) must be situate within the loop or within the triangle; and recollecting that at the highest point of the loop we have y = a p t the condition ^-^s=+ is satisfied for every such point, and may therefore be omitted. The conditions therefore are /=-, (a?, y) within the loop, that is, in the region T t or within the triangle, that is, in the region P or the region T\ but for any point of T the general theory gives J=+, and the conditions are therefore ,7=-, (flf, y) within the region P \ which agrees with the character or. 2 - 2/ = +. if 1 OB, #) = +> that is, (IB, y) is within the upper region, that is, in the region Q or T \ and tyi_0 = +, (., y ) ^ill be within the portions of Q and 2' which lie beneath the line y = ^-; but J=~, and therefore (0, y) cannot lie in the region T\ hence the conditions amount to /=-, (#, y) within that portion which lies beneath the line y ~ *$. of the region Q. _ 3 - ff = +, ^-y = +, (, y) lies beneath the line y = ^, viz. in one of the regions P, Q or T; but J= + , (x, y) oannot lie in the region P or Q; hence the con- ditions give /=+, (as, T/) within the portion which lies beneath the line y = fy of the region T. 4 - 2/ = +. -?-- ttat is, (a;, ?y) lies above the line y = ^, and therefore in one of the regions T or Q; and by the general theory, according as (a;, y) lies in T or in Q, we shall have /= + or J=~, hence the conditions give J = t ( X} T/) within the portion which lies above the line y = ty, of the region Q, J = -f, (a;, y} within the portion which lies above the line 2/ = ^f, of the region T, 2 a , 3 D , and 4, each of them agree with the character r + 4i, and together they imply J"==-^(<e, y) anywhere in the region Q, or else J=+, (#, T/) anywhere in the region T-, which is right. Article Nos, 304 to 307". HERMITE'S third set of Criteria; comparison with No. 283, and remarks. 304. In the concluding portion of his memoir, M, Hermits obtains a third set of criteria for the character of a quintic equation ; this is found by means of the equation for the function 4D (0 a - 00 W. - 00 (04 - 0,) of the roots (# fl , lt 6 B , 8 S> 4 ) of the given quintic equation (a, b, o, d, e, /0, 1) B = 0. The function in question has 12 pairs of equal and opposite values, or it is determined 405] AN EIGHTH MEMOIR ON QUANTIGS. 175 by an equation of the form (tt a , 1) 13 =0, which equation is decomposable, not rationally but by the adjunction thereto of the square root of the discriminant, into two equations of the form ( 3 , 1) G = ; viz. one of these is + 10 (a+3 + " [i(a-VA) 9 +A] -u a d + w* [I (a 4 VA> + A] A + a (a-3V'A)A 3 + A 3 = 0, and the other is of course derived from it by reversing the sign of VA, I have in the equation written (a, d) instead of Hcrmite's writing capitals A, D ; the sign of the term in u 9 instead of +, as printed in his memoir, is a correction communicated to me by himself. The signification of the symbols is in the author's notation a = 5V1, d 4. 6" (.4.0-^ A), A = 6>0, whence, in the notation of the present memoir, the expressions of these symbols are a - o 4 /, A = 5"D. 305. From the equation in u t taking therein the radical VA as positive, M. Hermits obtains (d<0 a mistake for d>0) the following as the necessary and sufficient con- ditions for the reality of all the roots, A = +, a 4- 3 VA = -, d = +, character 5r (Hermite's third set of criteria). 306. It is clear that a+3VA = - is equivalent to (a = - and '" we have a 3 - 9A = 6" (125^-9.0), so that these conditions for the chi Now, writing as above, & 2"i-/ 3 D these are i/ = +, J = ~, ty* -&=>+> -2/ = -; the conditions </ (, y) is in the region P or the region Q ; and the condition 176 AN EIC4HTH MEMOIR ON QUANTICS. [405 line # y lies between the lines # + ?/ = 0, a 2?/ = 0, and so does not cut either the region P or the region Q) restricts (tc, y) to the region P ; and for every point of P y is at most =1, and the condition IJA y = + is of course satisfied. The con- dition, 125 J a 9D = +, is thus wholly unnecessary, and omitting it, the conditions are = 0, character 5r, which, being an admissible value of /*, agrees with the result ante, No. 283. 307. It may be remarked in passing that if 12345 is a function of the roots (ft 1 ,, #, a' 3l A' 4) d' B ) of a qiuntic equation, which function is such that it remains unaltered by the cyclical permutation 12345 into 23451, and also by the reversal (12345 into 15432) of the order of the roots, so that the function has in fact the 12 values a, = 12345, # = 24135, cr a = 13425, ft = 32145, a 3 = 14235, # = 43125, 4 = 21435, # = 13245, a B = 31245, #=14325, /= 41325, # = 12435, then <$ (a t /3) being any unsym metrical function of (a, /3), the equation having for its roots the six values of <(, j8) (viz. <(a u #), 0(o a , #)...< ( BJ #)) can be expressed rationally in terms of the coefficients of the given quintic equation and of the square root of the discriminant of this equation. In fact, v being arbitrary, write , i-IIa {*-(, 0)1, Jf=II 8 [<;-(# a)}, then the interchange of any two roots of the quintic produces merely an interchange of the quantities L, M\ that is^ L + M and (Z -Jlf ) + &(&' * a , # 3 , es tt a' n ) are each of them unaltered by the interchange of any two roots, and are consequently expressible as rational functions of the coefficients; or observing that $(aj lt K. I} # 3 , es it a; B ) is a multiple of VD, we have L a function of the form P-fQVI)'; the equation i = 0, the roots whereof are v = < (a, t #) ...v = ^ (a a , #), is consequently an equation of the form P + Q*/D = Q > viz. it is a sextic equation (*Jw, 1) B = 0, the coefficients of \vhich are functions of the form in question. Hence in particular i(? = 12345 = (a, - a; a ) a (x, - to,)* fa - a>$ fa - fl ) 3 (d? B - ^) a is determined as above by an equation (*][?, 1) G = 0. Another instance of such an equation is given by my memoir " On a New Auxiliary Equation in the Theory of Equations of the Fifth Order," Phil, Tmns, vol. OLT. (1861), pp. 263276, [268], 405] AN EIGHTH MEMOIR ON QUANTICS. 177 Article Nos. SOS to 317. HEEMITE'S Ganonioal form of the qirintic. 308. It was remarked that M. Hermite's investigations are conducted by means of a canonical form, viz. if A(=J, =G) be the quartinvariant of the given quintic (a, I, c, d, e, f$as, yf, then he in fact finds (X, F) linear functions of (a 1 , y) such that we have (a, 6, c, d, e, ffa, y)* = (X, & Vfc, \'Te, p', V,Y, 7) (viz. in the transformed form the two mean coefficients are equal; this is a convenient assumption made in order to render the transformation completely definite, rather than an absolutely necessary one) ; and where moreover the quadricovariant E of the trans- formed form is or, what is the same thing, the coefficients (\, /t, V&, V&, /*', \') of the transformed form are connected by the relations V - V Vft + 3ft = 0, \> - V Vft + 3ft = 0, the advantage is a great simplicity in the forms of the several oo variants, which simplicity arises in a great measure from the existence of the very simple covarianfc operator ~ . ~ r (viz, operating therewith on any oo variant we obtain again a covariant). rt-A d j. 309. Reversing the order of the several steps, the theory of M. Hermite's trans- formation may be established as follows : Starting from the quintic (a, b t o, d, e, and considering the quadricovariant thereof ((a, /3, 7) are of the degree 2), and also the linear covariaut Pa + Qy , Q) are of the degree 5), we have and moreover (, A 75ft --P)' V* viz. the expression on the left hand, which is of the degree 12, and wind an invariant, is = 0, where G is (ut suprti) C. VI. AN EIGHTH MEMOIR ON QU ANTICS. [4C The Jncobian Df the two forms, viz. P , aQ-0P) + is a linear covariant of the degree 7, say it is and it is to be observed that the detenninaub PQ'-P'Q of the two linear forms --2 (a, A 7$Q, -P) 3 , that is, it is = 2(7. 310. Hence writing whence also the determinant of substitution from (X, Y) to (T, U) is =2, that from (2\ (a;, y) is T7t%0 t = , and consequently that from (X, Y.) to (ai, y) is -1. We have .42 12 - E7 S = {(/3 3 - 407) (Pw + Q#) 2 - (Fat + Q'j/) 3 } ; 1 <v or putting for P', Q' their values, this is ~ -^ into 4 (a, /3, y^Q, - P) 2 (a* 3 + 2/3a;r/ + that is, we have A 2* a J7 fl = s# a + jSwy + 7#' 2 j and we have also AT 1 - tf 3 = i VI [(,Y + F) 2 - (A" - F)] = VI AT, consequently a a + /3a;y +>yy* = A 2" J ~-U z = ^AX F. 311. We have 405] AN EIGHTH MEMOIR ON QUA.NTICS. 179 so that, pausing a moment to consider the transformation from (a, y) to (T, If), we have (a, b, e, d, e,f~$a, t y}^ ~^(a t b, c, d, e,flQ'T-QU, - ' , c - d . e - f y > ^) r ' suppose, where (a, b, c, d, e, f) are invariants, of the degrees 36, 34, 32, 30, 28, 26 respectively;, it follows that b, d, f each of them contain as a factor the 18-fchio invariant J, the remaining 1 factors being of the orders 1G, 12, 8 respectively. 312. That (a, b, c, d, e, f) are invariants is almost self-evident ; it may however be demonstrated as follows. Writing [$} 9t + 263 ft + 3c9 rf + 4d9(, + 5e9/, = 8 suppose, {aS) v } = 569 + 4c9 ft + Md c + W d + fd c , = 8, then Ptc + Qy, P'<B+ Q'y being oo variants, we have SP = 0, 8Q*=P, 87 J ' = 0, SQ' = P', whence, treating T, U as constants, 8 (Q'T -QU) = FT -PU,B(- P'T +PU) = Q. Hence S(a, b, G, d, e, = 5 (a, 6, c, d, e&Q'T-QU, - P'T+ PU? .(-P'T+PU) + 6(0,6,0, d, J ) 4 -( P'T-PU) + 5 (6, c, d, e, /J Y 0, the three lines arising from the operation with S on the coefficients (a, b, c, d, e, f) and on the facients Q'T QU and -P'T + PU respectively; the third line vanishes of itself, and the other two destroy each other, that is, 8 (a, I, c, d, ifUQ'T-QU, - PT + P /)* = (), and similarly 8, (a, 6, c, d, e,fHQW-QU, -PT or the function (a, I, c, d, e,f&Q'T~QU, -PT + PU)*, treating therein T and U as constants, is an invariant, that is, the coefficients of the several terms thereof are all invariants. 813. The expressions for the coefficients (a, b, c, cl, e, f) are in the first obtained in the forms a= 2(i +6^ e- f B - 2 (i 7 - SJI/'O) ^l" 1 180 AN EIGHTH MEMOIR ON QU ANTICS, where, developing SI. Hermite'a expressions, [405 ni = 24J/= 24' = 3JJf = A'B + 1 /V/ - 1 ,U?7 + 1 7+1 A 6 C' 2 + 1 A 3 C - 1 (77 + 6 A 5 JF + 6 A-B^ - 3 A 4 BG - 24 jt^C' + 12 A 3 & + 9 s + 24 X 3 C a - 39 A-JPG + 9 ABC 1 + 108 C 11 + 72 and substituting these values, we find 3Ga = 36b = 36c = 36d = 36o = 30f= A J H + 1 /I'J/ - 3 A S J1 + 1 ABI - 3 A*JJ + 1 7J/ - 3 A B C* + 1 4C/ - 24 A S C + 1 C 1 /" - 12 A*G -i- 1 ^l ^ 3 + 6 yi J 5 3 + G ^i 8 JS* + 6 A*J3C - 39 ^5(7- 27 A'JIC -- 1C A*& + 9 yl a j5 3 + 9 yl/J s + 9 vi 3 ^ - 54 yPC 2 - 42 ylC a - 30 A*J3C - 36 JIG* + 144 WG + 36 ^l^C 2 + 288 C 3 + 1152 I have not thought it worth while to make in these formulae the substitutions A ~ J, B=^-K, C-SL + JK, which would give the expressions for (a, b, c, d, e, f) in terms of J, If, L. 314. Substituting for (#, i/) their values in terms of (X, Y), we have (a, 6, c, d, e, /$>, y? and by what precedes this gives and thence v, v, 7) s suppose, = 120-4 405] AN EIGHTH MEMOIR ON QUANTICS. 181 the left-hand side is a linear covariant of the degree 5, it is consequently a mere numerical multiple of P + Qy, and it is easy to verify that it is = 120 (Pa + Qy). (In fact writing b=d = e = Q, the expression is (3<fld tf s afdjfi^ (aa? + lOcafy 2 -\-ff), and the only term which contains x is a a / 3 .3i / s 9 a! 3 ,10ca? J y 3 = 120 a c/' a .iB ; but for &=tZ=e = 0, Table J gives Px = a?cf z , and the coefficient 120 is thus verified.) But Pa + Qy VO V(7 is -j-^r (X + Y\ and wo have thus Av Av f = -:-=.--, whence not only v = v' ) v/e v -4 v ^ 6' suppose, but we have further fc = -TT=^, a result given by M, Hermite. V A 6 315. Substituting for v v f the value V/j, we have (a, 6, c, d, e, /$>, ?/) fl * and we havo then aaP + fSwy + 72^ "JAXY, viz, the left-hand side being the quadri- covariant of (tt, &, Cj d, e, / jfls, 2/) B j the equation shows that fche quadricovariant of the form (\ /j,, V&, V, ^', X'JX, F) 5 is =^/A.XY, and we thus arrive at the starting-point of Hermite's theory. 316, The coefficients (\ /t, V&, ^, /i', X') of Hermite's form are by what precedes invariants; they are consequently expressible in terms of the invariants A, B, G (and I). M", Hermite writes XV==f/ f w' = h, and he finds or, what is the same thing, SAB+O , _AB + , G ' ~' ' which give g, h, k in terms of A, B, G, and then putting A = (9/c 2 + Uhk - ghy - 24Aft>, == -^ (the equation J B = X 7 A is in fact equivalent to the before-mentioned expression of 1 182 AN EIGHTH MEMOIR ON QUANTICS. [405 in terms of the other invariants), the coefficients (X, /*, //, X') are expressed in terms of y, h, k, that is of A, E, G, viz. we have 72 24 18*0" - 9A (.7 + 1GA) + (0 - IGfc) 1 6A A - /i - VA, these values of (X, p, ///, X') could of course be at once expressed in terms of (./, K, L) t but I have not thought it necessary to make the transformation. 317. It has been already noticed that the linear covariant (0, = P#-hQ?/), was = VI (Vifc, \lltQX, 7), it is to be added that the septic covariant (P'as + Q'y) is = VJ'(V, -V/^Y, 7), and that the canonical forms of the cubicovariants ^i (,*, y), &c. are as follows : ^(X, 7) =\'A(^ SVA, SVfc, XP'. 37. ^(Z, 7) = ^(^, V, - Jk.-pfS.X, Y)\ (X, Y)} = A> fa - 3 VE, 3 V& ( - X5^, r), -vo> 3( V& - 3( V& 5/i, V&, + 3 (8 VI" Vfc - 96 ( VP + /*/ Vft - 2/i'fe )), or, as the last formula may also be written, (X, Y) = A/I f (tfff - 53/i + HO/0 /* - 64V V/c),^ 405] AN EIGHTH MEMOIR ON QUAXTICS. 183 It is in fact by means of these comparatively simple canonical expressions that M. Hennito was enabled to effect the calculation of the coefficient VI. Article Nos. 318 to 326, Theory of the imaginary linear transformations which lead to a, real equation. 318. An equation (a, b, c, .-][#, y) n = is real if the ratios a : b : c, &c. of the coefficients are all real. In speaking of a given real equation there is no loss of generality in assuming that the coefficients (a, b, c, .,.) are all real; but if an equation presents itself in the form (a, 6, c, ...][#, y) n ~Q with imaginary coefficients, it is to be borne in mind that the equation may still be real ; viz. the coefficients may contain an imaginary common factor in such wise that throwing this out we obtain an equation with real coefficients, In what follows I use the term transformation to signify a linear transformation, and speak of equations connected by a linear transformation as derivable from each other. An imaginary transformation will in general convert a real into an imaginary equation ; and if the proposition were true universally, viz. if it were true that the transformed equation was always imaginary it would follow that a real equation derivable from a given real equation could then be derivable from it only by a real transfor- mation, and that the two equations would have the same character. But any two equations having the same absolute invariants are derivable from each other, the two real equations would therefore be derivable from each other by a real transformation, and would thus have the same character; that is, all the equations (if any) belonging to a given system of values of the absolute invariants would have a determinate character, and the absolute invariants would form a system of auxiUai's. But it is not true that the imaginary transformation leads always to an imaginary equation ; to take the simplest case of exception, if the given real equation contains only even powers or only odd powers of so, then the imaginary transformation a : y into ia : y gives a real equation, And we are thus led to inquire in what cases an imaginary transformation gives a real equation. 319. I consider the imaginary transformation a : y into (a + W) + (o + (M)y : (e +/) + (g 4- hi) y, or, what is the same thing, I write co = ( + U) X + (c 4- rfi) Y, and I seek to find P, Q real quantities such that Pu+Qy may be transformed into a linear function RX + 8Y t wherein the ratio E : S is real, or, what is the same thing, such that RX + SY may be the product of an imaginary constant into a real linear function of (X, Y). This will be the case if 184 AN EIGHTH MEMOIR ON QTJANTICS. [405 that is if which implies the relations 01; what is the same thing, and if the resulting value of P : Q be real, the last-mentioned equations give (ctff - ce) & - (ah + bg- e/- de) + bh-df= 0, and being known, the ratio P ; Q is determined rationally in terms of 6. 320. The equation in & will have its roots real, equal, or imaginary, according' sis (ah + bg-of- c?e) a - 4 (ag - ce} (bh - df), that is ct'fc 8 + W(f + tff* + d*tf - Zahbff - 2oAo/- Wide - 2bgof + kadfyh 4- is =4-, =0, or =-.; and I say that the transformation is subiinaginary, neutral, and superi magi nary in these three cases respectively. In the subimaginary case there are two functions P&s + Qy which satisfy the prescribed conditions ; in the neutral case a single function ; in the superim aginary case no such function. But in the last-mentioned case there are two conjugate imaginary functions, Pffl + Qy, which contain as factors thereof respectively two conjugate imaginary functions - UX + VT, 321. Hence replacing the original ce r y t X, Y by real linear functions thereof, the subirn aginary transformation is reduced to the transformation cs \ y into JeX : K, where k is imaginary ; and the superimaginary transformation is reduced to at + iy : co iy into k(X+iY) : (X iY}> where k is imaginary. As regards the neutral transformation, it appears that this is equivalent to cs = (a + bi) X + (G + di) 7, 7, with the condition = (aK + &/) a 4>agbh, ~ (ah - l)g}\ that is, we have ah bg~Q, or without any real loss of generality ff = a, h = b, or the transformation is ss-(a + bi) X + (c + di) Y, 2/= (a+bi)7 t that is, K. : y**X+kY ; 7, where k is imaginary. 5 J AN EIGHTH MEMOIR ON QUANTIC3. of thf LJ he rigiUal eqUati n aftei ' aUy iml transformfltion ^ofi ^ till an equation (a, ...J^ #) = Q ; and if we consider first the neutral transformation, the transformed equation is (a, ...QX+kY, 3 7 ) = 0j tins is not a real equation except in tho case where k is iml. tn T ^ A th !i - aU P erii ; m inai T transformation, starting iu like manner from (a, ... &#, 2/) )l =0, this may be expressed in the form (a + /3i, y + Si, . . . , 7 _ S a _ ^ j ffl _ h ^ , G _ ^ n = ^ viz. when in a real equation (, y) we make the transformation a, : into a-' + iy : 0-iy, the coefficients of the transformed equation will form as above pairs of conjugate imaginaries. Proceeding in the last-mentioned equation to make the trans- formation +;/ - a-iy into k(X+iY) : X-iY, I throw k into the form cos 20 + i sin 2<, = (cos <f> + i sin 0) - (cos ^ - i sin <^) (of course it is not here assumed that 4, is real), or represent the transformation as, that of + ty : - *y into (cos + ,' s in <,!,} (A' + iY) : (co S - i sin </,) (,Y - i Y) ; the trans- formed equation thus is +iy), (ooa ^-4' sin The left-hand side consists of terms such as (A' 2 -f F)-2* into (y + K) (cos ^ + i sin s # (,Y + iY) + ( 7 _ Si) ( cos ^ viz. the expression last written down is = (7 cos s0 - S sin s <) ({ jf + 1 F) s -I- ( A r - i sin ^ + 8 cos and observing that tho expressions in [ ) are real, tho transformed equation is only real if ( 7 cos_^ - 8 sin . 8 & + ( 7 sill S{/> + fi oos s</)) bo realj that - S| - n oi , der fchafc fchfl ^ formed equation may be real, we must have tan ^= real; and observing that if tansd, be equal to any given real quantity whatever, then the values of tan A are all of them real, and that tan A real gives cos A and sin A each of them real, and therefore also A. real, it appears that the transformed equation is only real for the brans formation + ;r) : (cos A - i sin A) (X - iT), wherein ^ is real; and this is nothing else than the real transformation * : into- X OOB^ - 1 sin A X sin A + Y cos A. Hence neither in the case of the neutral trans- formation or in that of the superimaginary transformation cau we have an imaginary transformation leading to a real equation y ' VI - ' ' 24 186 AN EIGHTH MEMOIR ON QUANTICS. [405 324. There remains only the subimaginary transformation, viz. this has been reduced to at : y into kX : Y, the transformed equation is 17=0, and this wilt be a real equation if some power & of * (p not greater than ) be real, and if the equation (,..$*. 2/) = contain only terms wherem the index of a (or that of ) is a multiple of p. Assuming that it is the index of y winch is a multiple, the form of the equation is in fact (^, 2/*) w =0, (n-mp + a), and the transformed equation is ,Y(MP 7^ = 0, which is a real equation. 325 It is to be observed that if p be odd, then writing &=K (K real) and taking k 1 the real ji-th root of K, then the very same transformed equation would be obtained by the real transformation at : y into Jc'X : F; so that the equation obtained by the imaginary transformation, being also obtainable by a real transfor- mation, has the same character as the original equation. 326. Similarly if p be even, if K be real and positive, the equation &P = K has a real root jf which may be substituted for the imaginary k t and the transformed equation will have the same character as the original equation ; but if K be negative, say /f = -l (as may be assumed without loss of generality), then there is no real transformation equivalent to the imaginary transformation, and the equation given ^ by the imaginary transformation has nob of necessity the same character us the original equation; and there are in tact cases in which the character is altered. Thus if p = 2, and the original equation he x(tf, f) m = Q, or (a?, py=Q, then making the transfor- mation re : y into iX : F, the transformed equation will be X (X\ - Y*) m = or (X*, - P)" 1 = 0, giving imaginary roots X* + of* = corresponding to real roots s? - aif = 0. Article No. 327. Application to the availiars of a quintic, 327. Applying what precedes to a quiutic equation (a,. . . .J0, 2/) B =0> tu is after any real transformation whatever will assume the form (of,. . .Jo 7 , 2/') 6 = 0; f 1 ^ ne on ty cases in which we can have an imaginary transformation producing a real equation of an altered character is when this equation is (a', 0, o', 0, e' t OJJV, i/') 5==0 (cf not =0), or when it is (a 1 , 0, 0, 0, d, OjJV, yJ = Q, viz. when it is a/( a '' 4 + 1( Wy a + 5y) = 0, or *(oV 4 + 5e'y' 4 )=0, In the latter case the transformation to', y' into X\/ -l\Y gives the real equation Z(a'Jf 4 -5eT 4 )=0. I observe however that for the form 0, 0, 0, e', OJX 2/) 4 , and consequently for the form (a, ... $<B, i/) B from which it is derived we have J = ; this cose is therefore excluded from consideration, The remaining case is (a', 0, o', 0, e', 0][a)', y 1 } 6 = 0, which is by the imaginary transformation of : y' into iX : Y converted into (a', 0, -c', 0, e', OjZ, 7) 5 = 0; for the first of the two forms we have J' = 16a'cV a ) and for the second of the two forms J"= 16a'oV 9 , that is, the two values of J have opposite signs. Hence considering an equation (a, I, c, d, e, /}[ffi, /)" = for which J is not = 0, whenever this is by an imaginary transformation converted into a real equation, the sign of J is reversed ; and ifc follows that, given the values of the absolute invariants and the value of J (or what is sufficient, the sign of J), the 405] AN EIGHTH MEMOIR ON QUANTICS. 187 different real equations which correspond to these data must be derivable one from another by real transformations, and must consequently have a determinate character ; that is, the Absolute Invariants, and J, constitute a system of auxiliars. ANNEX. Analytical Theorem in relation to a Smart/ Qituntw of any Order. The foregoing theory of the superimaginary transformation led me to a somewhat remarkable theorem. Take for example the function or, ns this may "be written, (a, b, c$ir, + k, 1 - Aw) 3 , ft or ( c, 26, 26, 2 - 2c, ft, - 26, a 26 c a 3 c, 26, oj 26, 2ft - 2c, - 26 1 ft, -26, c, then the determinant c, 26, a 26, 2ft - 2c, - 26 ft, 26, o is a product of linear functions of the coefficients (a, b, c); its value in fact is = - 2 (ft + c) (ft + 2W + ci a ) (a - 26i + a a ), = - 2 (a + c) [(ft - c) 2 + 46 9 ]. To prove bhis directly, I write a a ct' J , and \ve then, have c' re 4- ( c, 26, , (26, 2ft- 2c, - ( a, -26, c, 26, 26) 26, a 1, 2 , 1 2a-2c, -26 i, , -i ' ^V, fr , ~~ ' J\t , 1 n ? ^ ^9 o 9^1 n / ? 3 \ V J i t, t y, ^, u, ^t ) } \i t k, i> ) 2m' f 06', - 2-j'c' 2, , - 2i ft',, 26', c' :- 1, 2 , 1 242 188 AN EIGHTH MEMOIR ON QU ANTICS. whence observing that the determinants 1, 2 , 1 i\ -21* i a , " W j ^^ CH 2i, , -a &, - 2i a , i 2 1, 2 , i -are as 1 : -2, we have the required relation, c, 2ft, 26, 26 c = - 2'6V, = - 2 (a + c) {(a ~ c) 2 + 4i 3 }. It is to be remarked that the determinant 2 , 1 , -i , taken as the multiplier of c, 26, 26, 2a-2c, -26 a, - 26, c [405 is obtained by writing therein = 6 = C( =1; and multiplying the successive lines thereof by 1, #, ^ (1, ^ 1 are the reciprocals of the binomial coefficients 1, 2, 1), the proof is the same, and the multiplier is obtained in the like manner for a function of any order; thus for the cubic (a, fr, Cj dp + 0, 1 -&), = a.- 3 a; 2 iw 1 rf> Sc, -36, a 3c, -66-^-3c^ J 3ft -6c t 36 36, Srt-Gc, 86-St?, 3c a, 36, So, (2 the multiplier is obtained from the determinant by writing therein o = 6 = o = d=l > and multiplying the successive lines by 1, fa tf, $, viz. the multiplier is 1, 3, -3, 1 i, -*, -, i i\ ~P, i\ i? and the value of the determinant is found to be - 9 ((a 405] AN EIGHTH MEMOIB ON QUANTICS. 189 But the theory may be presented under a better form ; take for instance the cubic, vis;, writing '- and j for x and k respectively, we have (a, b, c, d$ky + lx, Iy - k) a i/ ... d, -3c, - 36, a 3c, - 66 + 3d, 3a - So, 36 -36, 3a-6c, 66 - 3d, 3c f a, 36, So, a bipartite cubic function (#][/<;, l) s (x> 2/) 3 ! an( ^ tne determinant formed out of the matrix is at once seen to be an invariant of this bipartite cubic function. Assume now that we have identically viz. this equation written under the equivalent form (a f , 6', o', d'\X, Y) 3 = (, 6, c, dQX + Y, i (X - F)) 3 , determines (a', b', G', d') as linear functions of (o, 6, c, d), it in fact gives of = (a, 6, c, dl, - i) 3 = a - 36t 4- 3ci fl - di 3 , &' = (, 6, c, c^l, - t) a (1, t) = a - 6i ci 2 4- dt 3 , o' = (a, 6, c, d$l, i) (1, *) a ~a4-' &i~ ci a di 3 , then observing that % + fo? i (fy - few) = ( *y) (T ^ + 0. we liave (a, 6, c, d$7c2/ 4- lx, Iy- kaf^ = (of, b', c', d'$ (a) 4- iy) (- ik + 1), $ (so - iy} (ilc + I)) 3 , and if in the expression on the right-hand side we make the linear transformations V2, - ik + 1 = k' V2, -iy~- iy' 2, ifc + i = - 2 f which are respectively of the determinant 4- 1, the transformed function is = (a', 6', c', d'&V, -iyy, that is, we have (a, b, c, djty + to, ^-feBy = (tt', 6', o', d'Jfe'a/, -2Y) 3 . 190 AN EIGHTH MEMOIR ON QU ANTICS. The last-mentioned function is [405 k'* I" . -36' d 1 and (from the invariantive property of the determinant) the original determinant in equal to the determinant of this new form, viz. we have - d t 3o, -36, a So, -Qb + 3d, 3a - -Gc, 36 -36, 8a- Gc, G&- -3d, 3c a, 3&, 8c, d [(a - 3c) 2 + (36 - rf) a ] [(a + c) 3 + (6 + rf)' J ] , which is the required theorem. And the theorem is thus exhibited in its fcrur connexion, as depending on the transformation (a, ...TJX y) n = (', ADDITION, Tift October, 1867. Since the present Memoir was written, there has appeared the valuable pnpor by MM, Clebsch and Gordan "Sulla rappresentazione tipica delle forme binarie," Anmdi di Matematica, fc, I. (18(i7) pp. 23 79, relating to the binary quinfcic and sextic. On reducing to the notation of the present memoir the formula 95 for the representation of the quintic in terms of the eovariants a, /3, which should give for (a, b, c, d, o, f) the values obtained ante, No. 312j I find a somewhat different system of values; viz. these are 36a = 36b = 36o = 36d = 36e = 35f A 7 + 1 */I 4 / - 1 A + 1 *A 3 I ~ I A S J1 + 1 *^' J / - 1 A 6 C + \ A*BI- 3 A B C + 1 ABI - 3 ^(7. + 1 ASf --3 A 6 JP + 6 *AGI + 24 jl-'J? 2 + 6 *CI + 12 A 3 JP + 6 A*SC - 39 A 3 $C - 27 / 9 B(7 - 15 A*B 3 4- 9 ^J 1 + 9 AJP + 9 A*C* - 54 ^(7 2 - 42 AC* ~ 30 A*JPC- 126 *AffC 90 '^C - 64 AJJC* + 288 -flC 1 '' +144 G a +1162 where I have distinguished with an asterisk the terms which have different coefficient-* in the two formulas, I cannot at present explain this discrepancy. Plate IH Mc/.Z. The lower cusp of (he Bicomis drawn out of its true fosilion, which ismuchfitt-theroff(dorigtfieasyji/j)Me.,l}iec0-6rcliiiates in fact are x=-76j$-, y^-<!4i (tlwco-ordiiudts of the. upper or nod&-cusp'beiny -1, 1. \* ,0.,-- Ccuyley's Papers. VI 406] 406. ON THE CUBVES WHICH SATISFY GIVEN CONDITIONS. [From the Philosophical Transactions of the Royal Society of London, vol. CLVIIL (for the year 1868), pp. 75143. Received April 18, Read May 2, 1867.] THE present Memoir relates to portions only of the subject of the curves which satisfy given conditions; but any other title would be too narrow: the question chiefly considered is that of finding the number of the curves which satisfy given conditions ; the curves are either curves of a determinate order r (and in this case the conditions chiefly considered are conditions of contact with a given curve), or elsi3 the curves are conies; and here (although the conditions chiefly considered are conditions of contact with a given curve or curves) it is necessary to consider more than in the former case the theory of conditions of any kind whatever. As regards the theoiy of conies, the Memoir is based upon the researches of Chasles and Zeuthen, as regards that of the curves of the order ?, upon the researches of De Jonquieres: the notion of the quasi-geometrical representation of conditions by ' means of loci in hyper-space ^ is employed by Salmon in his researches relating to the quadric surfaces which satisfy given conditions. The papers containing the researches referred to are included in the subjoined list. I reserve for a separate Second Memoir the application to the present question, of the Principle of Correspondence. List of Memoirs and Works relating to the Curves which satisfy given conditions, remarks. De Jonquieres: "The'oremes generaux concernant les courbes gdomdtriques planes d'un ordre quelconque," Liouv. t. vi. (1861), pp. 113134. In this valuable memoir is established the notion of a series of curves of the index JV; viz. considering the curves of the order n which satisfy in( + 3)-l conditions, then if N denotes how many there are of these curves which pass through a given arbitrary point, the series is said to be of the index N, In Lemma IV it is stated that all the curves G u of a series of the index JV can 192 ON THE CUBVES WHICH SATISFY GIVEN CONDITIONS. [406 be analytically represented by an equation F(y, &)=$, which is rational and integral of the decree N in regard to a variable parameter X : this is not the case ; see Annex No. 1. Chasles: Various papers in the Gomptes Rendw, t. LVlli. et seq. 1864 67, The first of them (Feb. 186*), entitled " Determination du nombre des sections coniques qui doivent toucher cinq courbea donnees d'ordre queloonque, on satisfaire a diverses autres conditions," establishes the notion of the two characteristics (/t, v) of a system of conies which satisfy four conditions ; viz. /t is the number of these conies which pass through a given arbitrary point, and v the number of them which touch a given arbitrary line. The Principle of Correspondence for points on a line is established in the paper of Jime July 1864. Many of the leading points of the theory are repro- duced in the present Memoir. The series of papers includes one on the conies in space which satisfy seven conditions (Sept. 1865), and another on the surfaces of the second order which satisfy eight conditions (Feb. I860). Salmon: "On some Points in the Theory of Elimination," Quart, Math. Journ. t. vn. pp. 327337 (Feb. 1866); "On the Number of Surfaces of the Second Degree which can be described to satisfy nine Conditions," Ibid. t. vm. pp. 1 7 (June 18(i6), which two papers are here referred to on account of the notion which they establish of the quasi-geometrical representation of conditions by means of loci in hyper-spaco. Zeuthen: Nyt Bidrag... Contribution to the Theory of Systems of Conies which satisfy four conditions, 8. pp. 1 97 (Copenhagen, Cohen, 1865), translated, with an addition, in the Nouvelles Annalss. The method employed depends on the determination of the line-pairs and point- pairs, and of the numerical coefficients by which these have to be multiplied, in the several systems of conies which satisfy four conditions of contact with a given curve or curves. It is reproduced in detail, with the enumeration called "Zeutheu's Capitals," in the present Memoir. Cayley: "Sur les coniques de'termine'es par cinq conditions d'intersection avec une courbs donne'e," Gomptes Ecndus, t. LXIII. pp. 912, July 18G6. Results reproduced in the present Memoir. De Jonquieres: Two papers, Gomptes Rendtis, t. LXIII. Sept. 186G, reproduced and further developed in the "Me'moire sur les contacts multiples d'oidre quelcouquo des courbes du degre* r qui satisfont a des conditions. clonne'es de contact avec une courbo fixe du degre' m; suivi de quelques reflexions sur la solution d'un grand nombre do questions concernant les proprie'te's projectives des courbes et des surfaces alge"bric|ues," Qrelle, t. LXVI. (1866), pp. 289 322, contain a general formula for the number of curves & having contacts of given orders a, 1), c, . , with a given curve U m , which formula is referred to and considered in the present Memoir, De Jonquieres: Recherches sur les aeries ou systemes de courbes et do surfaces algtSbriques d'ordre quelconque ; suivies d'une reponse &c. 4. Paris, Gauthier Villars, 1866 ('). 1 The foregoing list is not complete, ami the remarks aro not intended to give even a sketch of the eon- tents of the worts comprised therein, but only to show their bearing on the present Memoir. 406] ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 193 Article Nos. 1 to 23. On the quasi-geometrical representation of Conditions. 1. A condition imposed upon a subject gives rise to a relation between the parameters of the subject; for instance, the subject may be, as in the present Memoir, a plane curve of a given order, and the parameters be any arbitrary parameters con- tained in the equation of the curve. The condition may be onefold, twofold,... or, generally, k-fold, and the corresponding relation is onefold, twofold,... or A-fold accord- ingly. Two or more conditions, each of a given manifoldness, may be regarded as forming together a single condition of a higher manifoldness, and the corresponding relations as forming a single relation ; and thus, though it is often convenient to con- sider two or. more conditions or relations, this case is in fact included in that of a /c-ibld condition or relation. In dealing with such a condition or relation it is assumed that the number of parameters is at least =&; for otherwise there would not in general be any subject satisfying the condition; when the number of parameters is Ic, the number of subjects satisfying the condition is in general determinate. 2. A subject which satisfies a given condition may for shortness be termed a solution of the condition ; and in like manner any set of values of the parameters satisfying the corresponding relation may be termed a solution of the relation. Thus for a &-fold condition or' relation, and the same number k of parameters, tho number of solutions is in general determinate. 3. A condition may in some cases be satisfied in more than a single way, and if a certain way be regarded as the ordinary and proper one, then the others are special or improper: the two epithets may be used conjointly, or either of them separately, almost indifferently. For instance, the condition that a curve shall touch a given curve (have with it a two-pointic intersection) is satisfied if the curve have with the given curve a proper contact; or if it have on the given curve a node or a cusp (or, more specially, if it be or comprise as part of itself two coincident curves) ; or if it pass through a node or a cusp of the given curve : the first is regarded as the ordinary and proper way of satisfying the condition ; the other two as special or improper ways ; and the corresponding solutions are ordinary and proper solutions, or special or improper ones accordingly. This will be further explained in speaking of the locus which serves for the representation of a condition. 4. A set of any number, say <a, of parameters may be considered as the coordi- nates of a point in w-dimensional space ; and if the parameters are connected by a onefold, twofold,.,, or k-fold relation, then the point is situate on a onefold, twofold,.,, or Ar- fold locus accordingly ; to the relation made up .of two or more relations corresponds tho locus which is the intersection or common locus of the loci corresponding to the several component relations respectively. A locus is at most w-fold, viz. it is in this case a point-system. The relation made up of a- Mold relation, an 2-fold relation, &c., is in general (k + 1 4- fee.) fold, and the corresponding locus is (& + I + &c.) fold accordingly. 5. The order of a point-system is equal to the number of the points thereof, where, of course, coincident points have to be attended to, so that the distinct points of the system may have to be reckoned each its proper number of times. The locus c. VI. 25 194 ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. [406 corresponding to any linear j-fold relation between the coordinates is said to be a j-fold omal locus; and if to any given /c-fold relation we join an arbitrary (a -A) fold linear relation, that is, intersect the ft-fold locus by an arbitrary (w - k) fold omal locus, so as to obtain a point-system, the order of the &-fold relation or locus is taken to be equal to the number of points of the point-system, that is, to the order of the point- system. And this being so, if a ft-fold relation, an J-fold relation, &c. are completely independent/ that is, if they are not satisfied by values which satisfy a less than (k + l+ &c.) fold relation, or, what is the same thing, if the fc-fold locus, the Z-fold locus, &c., have no common less than (k + 1 + &c.) fold locus, then the relations make up together a (jfc + 1+ &c.) fold relation, and the loci intersect in a (k -\- 1 + &c.) fold locus, the orders whereof are respectively equal to the product of the orders of the given, relations or loci. In particular if we have A+Z + &C. o>, then we have an w-fold relation, and corresponding thereto a point-system, the orders whereof are respectively equal to the product of the orders of the given relations or loci. 6. A Mold relation, an Z-fold relation, &c., if they were together equivalent to a less than (k + 1 + &c.) fold relation, would not be independent; but the relations, assumed to be independent, may yet contain a less than (& + Z + &o.)fold relation, that is, they may be satisfied by the values which satisfy a certain less than (k + I-}- &c.) fold relation (say the common relation), and exclusively of these, only by the values which satisfy a proper (fc + 1 + &c.) fold relation, which is, so to speak, a residual equivalent of the given relations. This is more clearly seen in regard to the loci; the &-fold locus, the J-fold locus, &c. may have in common a less than (k + 1 + &o.) fold locus, and besides intersect in a residual (k 4- 1 + &o.) fold locus. (It is hardly necessary to remark that such a connexion between the relations is precisely what is excluded by the foregoing definition of complete independence.) In particular if & + J+&Q. = to, the several loci may intersect, say in an (a) j) fold locus, and besides in a residual w-fold locus, or point-system. The order (in 'any such case) of the residual relation or locus is equal to the product of the orders of the given relations or loci, less a reduction depending on the nature of the common relation or locus, the determination of the value of which reduction is often a complex and difficult problem, 7. Imagine a curve of given order, the equation of which contains w arbitrary parameters ; to fix the ideas, it may be assumed that these enter into the equation rationally, BO that the values of the parameters being given, the curve is uniquely .determined. Suppose, as above, that the parameters are taken to be the coordinates of a point in -dimensional space ; so long as the curve is not subjected to any condition, the poiut in question, say the parametric point, is an arbitrary point in the CD -dimensional space; bub if the curve be subjected to a onefold, twofold,,., or /c-fold condition, then we have a onefold, twofold,,., or Mold relation between the parameters, and the parametric point is situate on a onefold, twofold,... or &-fold locus accordingly: to each position of the parametric poiut on the locus there corresponds a curve satisfying the condition, that is, a solution of the condition. In the case where the condition is w-fold, the locus is a point-system, and corresponding to each point of the point-system we have a solution of the condition ; the number of solutions is equal to the number of points of the point-system. 406] ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 195 8. Considering the general case where the condition, and therefore also the locus, is Mold, it is to be observed that every solution whatever, and therefore each special solution (if any), corresponds to some point on the Mold locus; we may therefore have on the Mold locus what may be termed "special loci," viz. a special locus is a locus such that to each point thereof corresponds a special solution. A special locus may of course be a point-system, viz. there are in this case a determinate number of special solutions corresponding to the several points of this point-system. "We may consider the other extreme case of a special A-fold locus, viz. the Mold locus of the parametric point may break up into two distinct loci, the special Mold locus, and another Mold locus the several points whereof give the ordinary solutions: we can in this case get rid of the special solutions by attending exclusively to the last-mentioned Mold locus and regarding it as the proper locus of the parametric point. But if the special locus be a more than Mold locus, that is, if it be not a part of the Mold locus itself, but (as supposed in the first instance) a locus on this locus, then the special solutions cannot be thus got rid of: we have the Mold locus of the parametric point, a locus such that to every point thereof there corresponds a proper solution, save and except that to the points lying on the special locus there correspond special or improper solutions. It is to be noticed that the special locus may be, but that is not in every case, a singular locus on the Mold locus. 9. Suppose that the conditions to be satisfied by the curve are a Mold condition, an Z-fold condition, &c. of a total manifoldness =ro. If the conditions are completely independent (that is, if the corresponding relations, ante, No. 5, are completely indepen- dent), we have a &-fold locus, an /-fold locus, &c,, having no common locus other than the point-system of intersection, and the number of curves which satisfy the given conditions, or (as this has been before expressed) the number of solutions, is equal to the number of points of the point-system, or to the order of the point-system, viz. it is equal to the product of the orders of the loci which correspond' to the several con- ditions respectively; among these we may however have special solutions, corresponding to points situate on the special loci upon any of the given loci; but when this is the case the number of these special solutions can be separately calculated, and the number of proper -solutions is equal to the number obtained as above, less the number of the special solutions, 10. If, however, the given conditions are not completely independent (that is, if the corresponding relations are not completely independent), then the Mold locus, the 2-fold locus, &o, intersect in a common (o>-j)fold locus, and besides in a residual point-system. The several points of the (w - j) fold locus give special solutions in fact the very notion of the conditions being properly satisfied by a curve implies that the curve shall satisfy a true (k + 1 + &o.) fold, that is, a true co-fold condition ; the proper solutions are therefore comprised among the solutions given by the residual point- system, and the number of them is as before equal to the order of the point-system, or number of the points thereof, less the number of points which give special solutions: the order of the point-system is, as has been seen, equal to the product of the orders of the Mold locus, the Z-fold locua, &o,, less a reduction depending on the nature of 252 19G ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. [406 the common (a>-j)fold locus, and the difficulty is in general in the determination of the value of this reduction. 11. In all that precedes, the number of the parameters has been taken to bo o> ; but if the parameters are taken to be contained in the equation of the carve homo- geneously, then the parameters before made use of are in fact the ratios of these homogeneous- parameters ; and using the term henceforward as referring to the homo- geneous parameters, the numbers of the parameters will be =o + 1. 12. I assume also that the equation of the .curve contains the parameters linearly ; this being so, the condition that the curve shall pass through a given arbitrary point implies a linear relation between the parameters; and the condition that the curve shall pass through j given points, a j-fold linear relation between the parameters. It follows that the number of the curves which satisfy a given /c-fold condition, and besides pass through m k given points, is equal to the order of the ft-fold relation, or of the corresponding jfc-fold locus ; and thus if we define the order of the /c-fold condition to bo the number of the curves in question, the condition, relation, and locus will be all of the same order, and in all that precedes we may (in place of the order of the relation or of the locus) speak of the order of the condition. Thus, subject to the modifications occasioned by common loci and special solutions as above explained, the order of the (k -f I + &o.) fold condition made up of a ft- fold condition, an 2-fold condition, &c., is equal to the product of the orders of the component conditions; and in particular if k + 1 + &c. = a), then the order of the w-fold condition, or number of the solutions thereof, is equal to the product of the orders of the component conditions. 13. The conditions to be satisfied by the curve may be conditions of contact with a given curve or curves. In particular if the curve touch a given curve, the para- metric point is then situate on a onefold locus. It is to be noticed in reference horobo that if the given curve have nodes or cusps, then we have special solutions, viz. if the sought for curve passes through a node or a cusp of the given curve ; and each such node or cusp gives rise to a special onefold locus, presenting itself in the first instance as a factor of the onefold locus of the parametric point ; this is, however, a case where the special locus is of the same mauifoldness as the general . locus (ante, No. 8), and is consequently separable; throwing off therefore all these special loci, we have a onefold locus which no longer comprises the points which correspond to curves passing through a node or a cusp of the given curve; the onefold locus, so divested of the special onefold factors, may be termed the "contact-locus" of the given ourvo. To each point of the contact-locus there corresponds a curve having with the given curve a two-pointic intersection, viz. this is either a proper contact, or it is a special contact, consisting in that the sought for curve has on the given curve a node or cusp, or (which is a higher speciality) in that the sought for curve is or contains as part of itself two or more coincident curves (ante. Nn. RY Tn n. nnint. nnwil an 406 ] ON THE CURVES WHICH SATISFY GIVEN CONDITIONS, 197 the order r which touch a given curve of the order in and class n, the order of the contact-locus is = n 4- (2r - 2) m. 14. If, then, the curve touch a given curve, the parametric point is situate on the contact-locus of that curve. If it touch a second given curve, the parametric point is in like manner situate on the contact-locus of the second given curve, that is, it is situate on the twofold locus which is the intersection of the two contact-loci; and the like in the case of any number of contacts each with a distinct given curve. But if the curve, instead of ordinary contacts with distinct given curves, has either a contact of the second, or third, or any higher order, or has two or more ordinary or other contacts with the same given curve, then if the total manifoldncss be =k t the parametric point is situate on a A-fold loous, which is given as a singular locus of the proper kind on the onefold contact-locus ; so that the theory of the contact-locus corresponding to the case of a single contact with a given curve, contains in itself the theory of any system whatever of ordinary or other contacts with the same given curve, viz. the last-mentioned general wise depends on the discussion of the singular loci which lie on the contact- loons, And similarly, if the curve has any number of ordinary or other contacts with each of two or more given curves, wo have here to consider the inter- sections of singular loci lying on the contaot-loei which correspond to the several given curves respectively, or, what is the same thing, to the singular loci on the intersection of these contact-loci; that is, the theory depends on that of the contact-loci which belong to the given curves respectively. 15. Suppo.se that the curve which has to satisfy given conditions is a line; the equation is ats -\-by + cz Q, and the parameters (a, b, c) are to be taken as the coordinates of a point in a plane. Any onefold condition imposed upon the lino establishes a onefold relation between the coordinates (a, b, c), and the parametric point is situate on a curve; a second onefold condition imposed on the line establishes a second onefold relation between the coordinates (, 6, c), and the parametric point is thus situate on a second curve; it is therefore determined as a point of intersection of two ascertained curves. In particular if the condition imposed on the lino is that it shall touch a given curve, the locus of the parametric point is a curve, the con- tact-locus; (this is in fact the ordinary theory of geometrical reciprocity, the locus in question being the reciprocal of the given curve ;) and the caae of the twofold condition of a contact of the second order, or of two contacts, with the given curve, depends on the singular points of the contact-locus, or reciprocal of the given curve ; in fact according as the line has a contact of the second order, or has two contacts with the given curve (that is, as it is an inflexion- tangent, or a double tangent of the given curve), the parametric point is a cusp or a node on its locus, the reciprocal curve: this is of course a fundamental notion in the theory of reciprocity, and it is only noticed here in order to show the bearing of the remark (ante, No. 14) upon the case now in hand where the curve considered is a line. 16. If the curve which has to satisfy given conditions is a conic (a, b, o, f, ff, Jigai, y, *) = 0, we have here six parameters (ft, b t c, f, g, Ji) f which are taken as the coordinates of a 11)8 ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. [406 point in 5 -dimensional space. It may be remarked that in this 5-dimensional space we have the onefold cubic locus ale - a/ 3 - Iff - c# -t- 2/</A, = 0, which is such that to any position of the parametric point upon it there corresponds not a proper conic but a line-pair; this may be called the discriminant- locus. We have also the threefold locus the relation of which is expressed by the six equations (Jo_/ = 0, ca-(f = Q, ai-A a = 0, gh-af=Q, 1tf-lg = Q, fg- G h=*Q) t which is such that to any position of the parametric point thereon, there corresponds not a proper conic but a coincident line-pair. I call this the Bipoint-Iocus( 1 ), and I notice that its order is = 4 ; in fact to find the order we must with the equations of the Bipoint combine two arbitrary linear relations, (* ~$a, b, c, f, g, A) - 0, (*'$, 6, c,/ g, A)i=0; the equations of the locus are satisfied by a : & : c ; / : g : h = a a : {3* ; 7' : 7 ; yet : a/3 (where : /3 : 7 are arbitrary) ; and substituting these values in the linear relations, we have two quadric equations in (a, /?, 7), giving four values of the set of ratios (a : /3 : 7) ; that is, the order is = 4, or the Bipoint is a threefold quadric locus. 17. The discriminant-locus does not in general present itself except in questions where it is a condition that the conic shall have a node (reduce itself to a line-pair) ; thus for the conies which have a node and touch a given curve (?ft, n\ or, what is the eame thing, for the line-pairs which touch a given curve (m, n), the parametric point is here situate on a twofold locus, the intersection of the discriminant-locus with the con- tact-locus. It may be noticed that this twofold locus is of the order 3 (n + 2m), but that it breaks up into a twofold locus of the order 3?i, which gives the proper solutions; viz. the nodal conies which touch the given curve properly, that is, one of the two lines of the conic touches the curve ; and into a twice repeated twofold locus of the order 3m which gives the special solutions, viz. in these the nodal conio lias with the given curve a special contact, consisting in that the node or intersection of the two lines lies on the given curve. By way of illustration see Annex No, 2. But the con- sideration of the Bipoint-locus is more frequently necessary. 18. Suppose that the conic satisfies the condition of touching a given curve; the parametric point is then situate on a onefold contact-locus (a, &, c, /, g, A)* = (to fix the ideas, if the given curve is of the order m and class n, then the order q of the contact-locus is = ?i+ 2m). The contact-locus of any given, curve whatever passes through the Bipoint-locus; in fact to each point of the Bipoint-locus there corresponds a coincident line-pair, that is, a conic which (of course in a special sense) touches the given curve whatever it be; and not only so, but inasmuch as we have a special 1 In framing tho epithet Bipoint, the coincident Hue-pair is regarded as being really a point-pair: aeo jwit, No. 30. iii 406] ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 199 contact at each of the points of intersection of the given curve with the coincident line-pair regarded as a single line, that is, in the case of a given curve of the m-th order, m special contacts, the Bipoiut-locus is a multiple curve on the corresponding contact-locus. 19. If the conic has simply to touch a given curve of the order m^ and class n 1} then the order of the condition (or number of the conies which satisfy the condition, and besides pass through four given points) is equal to the order of the contact-locus, that is, it is =n 1 +2m 1 . If the conic has also to, touch a second given curve of the order m* and class a , then the order of the twofold condition (or number of the conies which satisfy the twofold condition, and besides pass through three given points) is equal to the order of the intersection or common locus of the two contact-loci; and these being of the orders w, + 2^ and 2 + 2m 3 respectively, the order of the intersection and therefore that of the twofold condition is =(n, + 2m 1 )(nj + 2m a ). But in the next succeeding case it becomes necessary to take account of the singular locus. 20. If the conic has to touch three given curves of the order and class (m^ %), (Mm a), (m a , ih) respectively, we have here three contact-loci of the orders nj^^m,, s+2ma, n 8 + 2m 8 respectively; these intersect in a threefold locus, but since each of the contact-loci passes through the threefold Bipoint-locus, this is part of the intersection of the three contact-loci; and not only so, but inasmuch as they pass through the Bipoint-locus 7^, m a , m, times respectively, the Bipoint-locus must be counted m 1 9n a m 3 times, and its order being =4, the intersection of the contact-locus is made up of the Bipoint reckoning as a threefold locus of the order ^TH^BO and of a residual three- fold locus of the order = iWas + 2 (njngWia + &c.) 4- 4t (?? a THjm, + &c.) and the order of the threefold condition (or number of the conies which touch the three given curves, and besides pass through two given points) is equal to the order of the residual threefold locus, and has therefore the value just mentioned. 21. In going on to the cases of the conies touching four or five given curves, the same principles are applicable; the contact-loci have the Bipoint (a certain number of times repeated) as a common threefold locus, and they besides intersect in a residual fourfold or (as the case is) fivefold locus, and the order of the condition is equal to the order of this residual locus; but the determination of 'the order of the residual locus presents the difficulties alluded to, ante, No. 10. I do not at present further examine these cases, nor the cases of the conies which have with a given curve or curves contacts of the second or any higher order, or more than a single contact with the same given curve, 22, The equation of the conic has been in all that precedes considered as con- taining the six parameters (a, b, o, f> g, Jt) ; but if the question as originally stated relates only to a class of conies the equation whereof contains linearly 2, 3, 4, or 5 parameters, or if, reducing the equation by means of any of the given conditions, it 200 ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. [406 can be brought to the form in question, then in the latter case we may employ the equation iu such reduced form, attending only to the remaining conditions; and in either case we have the equation of a conic containing linearly 2, 3, 4, or 5 parameters, which purametera are taken as the coordinates of a point in 1-, 2-, 3-, or 4-dimensional space, and the discussion relates to loci in such dimensional space. This is in fact what is done in Annex No. 2 above referred to, where the conies considered being the conies which pass through three given points, the equation is taken to be fi}3+gz.v+hxy = Q, and we have only the three parameters (/, g, A); and also in Annex No. 3, where the conies pass through two given points, and are represented by an equation containing the four parameters (a, o, c, A): I give this Annex as a ^ some- what more elaborate example than any which is previously considered, of the application of the foregoing principles, and as an investigation which is interesting for its own sake. See also Annexes 4 and 5, winch contain other examples of the theory. The remark as to the number of parameters is of course applicable to the case where the curve which satisfies the given conditions is a curve of any given order r; the number of the parameters is here at most = ()' + 1) (r + 2), and the space therefore at most |r(r-h3) dimensional; but we may in particular cases have <u + l parameters, the coordinates of a point in a -dimensional space, where w is any number less than ir(r + 3). 23. I do not at present consider the case of a curve of the order r, or further pursue these investigations; my object has been, not the development of the foregoing quasi-geometrical theory, so as to obtain thereby a series of results, but only to sketch out the general theory, and in particular to establish the notion of the order of con- dition, and to show that, as a rule (though as a rule subject to very frequent exceptions), the order of a compound condition is equal fco the product of the orders of the component conditions, The last -mentioned theorem seems to me the true basis of the results contained in a subsequent part of this paper in connexion with the formula* of De Jonquieres, post, No. 74 et seq. But I now proceed to a different part of the general subject Article Nos. 24 to 72, Reproduction and Development of the Researches of CHABLES and ZEUTHEN. 24. The leading points of Ghasles's theory are as follows: he considers the conies which satisfy four conditions (&), and establishes the notion of the characteristics (fi t v} of such a system, viz. p, = (4X ), denotes the number of conies in the system which paas through a given (arbitrary) point, and v t =-=(417), the number of conies in the system which touch a given (arbitrary) line. We may say that p is the parametric order, and v the parametric class of the system. 25. The conies CO, (.'./X O//). ('//A (////) which pass through four given points, or which pass through three given points and touch a given line, &a, ... or touch four given lines, have respectively the characteristics (1, 2), (2, 4), (4, 4), (4, 2), (2, 1). 406] ON THE CURVES -WHICH SATISFY GIVEN CONDITIONS. 201 26. A single condition (A') imposed upon a conic lias two representative numbers, or simply representatives, (a, ) ; viz. if (4#) be an arbitrary system of four conditions, and (& v) the characteristics of (42), then the number of the conies which satisfy the five conditions (X, 4>Z) is ~ ap, + /3v. 27. As an instance of the use of the characteristics, if X, X', X", X"' t X"" be any five independent conditions, and (a, /3), ... (a"", /3"") the representatives of these conditions respectively, then the number of the conies which satisfy the five conditions (X t X', X", X'", X""} is = (1, 2, 4, 4, 2, 1J, )(', ') (a", /3")('", viz. this notation stands for laVY"""+ 2Saa'a"a"'/3"" ... 4- 28. In particular if X be the condition that a conic shall touch a given curve of the order m and class n, then the representatives of this condition are (n, m), whence the number of the conies which touch each of five given curves (m n) . (m"", n"") is = (1, 2, 4, 4, 2, IJn, m)<X m') (", m")(n'", in"')(n"", m""). 29, A system of conies (4/Y) having the characteristics (ft, v), contains 2y - n line-pairs, that is, conies each of them a pair of lines ; and 2^~y point-pairs, that is, conies each of them a pair of points (coniques infimment aplaties). aO. I stop to further explain these notions of the line-pair and the point-pair; and also the notion of the line-pair-point. A conic is a curve of the second order and second class; qu& curve of the second order it may degenerate into a pair of lines, or line-pair (hut the class is then =0): qua curve of the second class it may degenerate into a pair of points, or point-pair (but the order is then = 0). The two lines of a line-pair mny be coincident, and we have then a coincident line-pair; such a line-pair (it must I think be postulated) ordinarily arises, not from a line-pair the two lines of which become coincident, but from a proper conic, flattening by the gradual diminution of its conjugate axis, while its transverse axis remains constant or approaches a limit different from zero; the conic thus tends (not to an indefinitely extended but) to a, terminated line^); in other words, the tangents of the conic become more and more nearly lines through two fixed points, the terminations of the terminated line; and these terminating points, which continue to exist up to the instant when the conjugate axis takes its limiting value =0, are regarded as still existing at this- instant, and the coincident line-pair aa being in fact the point-pair formed by the two terminating points. Similarly the two points of a point-pair may be coincident, and we have then a coincident point- * A line is regarded aa extending from any point A thereof to B, and then in the same diraition, from B through infinity to A; it thus consists of two portions separated by those points; and considering either portion as removed, the remaining portion is a terminated lino. o. VI. 26 202 ON THE OUSVES WHICH SATISFY GIV.EN CONDITIONS. [4QG pair; such a point-pair (it must in like manner be postulated) ordinarily avisos, not from a point-pair the two points of which become coincident, but from a proper oonio sharpening itself to coincide with its asymptotes, and so becoming ultimately a pair of lines through the coincident point-pair; and the coincident point-pair is rcgai'dinl as being in fact the line-pair formed by some two lines through the coincident point- pair. 31. In accordance with the foregoing notions we may with propriety, and it will in the sequel be found convenient to speak of a point-pair as a line terminatoil by two points on this line, and similarly to speak of a line-pair as a point torminutnil (that is, the pencil of lines through the point is terminated) by two lines through thn point. 32. If in a point-pair, thus considered as a line terminated by two point** 1.1m two points become coincident (the line continuing to exist as a definite lino), or, what is the same thing, if in a line-pair thus considered as a point terminated by Mvo lines, the two lines become coincident (the point continuing to exist as a doliiiitn point), we have a line-pair-point ;" viz. this is at once a coincident line-pair ami a coincident point-pair; it may also be regarded as the limit of a conic the axnw of which, and the ratio of the conjugate to the transverse axis, all ultimately vanish : it may be described as a line terminated each way at a point thereof, or as a point terminated each way at a line through it. The notion of a lino -pair-point flrt presents itself m Zeuthen's researches, as will presently appear; but it may bo nobmuil here that line-patr-points, and these the same line-pair-points, may present bhomsolv among the 2i/-/* line-pairs, and among the 2^-p point-pairs of the system of couias 4 A". 88. Returning to the foregoing theory of characteristics, I remark that the flnwln- mental notion may be taken to be, not the characteristics fa ) O f the conies whidi satisfy four conditions but in every case the number of the conies whieh satisfy fivo conditions. Thus for the comes not subjected to any condition, wo may consider tho syin DO is OU (::/), (A//), (:///), (.////), denoting the number of the conies which pass through fivo given points, or which Ten nZ S rt g 'T P mtS a ' ld t0ll0h a giV6n HM < *" " ^ioh touch v given lines i these numbers are respectively = 1, 2, 4, 4, 2, l. So for the conies which satisfy a given condition X, or two conditions 2X, .... or five conditions 6Z, we have respectively the numbers 406] ON THE CURVES WHICH SATISFY G-IVEN CONDITIONS. '203 where the X, 2X, c. belong to the symbols which follow: read (X::), (Z/./), &c., or, as we may for shortness represent them, X", j/", p'" t a"', T' P", *"> p", *" fi , v t p tl- , V viz, ^the single condition X has the five characteristics (p"', ... T'"), ... ; the four conditions 4<X, the characteristics (p, v) as in the original theory; and the five conditions 5X a single characteristic p, . 34. We thus see the origin of the notion of the representatives (a, ) of a single condition X; for considering the arbitrary four conditions 4>Z t the characteristics whereof are . (p t v], and assuming that the single characteristic, or number of the conies (X, 4<Z), is =a^, + /3y, and taking for (4#) successively the conditions 00. (*./), (://), (///>. (////). having respectively the characteristics (1, 2), (2, 4), (4, 4), (4, 2), (2, 1), we have ^'"la + 2ft i/" - 2a 4 4/3, p"' = 4a + 4/9, o-'" 4a + 2/3, that is, the characteristics (/", i/", /)"', <r r ", r'") of a single condition X are not independent, but are represontable as above by means of two independent quantities (a, /9); or, what is the same thing, we have which being satisfied, tho representatives (a, /3) are given by 35. I find that a like property exists as to the characteristics {/*", y", p" t a fl ) of the two conditions 2Z, viz. these are not independent but are connected by a single linear relation, X'-f i/' + /"- o^O. This may be proved in the case where the conditions 2-3T ore two separate conditions (X t X'}\ viz. let the representatives of these be : (a, 0), (a', ^') respectively, then 26 -2 204 OX THE CURVES WHICH SATISFY GIVEN' CONDITIONS. [400 combining with them the three arbitrary conditions X", X'", X'" having respectively the representatives (a", /3"), (a'", #"'), (a"", ""), we have the general equation (X, X', X", X'", T' ") = (!, 2, 4, 4, 2, !$, /3)(', /?')(", /TXa'", /3"')("", /3"")j taking herein successively, and observing that the representatives of (.) are (1, 0) and those of (/) are (0, 1), we thus obtain for (^", ,/', p " t ff "), characteristics of (X, X'), the values /' = (!, 2,4$, /3) (a, '), p"=(4, 4, 2J, )(', ff), "-(*, 2, !$, /9)/9'), (viz. /i" = laa' + 2(^' + R ^) + W, &&), and these values give identically 2/i"-3i/' + 3 /) "-2<r":=0, which is the foregoing equation. And I assume that the theorem extends to the case of two mseparable conditions 2X, but in this, case I do not oven know vrhoro the proof is to be sought for. The characteristics &, v >, p ' } O f the three conditions 3X are in general indopondoufc. the oonditioii X and t ' A, and (p, v ) the characteristics of the conditions 4,2, then (X, *Z) = a fJ , + /3v ) this is the most convenient form of the theorem bnf /* o\ i * ,- of the characteristics ('", ,'" ,'" ,'" ^0 H' ,- ( ' ^ kn Wn ftlllotlona effect an expression for (X lz) in 'terms of \l T ^ Z ^ ^ nfltio11 ifi in F ^, w; in teims of the characteristics of X and 4 reapootivoly. There is, similarly, an expression for (9Y i?\ * n , <W < P', O of M (satisfying Kto X^V + i; 11 T m it amCtOL " b i C9 (ft P, /)) of 2JT, viz. we have t"+l/-r0)aiid the elmractoristics e ^ h ^ d t t t;::rfVr^ " ara - io 3^ three separable conditions Z" ?'" 7>J' ' f } res P ectlvel y> and tlio (a"", /n respectively; we have, b 1't ^presoatatives (, ^ 406] ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. and with these values the function (X, X\ Z", Z'", Z""), =(1, 2, 4, 4, 2, l 205 is found to be expressible as above m terms of (& v, p), (&', v ', p', </) ; but I do not know how to conduct the proof for the inseparable conditions 2X and 3Z. 37. It may be remarked by way of verification that writing successively W = (.'.), (:/>,<//),(///>, that is, 0*, v, p) = (l, 2, 4), (2, 4, 4), (4, 4, 2), (4, 4, 1), we have in the first case (2AV.)= -ip' + i* 7 and similarly in the other three cases, (2Z- //) = ,', 38, Let O, y, p, ff ) be the characteristics of (X, i/, p', o-') the characteristics of 2X (M'~f / + /?' for (2JK", 3^), writing successively for 3^ , (/*-$ y 4-f p - o- = 0), and ' = 0). Then in tho formula and (2J5T ), characteristics (^ y, we obtain expressions for the characteristics (2Z, 2^.) and (2Z, eliminating from the formulas, first the (a-, o-') and secondly the may be expressed in two different forma as follows: of (2X, 2.2T), viz. /), each of these i (p"' -i- pv) ' + IPP' 7 + 206 ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. [406 the two expressions of the same quantity being of course equivalent in virtue of the relations between (p, v, p, <r) and (X, v\ p' t a') respectively. The characteristics of (X> Z}, (X, 2#), (X, 3Z) are at once deducible from the before-mentioned expression up + $v of (X t 4>Z). 39, Zeutheu's investigations are based upon the before-mentioned theorem, that in a system of conies (4-), characteristics (p, v), there are 2^ v point-pairs and 2i> ft, line-pairs. If in the given system the number -of point-pairs is ~\ and the number of line-pairs is = vr, then, conversely, the characteristics of the system are v ~ And by means of this formula he investigates the characteristics of the several systems .of conies which satisfy four conditions (4.X) of contact with a given curve or curves, viz. these are the conies <1,1) (!)(!), (1,1) (1,1), (1,1,1X1), (1,1.1,1), (2)(1)(1) , (2)(i, 1) , (2,1)(1) , (2, 1,1), (2) (2) , (2, 2) ()(!) , (3, 1) (4) where (1) denotes contact of the first order, (2) of the second order, (3) of the third order, (4) of the fourth order, with a given curve; (1)(1) denotes contacts of the first order with each of two given curves, (1, 1) two such contacts with the same given curve, and so on. A given curve is in every case taken to be of the order m and class n, with S nodes, cusps, r double tangents, and i inflexions (m^ n^ B lt K lt r lt tij ?tt a , n, foe., as the case may he). The symbols (1), &c. might be referred to the corresponding curves by a suffix'; thus (l) m would denote that the contact is with a given curve of the order m (class n, c.) ; but this is in general unnecessary, 40. In a system of conies satisfying four conditions of contact, as above, it is comparatively easy to see what are the point-pairs and line-pairs in these several systems respectively; but in order to find the values of \ and or, each of these point- pairs and line-pairs has to be counted not once, but a proper number of times; and it is in the determination of these multiplicities that the difficulty of the problem consists. I do not enter into this question, but give merely the results. 41, For the statement of these I introduce what I call the notation of Zeuthen's Capitals. We have to consider several classes of point-pairs and the reciprocal classes of line-pairs. A point-pair may be described (ante, No. 31) as a terminated line, and a- line-pair as a terminated point; and we have first the following point-pairs, viz.: A t line terminated each way in the intersection of two curves or of a curve with itself (nocle), J3, tangent to a curve, terminated in a curve, and in the intersection of two curves or of a curve with itself. . 406] ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 207 G, common tangent of two curves, or double tangent of a curve, terminated each way in a curve. -D, inflexion tangent of a curve terminated each way in a curve : and the corresponding line-pairs, viz. : A', point terminated each way in the common tangent of two curves or the double tangent of a curve. B', point of a curve terminated by the tangent of a curve, and by the common tangent of two curves or double tangent of a curve. G', intersection of two curves, or of a curve with itself (node), terminated each way by the tangent to a curve. .D', cnsp of a curve terminated each way by the tangent to a curve : all which is further explained by what follows ; thus in the case (1) (1) (1) (1), - (Ik (!)!, (Ik 0-k> tne value of 4 is given as Smim s .w(aft 4 (=3m 1 am B ffi,). Here A is the number of the point-pairs terminated one way in the intersection of any two mi, m 2 of the four curves, and the other way in the intersection of the remaining two m 3 ,m, of the four curves. But in the case (1, 1)(1)(1), =(1, l) m (l) W| (!),,, the value of A is given as =Bm i m 3 + m f rn 1 .mm it Here A denotes the number of the point-pairs, which are either (7%?%) terminated one way at a node of TO, and the other way at an intersection of m,, m 2) or else (mm, . mit^) terminated one way at an intersection of in, w^, and the other way at an intersection of m, m a : and so in other cases. 4<2. This being so, we have G (1, 4- D = 1Hi /_ 3 fHjTO^mj 2 A. = Sjtt^ij . 7lg?t4 (=3 WiWa?isU4 ), ,=(!,!) dk dk- + wm 1 .mm a , 1 ^U. = T)ii?Z-a 4- HHj . W)J 3 , + Sn a 7?h 2 J9 = TWl a ?2 2 + TWiijJl! 2) m, + mwij, (ft - 2) ?ft, + ?m] (m 2) a + mis (m 2) ?^ 1) + mtn$ii (m 1) + Whm,(-l)+wn a m a (-l) -2), + ?i iigTn (?i 2), 4 G' = S^ns ^+nn a(W -2) W , + tfwih (n 2) WB + mm a (ft 2) 7 3 D'= wi.nl 208 ON THE CURVES AVHICEL SATISFY GIVEN CONDITIONS. [40G , ix =<i, iva, iv,. ^.= 88, + wmij (mm,! 1), S = 8?u (mi - 2) + Sjn (m - 2) (n 2) (TOT I) + mm,, (n, 2) (7*1 1), = T . J flli (ffl, 1) + Tj . 4 TO (m - 1) + K! (m ~ 2) (nil 2), D i . ^ TO! (mi 1) 4- (i . | m ( m ~ !) TT, 2) , (m (^ (n 1) + + 2), (-2) m,- a) <-i), (1, 1, !)(!) (1,1, 1) M (!),, -4 = Swini,, 5 = S (n - 4) ii + S?h (m - 2), + 0wi (w - 2) (t 3), (7 T (m 4) 1! + 7Mi! . ^ (m 2) (m 3), 4) tii + T7! (?i 2), = 8 ( n - 4) ?i, + mm, 4 ( - 2) (? (1, 1, 1, 1), =(1, 1, 1, = i8(S~l), = 8 (n - 4.) (m - 4), E'= 43, Secondly, we have the point -paiis : E, tangent to curve from intersection of two curves or of a curve with it? (node), and terminated at the point of contact and the last- mentioned point. J 1 , tangent to a curve at .intersection with another curve or with itself, and terminated there and at a curve. (?, common tangent of two curves or double tangent of a curve, terminated ftl one of the points of contact and at a curve. D, ut supr&. lining cusp of a curve with intersection of two curves or of a ciu'vu If, and terminated at these points. /, line from cusp of a curve touching a curve, and terminated at the cusp and at a curve. J, Inflexion tangent of a curve, terminated there and at a carve : 406] ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 209 and the corresponding line-pairs, viz. E', point on a curve in common tangent of two curves or double tangent of a curve, and terminated by this tangent and by tangent to a curve. F, point on a curve in common tangent of this and another curve or in double tangent of this curve, and terminated by this tangent and by tangent to a curve. D', ut suprcl, H', intersection of inflexion tangent of a curve with common tangent of two curves or double tangent of a curve, and terminated by these lines. /', intersection of inflexion tangent of a curve with a curve, and terminated by this tangent and by tangent of a curve : and this being so, E ~ n . F = minj , m s + mtn. 2 . in l} D = H~K.m l m 2 , E = Sji, F ~ m . mi (mi ~ 1), G = ntii (mi 2), I = ??! (wij 2). (2, F = mm, (m - 2) + 2Sm,, H = / M K (n - 3) mi + ?ii (m - 2), / = wn lt C. VI. E' = in , F ~ mi! , m, E' = (j - 2), ' = , in, (h - 1), /' = (HI 2). ^nik (n-2) +2TW,, 1 ' = mm, (n -2) -H H' 210 ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. (2, 1, 1), ={2, 1, l) w . E = & (n - 4), [406 Q = 2 T (m - 4), = t4(m-3)(m H= $ K , I = *(n- 3) (- J = tm ~3. = r(m-4>), = 2T( -3), -4), H = IT, I' = i(ro-3)(n-4), 44. Thirdly, we have the point-pairs: K, common tangent of two curves or double tangent of a curve, terminated at points of contact. L, line from cusp , of a curve touching a curve, and terminated at cusp and point of contact. M, line joining cusp of a curve with cusp of a curve, and terminated by the two cusps. N, inflexion tangent terminated each way at inflexion, viz. this is a line-pair-pomt, 0, cuspidal tangent terminated each way at cusp, viz. this is a Une-pair-point : and the corresponding line-pairs : K' t intersection of two curves or of curve with itself (node), and terminated by the two tangents. L' f intersection of inflexion tangent of a curve with a curve, and terminated by the inflexion tangent and the tangent at the intersection. M', intersection of inflexion tangent of a curve with inflexion tangent of a curve, and terminated by the two inflexion tangents. N' } = 0, Une-pair-point as above. 0', N } Une-pair-point as above: which being so, \ve have (2) (2), =(2) M (2) n!l . L =^1, + ^^ M=KK- i . 9 K' = mm lt 3 L 1 = mi + 1 M' = iti. 406] ON THE CURVES WHICH SATISFY GIVEN CONDITIONS, 211 (2, 2), =(2, 2) m . L = *(n-3), M = $K(K-1), N =L, =K. K' = 8, L' = t(m-S), K , 0' = 45. Fourthly, we have the point-pairs : P, tangent of a curve at its intersection with another curve or itself, terminated each way at the point of contact line-pair-point. Q, common tangent of two curves or double tangent of a curve, terminated each way at one of the points of contact line-pair-point. J, ut supra. M, cuspidal tangent terminated at cusp and at a curve: and the corresponding line-pairs : P', = Q, line-pair-point. Q f , ~P, line-pair-point. J', lit 'suprh. R!, inflexion of curve terminated by the inflexion tangent and by tangent to a curve : ' which being so, we have.'. P = mm,, 2 T)/ Q-wh , 2 Q = m??^, / = , , 5 i/' = ICUi, R^Km, . 4 jy _ tn _ , IV. P = 2S, 2 <8'2T, Q =2r, 2 P' = 2S, J = t (m - 3), S r/ / JR = K (m - 3). 4 Ji'= *( 46. ^And lastly, we have the point-pairs N t (line-pair-points) and the line-pairs N', 0' (line-pair-points), ut suprti, and JV = , 0"t 272 212 ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. [406 47. Where in all cases the central column of figures gives the numerical factors which multiply the corresponding capitals, thus we have X = 2y - /* = A + 25 + 4(7, ar = 2i-y =:;i' + 25' + 4a' for (1, X = 2y-/i = J 4 + 25+4(7+3.0, OT = 2/i- y = 4' + 25' + 4(7 + 35', and so on. 48. The elements (m, n, S, K, r, t) of a curve satisfy Pltioker's six equations, and Zeuthen uses these equations, m a somewhat unsystematic way, to simplify the form of his results. It is convenient in his formulas to write 3w+4, = 3 + , =a, and to express every- thing in terms of (m, n, a), viz. we have for this purpose 2S = m 2 - m + Sn-3a, 2T = ?t a -STO- w-3o. But I make another alteration in the form of his results; he gives, for instance, the characteristics of (1, 1) (1) (1) as V = V m y y li = fi'" w^m, + /*" (??!,n a + V where X =2m( m+ n-8)+ r, =(1, 1,-. ), /=/ =2/1 ( m + 2-6) + 2T, =(1, I:/ ), X" = " = 2 (2m + B - 5) + 28, =(1,1- //), i/" = 2w(m+ -3)+ S, =(1, I///), viz. the four components have really the significations (1, 1.-.) se t opposite to them respective ly -^ and accordingly, instead of giving the formula for tho two characterisbics ot (1, 1)(1)(1), I give those for the four characteristics (1, l.-.V & c . O f (1, 1) thus in every case obtaining formula which relate to a single curve only. Subject to the last- mentioned vanation of form I give Zeuthen's original expressions in Annex 6; but here m the text I express them as above in terms of (m, , a), viz. 49. We have the formula (1) 406] ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 213 0, 1) : /) - 2m a - m - 2 - 3a, , 1, 1) win" + n 8 - 2wi a n 3 - 4f m - ^ n + a (- 3m - Bn + 20), (1, 1, 1, 1) m a - - ?t -I- ^ m s n + wi 3 ft 3 + mn' + i (2) (.'.)= , (///)= ; (2,1) ( : ) = 12m + 12?i + (2m + n - 14) a, ( /) - 24i -|- 24 + (2m + 2w - 24) a, ( //) = 12m + 12n + ( m + 2w - 14) a ; (2, 1, 1) ( ) - 24m 3 + 36mn + 12?i 3 - 168m - 168n + ( m 2 + 2?n- ( / ) = 12m + 36m + 24n - 168m - 168n + a (i-wt" + 2m (2,2) ( ) - 27m + 24?i - 20a + fa\ ( / ) 24m + 27n - 20a + a 3 ; (3) ( : ) = - 4m - 3w + 3, (./)- 8m - 8n + 6a, 214 ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. [406 (3, 1) ( . ) = - 8m a - 12wn - 3?i 3 + 56m 4- 53n + a (6m + 3n - 39), ( / ) - - 3m 3 - limn - S?t 2 -f 53??t + 5G + a (3 4 6 - 39) ; (4) ( ) - - lOwt - 8n 4- 6a, (/) = - 8m- I0+ 6a. 50. By means of the foregoing formula) I obtain, as will presently be shown, the following formula for the number of the conies which satisfy Eve conditions, viz. : (4, 1) = - 8m 3 - 20)ftn - S* 4- 104i 4- 104n + a. (6m + 6n - 66) ; (3, 2) = 120wi 4- 120/1+ a (- 4m- 4- 78) +3tf; (3, 1, 1) - f m 8 - lOm'H - 10mn - f n* 4- ^ m + H6m + -ifi - 434i - 4S4n (2, 2, l) 4- a (- 8m- - Sn 4- 327) 4- a a (|i 4- ^w - 12) ; (2, 1, 1, l) = 6m + 30m 9 H + 30wm 8 +0.*-l74m s -348mtt- + a (ft m 3 4- wte + inw 1 + ^ w ' - -yi wi 2 - 26nm - * n* + fifa m + 4|fi u _ 960) 4- 1 mw- ?i 3 + ^m 9 4- 23m + ^n-Afim-afi-w + 486) 51. I observe that by means of the above-mentioned expressions of (X, 4<Z) and (2Z, 3^), the foregoing result^ other than those for (5), (4, 1), &c., may be presented in a somewhat different form, viz. we have where (.) denotes (4Z .), (/) denotes (4^/) f and so in other cases, the understood terra being 3Z or 2Z, as the case may be. 1 In my paper in the Comytes Rendvs, I gftvo eiToneously the oooffioients - ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 215 (82) (2) = + (//)<-**-*"+) + (///)( m+fft-fa); 3m - 3 + a (- J m + ft + 2)} , 1,1) = ^m-^7i (- m+ t- 1)); in all which fonnulea it is to bo recollected that we have ('.)-* /) + (//)-(///)- o, to which may be joined where a, & are the representatives of the condition (#), and whore (4>X) is to bo con- sidered as standing successively for (4), (3, 1), (2, 2), (2, 1, 1), and (1, 1, 1, 1), the values of (43T-) and (4A7) being in each case given by the foregoing Table. 52, The formulas are very convenient for the calculation of the numbe conies which satisfy five conditions of contact with two given curves *** example, (8), =(3) tl . denotes the condition of a contact of the third given curve (wj), then writing for symmetry (2) m in place of (2), we have = a ( 4m, 4?t, 53. To obtain the foregoing expressions of (5), (4, 1), (3, 2), (3, 1, (2, 1, 1, 1), and (1, 1, 1, 1, 1), I assume that the given curvo breaks curves (in, n, ) and (m' f n', a'), or, as we may for shortness express it, in m and m'. 216 ON We have then THE CURVES WHICH SATISFY GIVEN CONDITIONS. [406 f e 'this contact with the curve '. Writing tins under the form observing that (5), is a Action * <* .. -X -1 that functional equation ^(w+m 1 , + < + )-*(w, , ) <M- < (m, n, a) = i, c ro arbitrary constants; but as the solution should be symmetrical in re. to t, , we have A or the solution is * (, , ) = < + -) + 54. Similarly we have vi, the C o,ics .vhich have ,vith the aggregate curve m + rt ^.^te ^ made up of the conies which have the two contacts 4 and 1 w tl the one cu 01 vUh the other curve, or the contact 4 with the one curve and the contact 1 with Ihf ottr curve, The expression on the right-hand ride is a known flmotion of ( n. ), (m f , n, a') ; hence the form of the functional equation is and any particular so.ution of this equation being obtained the general ^ s^tion in found by adding to it the term am + fc^ca. Assuming that the partial Boloi on ymmetlal in regard to ( n), then the term to be added a, before = a(m+n) + - And similarly far (8, 2), (8, 1, 1), fe; that is, in every CaS e we have a solution con- taming two arbitrary constants a, c, which remain to be determined. 55. Now in every case exce.pt (5) w the number of intersections of the conic with the curve is > 6 (viz, for (4, l) m and (3, 2) m the number is 7, for (3, 1, 1} and (2 2 1} it is 8 and for the remaining two cases it is 9 and 10 respectively); hence if' the riven curve be a cubic, the number of conies satisfying the preserved conditions is -0; and since a cubic may be the 8^^^^ * V 10 ? cuspidal cubic, we have the three cases (, n, )-(8, 0, 18), 8, 4, 18), and (3, 8, 10) We have thus in each case three conditions for the determination of the constants a, o ; so that there is in each case a verification of the resulting formula. 56 In the omitted case (5) W1 when the curve m is a cubic, the theory of the conies (5V is a known one, viz. the points of contact of these conies, or the "sextachc points of the cubic, are the points of contact of the tangents from the points ol inflexion; the number of the conies (6) m is thus (-8)*, viz. in the three case* 406] ON THE CUBVES WHICH SATISFY GIVEN CONDITIONS. 217 respectively it is = 27, 3, and 0. Hence for determining the constants we have the three equations 9 + 18c = 27, 7a-hl2c = 3, (j(t-i-10c= 0, which are satisfied by (( = -15, c = 9, and the resulting formula is (5)--15?H,-16n + 9. In the particular case of a curve without nodes or cusps, this is (5) = 12n - 15m, = m(127M-27), which agrees with the result obtained in my memoir "On the Sextactic Points of a Plane Curve," Phil. Trans, vol. CLV. (1865), pp. 545 oVS, [841]. 57. The subsidiary results required for the remaining cases (4, 1), &c. are at once obtained from the foregoing formulas for (4#) (1), (32) (2), &c.; for example, we have with like expressions for (3, l) m (!), , &e., (S) w (2) ffl . = t - 8m - fa' ) (- 8m - + (|i' a - ^j,') (- 3m - with like expressions for (2, l) w (2) m . ( (2, l) m (l, !),-, &c. &c. 58. Calculation of (4, 1), We have (*, l)w - (4-, 1) - (4, l) w . = (4), a (l) w . + (4) m - (1) M , = - 16mm' - 20 (mw' + m'ri) 16nri+ 6 (' + a'n) + 6 (cmi f + V), the integral of which is (4, l) m = - 8m? -.20wOT - 87z, a + a (m + n) + a (6m + 6n + c). The particular cases (m, n t a) (3, 6, 18), (3, 4, 12), (3, 3, 10) give respectively = 252 + 9ft + 18c, 0= 64 + 7a -I- 12c, 0= 36 + 6a + 10o, satisfied by a = 104, 0=3-66, 0. VI. 28 18 ox THE UUBVJJS WHICH SATISFY GIVEN CONDITIONS. W. Calculation of (3, 2). We have <*. 2) lll+)i , - (3, 2),., - (3, 2),,,, = (3) w (2),,, + (3),,, (2) w , = - 4 (ww' + m'a) - 4 (710' + n'ct) + fla / : '.|0<i (8, 2),,, = a (in + n) + a (- 4t - 4u + c) + Sa 11 , ' ' 324 + {) + 18c = 0, . . , . the Integra! w , , ami, as before, . , = , tinl by o 120, c = - 78. CO. For the calculation of (3, 1, 1) we have similarly (. I, 1U,H3, 1, IX. -(8. 1, ^^.(S cs, IV value, so obtained pt that thc- terms in d the auh.equeot eases In the Ft ,,nt ca.se thc more i "* 8101111 f (3 ' ^ ^' * ^ fc " and T- f g Ulg ex P ra8slon ! thp T " -" ^ + ) } '^ equabonfl which "'" "'" r (8, ", I >. 382 + 80 + 180*0, The remaining cases are (2, 2 1W9 i i }> ( ' lf X ' and - 1. 1). Wo have (2),, + (2, by flaa Again, 406] ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 219 and 1680 + 6 + 100 = 0, satisfied by a = 1320, c=-960; and finally, (i, i, i, i, iw-(i, i, i, i, i) m -(i, i, i, i, i )m , = (1| 1( lt i), Jt (i) Wld Hi.i,i)mai - 30618 + QO<E+180c = 0, - .14094 + 70fj+120c=0, - 10692 +60+ 1200 = 0, satisfied by 10ft = 6818, 10c = 4860, that is, a = -upi t c = 486. 62. The contacts of a conic with a given curve which have been thus far considered are contacts at unascertained points of the curve; hut a conic may have with the given. curve at a given point thereof a contact of the first order, the condition will be denoted by (2); or a contact of the second order, the condition will be denoted by (3), and so on. It is to^be observed that the conditions (2), (3), &c. are sibireciprocal, the contact at a given point of the_curve is the same tiling- as contact with a given tangent of the curve; but if we write (1) to denote the condition of passing through a given point of the curve, this is not the same thing as the condition of touching a given tangent of the curve ; and this last condition, if it were necessary to deal with it, might be denoted by (1), But I attend only to the condition (1). The expressions for the number of conies which satisfy such conditions us (1), (2), &c. are obtainable in several ways, G3. (1) When the total number of conditions is 4, the question may be solved by Zeuthen's method, viz. by determining the line-pairs and point-pairs of the system 4>Z> with the proper numerical coefficients, and thence deducing the values of the characteristics (4#.) and (4#/). A few cases are in fact thus solved in .Zeuthen's work. 64. (2) By the foregoing functional method, It is to be observed that there is a difference in the form of the functional equation, and that the general, solution is always given in the form, Particular Solution -f Constant, so that there is only a single constant to be determined by special considerations. To take the simplest example, let it be required^ to find the number of the conies (3) (1, 1) : writing for shortness in place hereof (1, 1), or (in order to mark the curve (TO) to which the symbol has. reference) (1, l) m , let_ the curve (m) be the aggregate of the curves (m) and (m'). Regarding the point 1 as a given point on the curve (m), that is, an arbitrary point in regard to the curve (m'), we have thus the equation where the right-hand aide is known j and .so in general the form of the functional equation is always 0(m + m')-^(m)- given, mine, that is, + ri, a + a.') - <j> (m, n , a) - given function of (m, n, a, nf r n f , o!)\ 282 220 ON THE oravffl waroH SATISFY GIVEN CONDITIONS. these cases (1. !)+,' - (I, l) M = B whence (I, !)*= + 2m + const. = (I, !)() and replaced CM, l) = be - mes - [ 406 have m the first of - 4, _ (I, T, I, I, 1) ^ , + 2wi - 8. and 2,1,1 2 1 1 eoiidein (2, 2, l) B = the fcnr given points on have reducn, Jf ^ V 2 *- -boide, that is, if U eoincide > there is uo further reduction, but we 1 I mjte indifferently (l)M il-\ J wi-.j, (i..j nr /..i\ i or (:;1). ftuc i so 406] ON THE CUHVES WHICH SATISFY GIVEN CONDITIONS. 221 GO, The expressions involving a single (I) may in every case be reduced by the foregoing method to depend upon other expressions; thus we have )(!,!) =(!) -2(2) )(!, 2) =(-2) -3(3) (I 1,1) =(-1, 1) -2(2, 1) , ( Z )(T, 1, 2) =(!, 2) -2(2, 2) -3(1, 3), (1, 1, 1, 1)=(-1, 1, l)-2(2, 1, 1), _ (1. S) =(-3) -4(4) (1, 4) =(*) -5(5) &c., where, comparing for example the equations for (2) (I, I, 2) and (2#)(1, 1, 1), it will be observed that in tho first case tlie contacts 1, 2 of tho symbol (1, 1, 2) successively coalesce with the point I, giving respectively 2 (2, 2) and 8 (1, 3), the exterior factor being in each case the barred number, whereas the second case, where tho contacts 1, 1 of the symbol (1, 1, 1) are of the same order, we do not consider each of these symbols separately (thus obtaining 2(2, 1)4-2(1, 2), =4(2, 1)), but the identical symbol is taken only once, giving 2(2, 1). Thus wo have also (1, 1, 1, 1, 1) = (-1, 1, 1, l)-2(2, 1, I, 1), (57. Tho value of a symbol involving (2), say tho symbol (&Z) (2), is 'connected with that of (3-/); but as an instance of the correction which is sometimes required I notice the equation (2, 1, 1, 1)(1, 1. l-/)-IiO-2)0-3) + i(n-2)(-3) + 3(3 l 1, 1)4-2(4, l)|, which I have verified by other considerations. 68, We obtain bhe series of results : (1) ( ") = 1, (/./) = 2, ( = //) = 4, ('///) ==4, (////)- 2 J (1, 1) (.'.)= + 2w-2, ( : / ) = 2 + 4m - 4, (.//) = 4<n -f 4??i 4, ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. (1.2) (:)=- 8, A 1, 1) ( : )2i a +2iH+^i s -Ow-^i+ 8 -fa, 12 - 3, 8 - 3a ; (T, 3) ( ) = -4m-3H-4 + 3a 1 ( / )=*- Sm-8B- 4 + G; (1,1,2) ( . )= (///) = !; (2.1) ( : ) = 2m+ -4, ( ' / ) - 2m + 2/i - 6 ] ( // ) = m + SH - 4 ; (2,2) (2, 1, 1) ( ' )= ^( / ) = (3) (0 = 1, 406] ON THE CUEVES WHICH SATISFY GIVEN CONDITIONS, 223 <3, 1) ( ) = n + 2m - 6, (/) ~2n+ m-6; (4) () =1, winch are the several cases for the conies which satisfy not more than four conditions, and 69, For the conies satisfying 5 conditions, we have (5) =1, -*- (3, 2) - 9 + a t (3, 1, 1) = | W " + 2nm + n * _ y m _ j^ 4. 27 _ (2, 3) (2, 2, 1) (2, 1, 1, 1) (, 4) = (I, 1, 3) = (1, 2, 2) = 27m + 24)i + 27 - 23 + a 3 , (T, 1, 1, 2) #wH30m + *ftt"-^w + (w 2 + 2nz, + i - 27m - (T, 1, 1, 1, l)-^w 4 + ^w a ft + m 5( rt 1l + i7 - JLWI' - 6m aR" + i + + 150 70, The given point on the curve to which the symbols I, 2, &c. refer may be a singular point, and in particular it is proper to consider the case where the point is a cusp. I use in this case an appropriate notation; a conic which simply passes through a cusp, in fact meets the curve at the cusp in two points ; and I denote the condition of passing through the cusp by l'/d ; similarly, a conic which touches the curve at the cusp, in fact there meets it iu three points, and I denote the condition by 21 J 1/cl, 21 are thus special forms of 1, 2, and the annexed T indicates the additional point of intersection arising ipso facto from the point 1 or 2 "being a cusp. Similarly, we should have the symbols 3/cl, 4/el, 51; bufc it is to be observed that ab a cusp of the curve there is no proper conic having a higher contact than ME SATISFY 2*1; thus if the symbol contains JvT n,- , to *- , " 6il n m, f , seo Annexes Kos. 4 and 5. J OTS f the Cilae f the ""s itself h -^ to ^ we have " e cin ' ve 1S an ordinary point. For exampk- in simply ol which only diffpr fmm th r fion, the e x pl , ssiom with in place of " + 2 '" ~ 3 ' 2 " + 4 " 1 ~ 6. * + * - 0, 4 m + 2 - S 2. We have (1*1) (-'/) = 2, (///) -4, 406] ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 225 (1*1, 1) (.'.)= +2m-3, (//) = 4. + 4m - 6, (///)== 4w + 2m-3; (1*3, 2) ( : )= a-4, (1*1, 1, 1) ( : )- 27W 9 + 2mn + n a -&-$+ 13 ( / ) = 2m" + 4m + ii 3 - 8i - 7n -f 18 ( // ) = m s + 4nm -F 2?i a - 4m - 8?i 4- 12 , 3) ( . ) = _ 4, m _ 3 n _ 5 4. 3 Hj 1, 1, 2) 4m + 8 , 1,1,1) (2*1, 1) ( : ) =s 2m + - 5, (// )= m+2n-4; , 2) 226 OX THE CURVES WHICH SATISFY GIVEN CONDITIONS. (2*1, 1, 1) ( ) = wi a +2ift + - a -fm 78. The remainder of this table, being the part where tho symbols (-) ami (/) ot occur, I present under a somewhat different form as follows : (8*1, 2) (3*1. I, 1) (2, 3) (2,2,1) & I. I.I) = 0, = 0, = 0, = 0, -(2*1, 3) f2*-l 9 i\ \ -A. i , & t Jl -(2*1, 1, 1, 1) I 4) =o, 1.1..) - eh ange of a cusp into an i fl n ^ h as fti ; but ; have a : o n :,; t contain this new symbo l. the investigation* oontamo.l in the ^ ' ld o theroforo ^ ivo ** nece8s ^'y to consider tho formula) fl/ o,, rv tho e of n th orders 406] ON THE CURVES WHICH SATISFY GIVEN CONDITIONS, conditions; viz. the number of the curves O 1 ' is = /*(a+ 1) (6 + 1) (o + 1)... into + [r-m - (a + 6 + c . .) -p - 2]'~ a (06 + ao + bo . .) [D] 2 where the curve t/'" is a curve without cusps, and having therefore a deficiency D = (in l)(m-2)-8; the numbers ft, b, G,.. are assumed to be all of them unequal, but if we have of them each = a, fi of them each = b, Ac., then the foregoing expression is to be divided by [a] a [/3] p . , . ; and p denotes the number of the curves G r which satisfy the system of conditions obtained from the given system by replacing the conditions of the t contacts of the orders a, b, c, &c. respectively by the condition of passing through a + b + c . . . arbitrary points. In order that the formula may give the number of the proper oitrves C" 1 which satisfy the prescribed conditions, it is sufficient that the ir(r + 3)-(tt -fi + c . .) p conditions shall include the conditions of passing through at least a certain number T of arbitrary points : this restriction applies to all the formula) of the present section. 7n. I will for convenience consider this formula under a somewhat less general form, vise. I will put p = 0, and moreover assume that the |r (r + 3) ~ (a + i + c . .) conditions are the conditions of passing through this number of arbitrary points ;. whence /* = 1. , We .have thus a curve G 1 ' having with the given curve ff m t contacts of the orders a, b, G,. respectively, and besides passing through r(i % + 3) (ft + 6+c..) arbitrary points; and the number of such curveH is by the formula =(+ 1) (b + l)(c+ 1),,.. into + [rm - (a + 6 + c . .) - I]'- 1 (a +6 + c . .) [DJ + [rm - (a + b + c . .) - 2]'-" (at + ac + &c . .) [D] a + [rm (ft + 6 + a . .) t ] (ct6c ... ) where, as before, in the case of any equalities between the numbers a, 6, o, ..., the expression is to be divided by []*[$]''.... 76. I have succeeded in extending the formula to the case of a curve with cusps: instead of writing down thp general formula, I will take successively the cases of a single contact a t two contacts a, b, three contacts a,, b t c, &c, ; and then denoting the numbers of the curves G 1 ' by (o), (a, b), (a, b t c), &c, in these cases respectively, I say that we have (a) = (a + 1) (rin - a" a 292 228 ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. (a, b) = (ft + 1} (b + 1} f [rm - a - ftp "40< 6 ' 0(6 ' "i fVf' J L K ,\ - [So (a + 1) (6 [ m - a _fi_ p + [ m _ ft - & _ _ ip ( ft -I- [rm - a - 6 - o - 2]' ( + (t [m _ f( _ b _ c _ : , ? + [ ~ - & - c - 2] 1 (a -I- uo + 60) [J3]" (a + 1) f [ m - a _ & _ c __ 1+ ' J i- .1 ^r ' (, i, o, rf), putting therein fo , + i + . + I.^ 4 + mte dow and also the formula for f/i /, "* for (a, i, , lor the combinations of fa b r rJ W have (a ' '' ' * _ putting (. b, c, rf), ( a> 6 f ) ) ( aj 6 j and ^^ ^ [rm - p 406] ON THE OUEVES WHICH SATISFY GIVEN CONDITIONS. 229 . (a + f [rm - a. - 1] 3 + [rm - a - 2] a ' [7)] 1 -3- [rm (t 3] 1 /3' [7J] a alod (a, 6, o, d, e) = [>-m a ]" + [?w - a - I]' 1 a [U] 1 -S- [rm - a - 2] 3 [D] 3 -f- [?'m - - 3] a 7 [D] 3 + [ m _ a _4 ( ]ig [jf)].> - e [D] G + [ rm a - 2J 3 a' [D] 1 + [rm ~ a - 4] 1 y [D] 3 + S' [!)]' J / [ m __2]> ^ ][*] J [ - aficde . [?'t - a 4]' 230 ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 7S. In all these formula thera is, as before, a numerical divisor in the case of any equalities among the numbers a, b, o, &c. And D denotes, as before, the deficiency, viz. its value now is D = %(m-l)(m-$)-%- K \ or observing that the class n is -w a -w-2S-3, we have D= $n-m + l +fa or say ^-3-m + ^n-^ = 1 + A if i\ = - DI + H + ^K. 79. It is to he observed with reference to .the applicability of these formula within certain limits only, that the formula* are the only formula) which are generally true; thus taking the simplest case, that of a single contact a, the only algebraical expression for the number of the curves G>' which have with a given curve Z7* a con- tact ,,f the order , and besides pass through the requisite number 4r(' + 8)- ' of arbitrary points, is that given by the formula, viz. (ft) = (a + 1) (rin -a + aD) - ttK , Considering the curve 0* and the order r of the curve G * given if s - but the e is no algebraic function of a which would give the number of tllC iMYinPi 1 nni'uoo f f r or. ,.,li i i . ".". 5<-vu unu llUHlUQr OI ioe. cuives C as well beyond as up to the foregoing limiting value of a f of the n 4 - given points, a, therefo e ,,o ol, , f 7 " " illaid<lnt line -l ir *8* *ho the nibe/of th S> ^ I 2 ^""7 f"^ ^ ^ """^ ' i coincident line-pair hfvin, at elVoflf,' , . hna J inin these P ointe P contact therewith, that I ha 1 , "J ""T* ^ the ^ our a%coial with tbe given curve; if the numW n f*/ m ~. 1 A'-2) ways three special contacts case any line whate e th ol ,Tlh 1 ^ P iUlS " 1 ' ttoo in the fl -* whatever, .^dec, as . coS^nt Hne^ C^ , ^ "^ '"" ^ U " e curve; and so in general there i, 7 , pe lal oolltaolls with the given )vMch vah,e the cUenf 7 1LT u"", V!*, """"^ '' giV611 P """ fo " imi^per curves 0', and for valued nfeS to iff ^v d9temillBta nnmbar f mfimte series O f irapl . opei , Olu . veg ' mei101 . to ll the oonditoons may be satisfied by De Jonquieres has determined the mini , ^ oomiderati "8 as these that to which the conditions shou d ^T I 7 "1 ? ' th numbOT of arbit 'y Point S I "for for his investigation and it ts n "?' " le f0 uto ^ be W^lo : remark that in t g e ou whT^l n P T* P '? XVH " ld XVUI f his >"-. can be corrected so L to riveT '' i Ilr0p " S luti n8 ia fcite - *e -give the number of proper solutions by simply ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 231 406] subtracting the number of the improper solutions: but this is not so when the improper solutions are infinite in number; the mode of obtaining the approximate formula is here to be sought in the considerations contained iu the first part of the present Memoir; see iu particular ante, Nos. 8, 9 and 10. HI. The expressions for (a), (a, b}, &c. may be considered as functions of rm, 1+A, and K, and they vanish upon writing therein ?m=0, A=0, = 0; they are consequently of the form (rm, A, /O' + tVm, A, ) s + &c., and I represent by [a], [a, b], &a the several terms (rm, A, K)\ which are the portions of (a), (a, i>), &c. respectively, linear in rm, A, and K. The terms in question are obtained with great facility ; thus, to fix the ideas, considering the expressions for (a, b, c, d), 1. To obtain the term in rm, we may at once write D = 1, = 0, the expression is thus reduced to and the factor in { } being =rm[rm~a- I] 3 , the coefficient of rm is which is = - (a + 1) (b + 1) (c 4- 1) (d H- !).(+ 1) (a -I- 2) (a + 3). 2. To obtain the term in A, writing rm=Q, = 0, and observing that &c. give the terms A, A, -A, 4- '2 A, - 6A, &c. respectively, the term in A is 1) C [--l] s a. 1} A 4- + ft + Y + 28 S. 2 (a + 2) (a + 3) A. 3. For the term in K S writing r = 0, D ~ 1, and observing that []', [] a , [] 3 ., [J* give respectively the terms K, , 2, 6/c, this is (a + 1) (6 - 1) {[- a - 2] 2 + [- a - 3] 1 ").-! -SM(a-l) ([_*_3] l + a'"}. 2 . + a&od . 6 . oqo ON THE nre where the terms in ( | nr that is, (+2--)( + a) and - respectively; whence the whole expression is (a+3) , 6 abed the expression multiplying ( + 2)(+ 3 ) j, K, and we have moreover Ihe series of fonmil * attain from completing the reduction aa s + ( + 1) O.K, and coeff . of . . C0effloi -* of - expressed in term, 406] ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. - /3 - 7 -23 233 -G where a = , [a, b> o, (- Se 2S 6e (c + rf + e + 3) (a 4- 1) (6 + 1) (6 + o + d + e +- 4) (a -{- 1) - ZAabcde rf+e, jS = &c., , . , e = A - 2 cd (c 4- d + 2) (a + 1) (I +1) ( + 3) ( + 4) K, (a + where a = 83. The complete functions (a), (a, &),.(> 6. c), &c. may be expressed by moans of the linear terms [a], [a, I], [a, b, c], &c. as follows, viz, we have (a) (a, b, c) = M ra k 6], (, 6,0, *) W[6][o]M 5, d] + [o][6, o, q + [a, 6, o, d], c. vi. 30 -34 OX THE CURVES WHICH SATISFY GIVEN CONDITIONS. and sti on; this is easily verified for (, &), and without much difficulty for (a, b, a) but in the succeeding cases the actual verification would be very laborious. S4. The theoretical foundation is as follows. Writing- for greater distinctness (<0 in place uf (it), we have ( ( t) M to denote the number of the curves O' which haw with a given curve 17'" a contact of the order a, and which besides pass through i-r(r + 3)-tt points. Let the curve U m be the aggregate of two curves of the orders >n, >: respectively, or say let the curve U m be the two curves m, in', then we have ()m+w' = (). +(tt) m ; u functional equation, the solution of which is * ()w -[]. where [a] is a linear fnuction of , ,, *, or, what is the .samo thing, of ,, A, AT. ;r " when the coefficients - "-' w- * >~ Similarly if (, J) m denote the number of the curve, G" which havo with the of h pomts - then lf the given where {() (6),,,.} is the number of the pnrwm (^ win'ni, i -LI order a and with ,' a contact of rt P S! , , 1 hflTO Wlth m a ntflfit o( ' thc points; and the lil , S ^ 1 en' nn * T ^""^ ^ ^^ + >"- * which are not too great ^^ of , "T'^' bnt ** Tnllie8 f ftnd order, of ^^ ^ and thence the functional equation (. aW*-(, i) H1 -( ff) fi But [a], rtl &c . being l ilieai . functions of % ^ , , M+*-M+M m ., and thence a particular solution of th, ,n f - - general solution is therefore q l n 1S at onoe seen to be [a] m [b] M ; the . W) present "de^^ A, . Henee, assuming fo, fc ho formula (, 6) = [ a ] [i] + W uM be fo ^ d to be -fc 6] , TO \ ave the 406] ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 235 The investigation of the expression for (, 6, c),,, depends in like manner on the assumption that we have and so in the succeeding cases; and wo thus, within the limits in which these assumptions are correct, obtain the series of formula) for (a, &), (a, b, c).... 85, It is to be observed in the investigation of (a, 6) that if a~b t the two terms [a] m [&], and [],[&], become equal, and the equal value must be taken not twice but only once, that is, the functional equation is and the solution, writing ^ [a, a],,, for the arbitrary linear function, is 0, a) m = ^ [a] m [a] m + % [d., a] m , in which solution it would appear, by the determination of the arbitrary function, that [a, a] has the value obtained from [a, b] by writing therein b = a. Writing the equation in the form (a, ft) -H] and comparing ' with this equation for (a, b), wo see that [a, b] is not to be considered as acquiring any divisor when b is put = a, but that the divisor is introduced as a divisor of the whole right-hand side of the equation in virtue of tlie remark as to the divisor of the functions (a, b), (a, 6, c).,. in the caso of any equalities between the numbers (a, 6, c...). This is generally the case, and the foregoing expressions for [., b], [a, b, c], &c. are thus to be regarded as trno without modification even in the case of any equalities among the numbers a, 6, c..,, 86, To complete according to the foregoing method the determination of the expressions for (a), (a, 6), . . , we have to determine the linear functions [a], [a, b], &o., which are each of them of the form fni + gk + kic, where (/, g t A) are functions of r and of a, b, &c.; and I observe that the determination can be effected if wo knofr tbe values of (cr), (a, 6), &o. in the cases of a unicursal curve without cusps 'and with a single cusp respectively. Thus assume that in these two cases respectively we have (a) (rt + l)(m a), (a) ~ (a + 1) (rm a) a. Waiting first A = -l, 0, and secondly A = -l, = 1, we have (a + 1) (i w - ft) - a =ftn g + h, whence /=(a-fl)r, 0(a + l)a, A-a, 302 -36 *>N" T!K CURVES WHICH SATISFY GIVEN CONDITIONS. |~4G(5 giving the foreg.-iing value [] - (a + 1} i-t + (a + 1) fflA - . Siiuilfirly, flu- two (.-ontacts assume that we have in the two cases respectively (, 1) = (ti + 1) (fi + 1) [ m - a - b]\ String h,i from the formula fo >] = (, 6) - [a] 0] -/fo +, A + A., and MU^., IU 'Jy A = - 1, = (j, and A = - 1, * = 1, we have ( + 1) (6 + 1) [m - fl - 6f - (( + 1) ( m _ fl)) ( (i + 1} (m _ fi)) =/m _ ^ ( + l)(Hl)[m--6p_ (0(6 + 1) + fi(a+1) j [m _ n _ fti-1]1 - |( + 1) (m - a) - a} {(b + 1) ( m _ 6) 6 j ^y^ - ff the fir,t of which, putting therein a + ft-, aft-ft ft fc onee redueed to ; wheilce h 87. The actual calculation of fa 4 d w,,,,i,i , , , . l>|uo,,t terms still more so; but it i' s 1 /, , .' labmo , and that of Oho the foregoing values, assuming them to be J!! u } pnu<riple a PP lia8 ' and >at for , uuicursal ourv, withou^cusp," that ' W UU be btabad if "^ know, the number of contacts a, I, \ ,, . that '-^ a " d fol a w, with a si ollap , . that the -liminution of ( o 6 , . , ( '' C '- )oCCaslmet "=y the single cusp i ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 237 88. Consider a unicuraal curve U m , and a curve & having therewith t contacts of ^ the orders a, b, c, ... respectively. The coordinates (at, y, z) of any poinfc of the umcursal curve are given as functions of the order in of a variable parameter 0; and substituting these values in the equation of the curve G 1 ', we have an equation of the degree rm in 0, but containing the coefficients of G f linearly; this equation gives of course the values of 6 which correspond to the nn intersections of the two curves. Hence in order that the curve G r may have the prescribed contacts with U m , the equation of the degree rni in 8 must have t systems of equal roots, viz. a system of a equal roots, another system of b equal roots, &c. ; this implies between the coefficients of the equation an ( + b + c ...) fold relation, which may be shown to bo of the order (u + l)(6 + l)(e+])...[rm-(a + & + c . ..)]'; and since the coefficients in question are linear in regard to the coefficients in the equation of the curve C f , the order of the relation between the last-mentioned coefficients has the same value ; that is, the number of the curves C" 1 which have the prescribed contacts with the nnicursal curve U m and besides pass through the requisite number of given points, is = (a + 1) (b + 1) (r, + 1.) . . , [rm - (a + b + c . ..)]'. Hi). The reduction in the case of a cusp appears to bu caused as follows : Consider on the curve lf m a points indefinitely near to the cusp, and let the condition of the curve C/ having the contact of the o-th order be replaced by the condition of passing through the a points; that is, consider the curves O 1 ' which have with the curve U m (t l) contacts of the orders b, c, ... respectively, which pass through the a points ori the curve U m in the neighbourhood of the cusp, and which also pass through the requisite number of arbitrary points. The number of these curves is =s(6 + l)(c + l)...[?w-a-(6+o + ..)] i - 1 (the term r-nt-a instead of r-m, on account of the given a points on the curve: compare herewith Do Jouquieres' formula con- taining nn~p\ Kach of these curves, in that it passes through a points in the neighbourhood of the cusp, will ipao facto pass through a + 1 points (viz, a curve which simply passes through the cusp of a cuspidal curve meets the cuspidal curve there in two points, a curve which touches the cuspidal tangent meets the curve in three points, &c,), and be consequently, in an improper sense, a curve' having a contact of the a-th order with the given curve U m , I assume that it counts as such curve a times, and this being so, we have, on account of the curves in question, a reduction = a (6+ 1) (c H- l)...[n?i - (a + b + c ...)]'"'. We have in like manner for the curves passing through b points in the neighbourhood of the cusp a reduction b(a 4-l)(o + l).., [nn (a + b 4- c ...IT" 1 . &c.. and hence when the ffiven unicnrsr 1 ~ T7 "" w '--- - - SR OX THE CTJBVES WHICH SATISFY GIVEN CONDITIONS. | .f * a .-.mi.-- (tlint i* j-=2) first in terms of in, n, K) ami finally in limns oC m, . (a ~-;j ( , .f A - ., s nVtvc). The results are for ,-='2, that is, ourvo O 1 ' n onnio. 2,jy + i, B+ 4w+ (IH+ 8 [1J- 2.- 2A- A i-]- aii-r A- 2 (!- 4n,i-f 12A- 3 HJ= i>j,.n- 20A- 4(r W- i;^..-i. SOA- U K , = _ 18ffl+ lfin + (1- 1J- - UV;i! - 20A+ IK ' "'" ( ' A " i " 16/f = 12iit- B0- ~ Jx = 56m - BSit - 4Gx \ = 140m - 180 - 144w- 82+ 'JlH- HH [1. 1,1,'J]=- W)Gj m _ 2(}7 Till 17 i^n on.- i each ca,e equal.) - In the case- of the (1 ::) (2 .-.) 10,. 628+ aH% = - 1184m + 1864M+ 780,, = - 1188,,,+ l8fifl B+ i2wi-i- m,, ... |,| a fi(im-|. -||,,., ;n, u HOWH- 132- Kir, WJH-I. dH- .f() r , = - 82wt = - 88m = - 118dm- lOWw-i- -- 118w*- lllfl,|. H = 820m-h SOlOw- ail = lOOfiflHH- lOlifif!,;. - 10018m- that in th iftHi , tho cooffioio.itH of cases i - 3fl+ 3fl (5) 406] ON THE CURVES WHICH SATISFY GIVEN CONDITION'S. 239 2 (1, 1 .-.) = (2i + )' 4m n 3a ; (1, 2 :) = (2m 4 ft) a (1, 3 .) (I, 4) 2 (2, 2 -) (2, 3) G (1, 1, 1 :) = (2m + ,) (- fan - 3ft -1- Sa) 56m + 49 -39; (2m + n) (- 10m - 8ft + Go) 140rn + 122 -84 [(m - 3) (- 12m - Gn + 60)] (= - 3) (4i + 2)]) ; a (~ 4m - 3n + 3a) 144m+126n-90a [24m + Qn + (n - 12) o] (= - [6r + ( - 3) ]) ; - 4m - n - 3a) 78a 2 (2, 1, 1 .) = 2 (3, 1, 1) 2 (2, 2, n) (law-*- i^. i.<*an- i w u- -336m -336ft 4 288a - [2a (m - 2) (m - 3)] (= - [6ft (m - 2) (m - 3) + 2* (m - 2) (m - 3)]) ; = (2m + iif (- 4m - 3?i + 3a) 4- 2 (2m + n) (56m + 49w - 39a) + ^_ 4 nt _ w _g)( 4m 3ft + 3a) -1184m -1094ft + 786a '-31Qwi-226ft~l; (= + (2m + n) (64m 4 48?t - 40a) -11887ft- 1110ft -820 S(2?i + 10m -38) 4 4 (6m 2 - 41m + 69) + T (8m - 32) OX THE CTBVES WHICH SATISFY GIVEN CONDITIONS. '406 2-* (1, -fa (- + IS ( wt - 4) r + -3) (4* + 2*) + 4 (2w + w) (- 32i - 58. + 78a) + 2208m +2610-2858a L-iiGfl (t-2)( t -3) (=-[(m-2)(m-3 14m (2, 1, 1, 1} = 3 (2m + w ) 68n + 78s 10650m + 10656?i-8016 1040m + 24 - 2256) where the correction is -(m-4) ' - (?B - 4) .+ * (+16n-06) 406] ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 120 (1,1,1,1,1)= (Sm + w)" + 1 (2m + n) s (~ 4m - n - 3a) + 10 (2m + tt)(- 32m - 68 + 78a) + 1 (- 4m - w - 3a) (- 32m - 58?i + 78a) + 5 (2m + n) (2208m + 2610 - 2358a) 241 102912m 31m 6 + - 810m 4 - - 235m 3 - + 10690m 3 +16080nm +960 + 2970m 3 + 900m -15630m - ^+ 28440 a a (135m -540), where the correction is = -(rn~4)f - 186m 8 -979m a + 6774m *) , + n (70?ft 8 - 180m 3 - 1750m + 9060) /~210m fl -h2130m-7110' 180tt m-1 which is = - ( m _ 4) / 31 (28 + 8/c) 3 + 110 (28 + S) (2r + 3t) + (-^m 3 + 1142m + 3174 )( *- t) + (-14m ~638- 1524 )(2S + 3/c) + (- 390m -I- 110?i + 4272 ) (2r + 3t ) + (- 210m 3 - 180mw + 2130m+ 990?i - 7110) but I have not sought to further reduce this expression, not knowing the proper form in which to present it. 92. The question which ought now to be considered is to determine the corrections or supplements which should be applied to the foregoing expressions (a), (a, 6), &c,, or to their equivalents [], [ti] [6] + [a, &], &o. in order to obtain formulae for. the cases O. VI. 31 242 ON THE OUBVE8 WHICH SATISFY OIVHK OOMDITIOJffl. [ 40 analogous to the formute I which "t L Vf H ^ M "" tra sfo ti< not considered. 2 P ' "' thls ls a question which I have which A,x No. ana ly tical Annex No, l to in the notice of D E JoNQUlto , memoil . of x - of i t i ' Wo b the '', L lse linearly and homogenco,,^ , f - + 8 T 1 these -"efflaientB so param6tera ' Ass me ' ""' " le m ' Ve should f ' of the pam.netes the pammetevs (, but ?t is oonvolL as to reduce as far as nossible number of the .series (that i s , botwaai the coordinates of a question must be shall pass through a (m that expressed b o a r therein as belonging To ft. Z" I !' t f m ' Ve "W^B the curve parses thrfugh a gTvo.fpoint'Z'r 'v **" " mtats)! that is ' when are given a s the LelS P 7l t iS l^ T^ f ^ P" number of the curves i US by an omal mefM lo s; the u on a noinf t . thlll & tak S parameters to be th. C T ""r^.^ th ^' "> Moreover ' the "dition that the cnrvo ^ pmmetm a HneM ' ,,,. ., . 406] ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 243 of the order n containing linearly and homogeneously the w -I- 1 coordinates of a certain (&) 1) fold locus of the order N. It is only in a particular case, viz, that in which the (to - 1) fold locus is unicursal, that the coordinates of a point of this locus can be expressed as rational and integral functions, of the order N of a variable para- meter 8 ; and consequently only in this same ease that the equation of the curves C n of the series of the index iV can he expressed by an equation (*$, y, z) n = 0, or (* JflJ, y, 1)" 0, rational and integral of the degree -A r in regard to a variable parameter 6. If in the general case we regard the coordinates of the parametric point as irrational functions of a variable parameter 8, then rationalising in regard to 6, we obtain an equation rational of the order N in 0, but the order in the coordinates instead of being = n, is equal to a multiple of ?z, say qn> Such an equation represents not a single curve but q distinct curves C n ., and it is to be observed that if we determine the parameter by substituting therein for the coordinates their values at a given point, then to each of the N values of the parameter there corresponds a- system of q curves, only one of which passes through the given point, the other q 1 curves are curves not passing through the given point, and having no proper connexion with the curves which satisfy this condition, Returning to the proper representation of the series by means of an equation con- taining the coordinates of the parametric point, say an equation (*$#, y, I)" 1 0, involving the two coordinates ((&, y), it is to be noticed that forming the derived equation and eliminating the coordinates of the parametric point, we obtain an equation rational in the coordinates (so, y\ and also rational of the degree N in the differential coefficient CL1I ~; in fact since the number of curves through any given point (# , y ) is =N t the differential equation must give this number of directions of passage from the point (a) , 2/ ) to a consecutive point, that is, it must give this number of values of ~~ , and (lit) must consequently be of the order N in this quantity. Conversely, if a given differential equation rational in <o, y, -p, and of the degree ft?/ JV in the last-mentioned quantity -~, admit of an algebraical general integral, the curves represented by this integral equation may be taken to be irreducible curves,, and this being so they will be curves of a certain order n forming a aeries of the index JV; whence the general integral (assumed to be algebraical) is given by an equation of the above-mentioned form, viz, an equation rational of a certain order n in the coordinates, and containing linearly and homogeneously the tu + 1 coordinates of a variable parametric point situate on an (to 1) fold locus. The integral equation expressed in the more usual form of an equation rational of the order N in regard to- the parameter or constant of integration, will be in regard to the coordinates of an order equal to a multiple of n, say =qn, and for any given value of the parameter will represent not a single curve G n , but a system of 5 such curves : the first- mentioned form ia, it is clear, the one to be preferred. 31 2 244 ON' THE CURVES WHICH SATISFY GIVEN CONDITION'S. [40G Annex No. 2 (referred to, No. 17). On, the line-pairs which pass through three given points and touch a given conic, Taking the given points to be the angles of the triangle formed by the linos (a= 0, y= 0, 3= 0), we have to find (/, y, h) such that the eonic (0, 0, O,/, g t hfa, y, 2f = 0, or, what is the same thing, /^ + ^' + % = 0, shall reduce itself to a lino-pair, and shall touch a given conic (1, 1, 1, \, /*, v \, y, *)=*0. The condition for a lino-pair is that one of the quantities/ g, h shall vanish, viz. it iafgh = Qi the condition for the contact of the two conies is found in the usual manner by equating to zero tlio discriminant of the t^i-fr + 8fy-(p+0g?-( v+ ehy + 2(\+ef)(r + $g)( v + 0h)**(a, b, G> snppo.se ; the values of a, b, c, d being = ~ -I (/" -I- c= v- Hence Considering (/ (J) h) as the coordinates of the parametric point, wo have blio discriminant-locus a = 0, and the contact-locus a?d* + -tac 3 + Wd - 36V - Gabcd = 0, and at the intersection of the two loci a = to fMij Q^\ A , , t ^i , , ' U) y t* 0ct ~ c ) = 0, eouationa brenkinn- mt " 8yStem (a = ' 6 = } "^ imd ^ S ^ m -O/^lsko; 'tha 2, of = 0, /H 5" + A' - A - 2HF - = 0, ^ noticed above thisto' " /l = ' "^ / a + ff a -2X/ = 0. A. The second system is =o, "" - + (* ~ ti g + (/A - ) A)' = 0, <*, s the second equation may also be mitten, -.Jffi ( tou tr . "(*+)-0. in which the Bne *v + -0 t^C ^ ' ^ ha hel ' S the Unfl -P odr and the like if , . 0, or if h = . T^ Jy te m 1 ^ 0m ( ' '' '' X ' "' "^ ' -LMS system it has been seen occurs only once. 406] ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 245 Annex No. 3 (referred to, No. 22). On the conies which pass through two given points and touch a given conic. Consider the conies which pass through two given points and touch a given conic. We may take #=0 as the equation of the line through the two given points, and then taking the pole of this line in regard to the given conic and joining it with the bwo given points respectively, the equations of the joining tines may be taken to be X**Q and F-0 respectively. This being so, we have for the given points (A r =0, #=0) and (F=0, #=0) respectively, and for the given conic aX'+ &P + 2hXY + c^ 2 - ; and since the required conic is to pass through the two given points its equation will be of the form = 0, where (x, y, z, w) are variable parameters which must satisfy a single condition .in order that the last-mentioned conic may touch the given conic. The condition is at once seen to be that obtained by making the equation bo considered as a cubic equation in X, have a pair of equal roots ; or if we write = - ax* - by* cz* + 2fe (icy zw), D = then the required condition is Hence the conic wX* + 2a; YZ 4- satisfies the prescribed conditions, if only the parameters (as, y, z, w) satisfy the last- mentioned equation, that is, if (to, y, z t w) are the coordinates of a point on the sextic surface represented by this equation. The surface has upon it a cuspidal curve the equations whereof are A, B, a =0; 5, (7, D '240 ON THE CUftVES WHICH SATISFY GIVEN CONDITIONS. /tQ<> this iriiiy be; considered as the intersection of the qnadric surface Ad -./&-() ami UK? cubic .surface AD- SC'= 0; and the cuspidal curve is consequently a soxfiic. The surface has^alao a nodal curve made up of two conies; to prove thin I writo lor shortness & = /<~V6, ^ = h + ^ab; the values of A, , 0, D then are A = ScA'AJ] , S ~ - kL\w - G = - ax 3 - b and it is in the first place to be shown that the surface contains the conic is : y : z : w = /a ' whore 6 is a variable parameter. Substituting these values, wo have and hence val.es which Sa ti sfy identic^ the equation rf fco subsfcitute M'.W:M = Q>fr.M t Z then the derived equation s )aa0 406 J ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 24 that is, It- I 14 tb I O-L or finally - k (A - 3.Bfc + mG - I<?D] = 0, which is satisfied by the foregoing values of A, B> G, D; hence the conic is a nodi curve on the sextic ; and by merely changing the sign of one of the radicals \/o, ^l (an therefore interchanging Jc, lc^) we obtain another conio which is also a nodal curve c the surface, that is, we have as nodal curves the two conies \a:y:z:wt=0/bi 6*fa, : 1 : -kP + j, and at : y : s : w^B^/b :-tfVa : 1 : -^5* + ^. It is to be remarked that each of the nodal conies meets the cuspidal curve in tv points, viz, writing for shortness ** r \/~ C ,-- . , = r \/ -- for the interae K v /C'i KI v 1C fcions of the first conic wo have x : y : z \ w = Va : V& : 1 c - and = - @ and for the intersections with the second conic K ; y : z : w~%^& >. ~%^b: .1 : r and =-^ 248 ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. [40(> of the sextic surface leads to the following values (agreeing with thoau obtained from the formnlro by writing therein m = n = %, a = 6), vk (1 ;:) =6, = 27/1+71, (1, 1 /.) = 4, = 2w- + Zmn + $n* - 2m - fri - {fa, (2.'.) = 0, = , (3:) =4, =-4m-3ft+3a. I remark that the section by an arbitrary plane is a sextic curve having cuaps and 4 nodes; it is therefore a wrimmd sextic; this suggests tho thoorom that bhu rn.xl.u- surface is also W ^d, viz. that the coordinates aro oxpi-oaaiblo lutionally in toniiH of two parameters; I have found that thia is in faet tho cruse. In doing fchi 8 Uioro is no loss of generahty m supposing that a-6-o-l; and aiming That thin IK .,, and puttmg also -1+A^, 1 + 4 = ^ and therefore 37^^ + ^ w h ^ vo The equation of the sextic surface being, as before, AW 3^0 - C>A80J) 0, .1 say that this equation is satisfied on writing therein e (, 0) are arbitrary, fc f act theae . ^ whence, being arbitrary, We hftve , Cf, J?^ j)t - - [* co 406] ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. viz. the equation (A t S, 0, D^ca, 1) 3 = 0, considered as a cubic equation in o>, has the twofold root w = -a, that is, we have the above relation between (A, B, 0, D). Whence o-\ -i _ \ a also writing sin^ = ^ _, cos< = , the equation of the surface is satisfied by the values - : z : w / 2 = A/- ^ (1 - /c,a) V ri.^ : 2 - - (1 - \) + 2 - r \ 7 \ Is or the coordinates are expressed rationally in terms of a, \. Annex No. 4 (referred to, Nos, 22 and 71). On the Gonics which touch a cuspidal cubic. In tho cuspidal cubic, if a = be the equation of the tangent at the cusp, y = that of the line joining the cusp with the inflexion, and # = that of the tangent at the cusp, then tho equation of the curve is f=a?z; the coordinates of a point on the cubic are given by as : y : z=>\ \ : 6 s , where 6 is a variable parameter; and we have, at the cusp #^co, at the inflexion = 0. In the cubic, m = = 3 ) a (= 8n + K) = 10. Considering now tho conic 250 ON TIIF, oimvKM wiiirn HATIMI-A* <;ivrv OI\HMHI\M. (int. ivtid wo lliim obtain Uio mjimtiioii ol' lh*< runi^'l Ixnr, in Hi- l>-itn ((to -'\llli -I- Hi:') 1 - 2V In.-,' | ^ftfil \nt- AV I '>' - H. lioh is a oniifnld locim of Mm unlnr li. It- lii!loW! llin? r hut,^ (UI, I ;.) '1. "Kivimif! with iK'l. 1 i- n -,,., ;i. whioh in a oiirlultl lntntH of Mm nnlcr ^; ir it! .MHT i'..]|.fiM i|,-,r (Id, I :/)> IU, ti^ivnitiK witli iKJ, | I..M. Tli is a iimUiir of Hitmn ililllmilty IM nlmiv rlmr v,< hit',.- (![, I //) :i |M, i^ivt'iiiK with tUl. 1 * . but 1 iirmiitt'il to oniidl, UIIH, |'H M |, rciiinrl{ih K (bit I .|M ,,MI of Itm valuu (||, I . //) , IK ; "Wn liavi) thn noxl.ii! lomin and ciiiiiliiiufd tluirnwilh f,wo (* }, s , fc, /, ac..,:i,^ tl f M/,/ , ,tah into,,!, in a thr^lol.l I.H...H ,,r rlin ,, ( |, r ,*, lt 'mtun, IIH rt ol M { ,, hn ,,! IV , . thrao turuiH, lcavin K a msidnal W may iiuaginn (,1m ,,,. ,, ( R( ,. f/ ft ,^,,,^^.^1 ,- four ciwnliimluH, and HII i-iidui'n iht* iu-,,1,1.,',./ rJ t 't!' Hlw i ii,vui,u |,|jp j)tMuh?ni lnnn i" --' - > HlOllttl H]HM'|t. Wn luiVM Uli a (Icivulopaliltt nnrf, wv tir ^ tf (a0, 60, Ucfl-^Mfh wl'n'li, '" ' l t ! U ' lrtk! l ' nrvi> I sin K Uirwuyh M. IWM \ v, ^ w, oca . MUJ, \vintsi! art( ]l(||(lU j , _ * t of tlioflo pomb counts ihwo Untw mnur,, M,,, n,,^^ ! ' !" uurvo, tiio nnmbor of it in to b 406] ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 251 is situate on the nodal line of the torse, and that the quartic curve touches there the sheet which is not touched by the tangent plane a = 0; for this being so the _ quartic curve touching one sheet and simply meeting the other sheet meets the torae m three consecutive points, or the two points of intersection count each of them three times. The torse has the cuspidal line fif = ae-46rf-l-3c a = 0, T = ace + Zbcd - ad?- fre- e a = 0, and the nodal line a and the equations of the nodal line are satisfied by the values (a = 0, &0, 3ce-U -0) of the coordinates of the points in question. To find the tangent planes at has points, starting from the equation ff-27P-0 of the torse, taking (A, S, 0, J), Z}- as current coordinates, and writing = then the equation of the tangent plane is in the first instance given in the form. SSs-lMW-O, which writing therein (a - 0. 6-0 B Cfl -W-0) assuine. as lt shouM do, the form = 0; the left-hand side is in fact foun^ Mo Proceeding to the second derived equation, tins is JSW+2S0S) - or substituting the values of the several terras, the equation is the terms in BO, BJ>, 0* vanish identically, that in * is (48-36 . .Ie7(to-W)i. wWoh also vanishes; hence there remain only the terms by A, giving first the tangent piano ,1 = 0, and secondly the other tangent plane, + ^.27o" =0- Taking the equations of the quadrio surfaces to be (\, p, v, p, o-, Tja', 6", o&, 806-2*. M (V, ^.^ p',^, 252 ON THE CUBVES WHICH SATISFY GIVEN CONDITIONS, tlie equations of the tangent planes are p (3oE + SeC- 4dD) + a- (eA - SdB) + T (dA - 12cB) = 0, "406 in all which equations we have 3ce 2tZ J =0; and if to satisfy this equation wo write c : d : e=2 : 3/9 : 3^, then the equations of the tangent planes become ($ (Aft - 8B) + 8 (30/9' - 4D/9 + 27?) = 0, (3GB - 4D/3 + 2) + (<r/3 + T) (,4/9 - 8.5) = 0, or the three tangent planes intersect in the line .A#-8# = 0, which completes the proof. Reverting to the sextic locus, (ae + 4bd - 3c 3 ) 2 - 27 (ace + Zbcd ~ ad? - fre - c 11 ) 2 = 0, considered as a locus in 4-dimensional space depending on the fivo cu.n-clinatiw (a, b, c, d, e), this has upon it the twofold locus ae - 4bd + 3c 3 = 0, ace 4- 26ccZ - CM? - b*e e 3 = 0, say the cuspidal locus, of the order 6, and the twofold locus '"' ;-6 3 ), S(ad-bc), ae + 2M-3c a , 3 (fie- erf), 6(ce-d a ) =0, say the nodal locus, of the order 4; there is also a threefold locus, a, b, G, d ~0, &, c, c, e say the supercuspidal locus, of the order 4. We thence at once infer (1*1, 2 ;) = 6, agreeing with (jl, 2 :) = a-4, [1 1 ,\ = A ,', r - . ' '' ' " t-i-*-*-) 1, J- :) = 2)/i a + 2m)i + -i)i B ~ 8??; ' } but I have not investigated the application to the symbols with ./ or //. the cusp, ! = of the cuspidal s ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 253 isumo the equation of the conic to be (d, 3&, 0, 0, a, ;}c.';, y, z)-=0. The equation 1 the contact-locus then is K. till is is a developable surface, or torse, of the order 4, and we at once infer (2*1, 1:) = 4, agreeing with (21, 1 ;) 2m + n 5. will nhow also that we have (2*1, 1 /) = 6, agreeing with (2*1, 1 /) = 2m. + 2w - 6, ml (2*1, 1 //) = 5, (2*1, 1 //) m + 2n - 4. 'ho condition that the conic may touch an arbitrary line KOI + /9y + 70 = 0, is in fact finch, considering therein (a, &, c, rf) as coordinates, is the equation of a quadric surface lassing fchrough the conic - 0, 4M-3c 2 =0; the quartic torse also passes through hiH conic; hence the quadric surface and the torse intersect in this conic, which is ,f tiho order 2, and in a residual curve of the order 6; and the number of the ionics (2*1, 1 /) is equal to the order of this residual curve, that is, it is =6. If the conic touch a second arbitrary line ' + &y + /* = 0, then we have in like imunur t.ho quadrio surface (0, -ft, f(4M-3c 3 ), fac, -fftft, OJ[', ff, 7) 2 = 0; ihat in wo have the quartic torse and two quadric -surfaces, each passing through the onie rt0 ^cZ-Bc a -0, and it is to be shown that the number of intersections not in this conk' IK - 5 The two quadric surfaces intersect in the come and in a second -rmic- this second conic meets the torse in 8 points, but 2 of these coincide with the noint'(t0 6=0 0=0, which is one of the intersections of the two comes (the point ^Tn Lo 1 = in in fact a point on the cuspidal edge of the torse, and, the conic passing through it, reckons for 2 intersections), and 1 of the 8 point, comcide 8 with tho other of tho intersections of the two comes; there remam therefoie 8-2-1, -5 in tor sections, or wo have (2*1, 1 //) = . Annex No. 5 (referred to, Nos. 22 and 71). On the Gonics which ^^f^^ of the cuspidal cubic be fc-!/'-0 (*-0 tangent at c 1( Hue joining cusp and inflexion; equation satisfied by and let the equation of the given conio be Z7=(a, 6, o,f, g, 254 OX TEE CUBVES WHICH SATISFY GIVEN CONDITIONS. [40G then writing - c^ + 2/0 1 + 2t/0 3 + 6# 2 -H 2A0 + c, thy equation of a conic having with the given cubic ab a given point (1, #, ^ a ) contact of the second order, and having double contact with the given conic, is = \t, viz. in the rational form this is a-, y, z Ve, i, 0, & (Vey . i, 30 s (V)" . . 60 V, i, 0, ^ 3 (V@y . i, 3^ . . 6* viz. this is (VI)" . 0, or developing and multiplying by * this is or, what is the same thing, and substituting for its value, this a) ^ , = , s 406] ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 255 The coefficients of the powers 16, 15, 14, 13 of 8 all vanish, so that this is in fact an equation of the twelfth order (*p, 1)" = 0; and putting, as usual, (fcc-/ 3 , ca- f f, ab-h*, gh-<tf, hf-l>ff,fg-oK)**(A, B t 0, F, G, H), the equation is found to be + SOoff I 'i' -|- IftJtt-U 6^ - 20/C - 36eS) - 22W?, + lOAffJ + 16/Ar + * 0ft "^ + 6hF] -lOcff" - ISO/iff /to Oa ' + 20M. \ + 40/F - 12rf, -60/B a + 33AJ5\ ~ 5;tC/ - 0* w = 0, here the form of the coefficients may be modified by means of the identical equations (A, H, fffa, kg) = K., (H,B, (ff, J 1 , There is consequently a conic answering to each value of given by this equation, or we have in all 12 conies. In the case where the given conic breaks np into a pair of lines, or say, (a, &, c, /, ff t lifa, y, 2) B - then, writing for shortness we have (A, B, G t F, <?, H) = (^, P, &, YZ, ZX, XY). 256 ON THE CUBVES WHICH SATISFY GIVEN CONDITIONS. [40(5 Substituting these values, but retaining (a, 6, c> f, g, h) as standing for their values 0.-2XV, &c., the equation in Q is found to contain the cubic factor 2X0 J - 3 Y&* + %, where it is to be observed that this factor equated to zero determines the values of 6 which correspond to the points of contact with the cuspidal cubic of the tangents from the point (X, Y, Z}, which is the intersection of the linos Tuw + /*# + y* = 0, ami ' i'*=0; and omitting the cubic factor, the residual equation is found to be icA- -12cF -8A- -12/F -106AT * 3//F :*M| -3- 17AF : ! s -i-a^ where the form of the coefficients may be modified by means of the identical equations The equation is of the 9th order, and there are consequently 9 cones. Annex No. C (referred to, No. 48).-tfontei fl r l with the variation referred to in the teat, ZEm'HEN's/oms/or the cAaraofemtfos of the conies which mfofy four conditions. (1) (1, 1) 2m, = 2m( 2 (2m T, (1. 1, 1) - tt ' - 80m- - 84m - + 3n - 26) 406] ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 257 + 4 (m 2 - 11m + 28) T -1-2 (n a - llw + 28) S + 2 (m 2 - llm + 28) T -1-4 (?i 2 - 11 + 28) + (4 (n - 4) (w - 4) - 1) (8 + 2r) + S a -1- 2r a } ; (2) (.-.)= 3m +t, (.//) = 2 (3m (2, 1) ( . / ) = 2 (3m 4- (m + n - 12) + 24 (m -H ft), ( // ) = 3 (m" + 2mn - 10m + 4n) + (m + 2 - 14.) (2, 1,1) , x . = (2m + n-7)(6T + (i-3)} 5) + T) (3m + * - 36) + ((n - m) (m + - 5) + $) (3m + 1 - 36) 2 2^ n 3") (.) 2(-4 (/)= 2( 3 (4) . = 107i- 10m + 6K= 8m- 0. VI. 258 ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. [406 .. Annex No. 7 (referred to, No. 93), In connexion with De Jonquieres' formula, I have been led to conaidm- tho following question. Given a set of equations : a = a (viz. b = b, c c, &c,), ab = ab /viz. ac= ao t &c., and the like in all tho aubHuquunt equations + ( ll)a.6\ + (H)a.c, abe = abc + ( 12) (a. be + 6.00 +o.a&) + ( 111) a.b.o, abed = abed + ( IS) (a. bed +&c.) .cd +&c.) + (1111) a.l.o.d, and so on indefinitely (where the ( ) is used to denote multiplication, and ab, aba, fed., and also ab, abc, &c. are so many separate and distinct flymbolH nob oxpixifmiblo in terms of a, b, o &o., a, b, c &c.), then we have conversely a sol; of tu a (viz. 6 =b, c=c &c.), /viz.ac= ac c., and the like in all tho Kubsoquont cquationw + [ 11] a.b\ +[ll]a.c, abc - abo + [ 12] (a . be + b . ac + e , ab) , +[ 111] a.b.o, abed - abed + [ 13] (a. bod +&c.) + [ 22](ab.cd +&c,) + [ 112](a.b.cd + [1111] a.b.o.d, 40G] ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 259. and so on; and it is required to find the relation between the coefficients < ) and [ ]; we find, for example, [11] = - (11), [12] - - = 3 (111), - (13), [22] = - (22), . - [112] 2 (IS) (12) + (22) (11) - . (112), [1111] = -12 (18) (12) (11) + 4 (13) (HI) . . - 3 (22) (11) (11) :, ' + 6 (112) (11) (1111) j and it is to bo noticed that, conversely, the ooofflotenW ( ) are given in torms of I he Efficients [ ] by the like equation, ,ith the very same ~al m fact from the last set of equations, this ia at once seen to be the case as feu as (112) ; and for the next term (1111) we have - 4 M [3 [12] [11] -[111]) + 4 [13] [HI] + 3 [22] [11] [H] +(3~fi- >- 3 [6][11][U] [11] \ +[22] [11] I -[112] J having the same coefficients -12, +4, -3, + 6, -I as in the formula for in terms of the coefficients ( ); it is easy ; to infer that the property hold generally. To explain the law for the expression of the coefficients of either set in t 260 ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. '400 the other set, I consider, for example, the case where the sum of the mimbora ia tho < ), or [ ] is = 5 ; and I form a kind of tree as follows : 13 11111 22 112 1111 H 111 12 HI 12 11 11 111 12 11 H 11 11 H Jl II [14] = - (14), [23] = - (23), [113]= 2 (14) (13) + (23) (11) (113), [122]= (14) (22) + 2 (23) (12) (112), [1112] = - 6 (14) (13) (12) - 3 (14) (22) (11) + 3 (14) (112) - 6 (23) (12) (11) + 3 (113) (12) + i (23) (111) + 3 (122) (11) - i (1112), 40 G] ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 261 [11111] = + 60 (14) (13) (12) (11) - 20 (14) (13) (111) + 15 (14) (22) (11) (11) - 30 (14) (112) (11) + 5 (14) (1111) + 30 (23) (12) (11) (11) - 10 (23) (111) (11) - 30 (113) (12) (11) + 10 (113) (111) + 10 - i (11111). To form tho symbolic parts, we follow each branch of the tree to each point of its course : thus from the branch 113 we have (113) belonging to [118], (113) (111) [111H], (113) (12) [1112], (113) (12) (11) [11111]; viss. (118) belongs to [113]; (113) (111), read 11 (3 replaced by) 111, belongs to [11111]; (113) (12), read 11 (3 replaced by) 12, belongs to 1112; (113) (12) (11), read 11 (3 replaced by) 1 (2 replaced by) 11, belongs to [11111]. And observe that where (as, for example, with the symbol 122) there are branches derived from two or more figures, we pursue each such branch separately, and also all or any of them simultaneously to every point in the course of such branch or branches ; thus for tho branch 122 we have (122) belonging to [122], (122) (11) ^ (122) (11) (11) [Hill]. Similarly for tho branch 23 we have (23) belonging to [23], (23) (111) .. [ m2 ]> (23) (12) [122], (23) (12) (11) (same as infrcl) [1112], (28) (11) (111) " [Hill], (23) (11) (12) (same as siqml) [1112], (23) (11) (12) (11) [Hill]- 262 ON THK OU.RVI'M WHICH HATIHKY (IIVKN ('O.NIHTln.VM. We than obtain thu Hyuiholio parts nf t.hn Hcvrral rNprrwinnM I'm' ||>||, |'J!I|.. , . 1 1 1 1 1 1 1 respectively: tho sign of ouch tonn IH + or - nrruriliittf in lli,< iuii.il, n ,!' la-i.n., in ( ) if* ovoii oi' odd; lilinn in tho oxpn-NHmii lur 1 1 1 1 I 1 1, lh,< |,.,ni ( M-K IM) { I'.!) ( 1 1 ) has four factor*, mul in UuTofnro .)., |,ho turn. (I !:)< li!)( 1 1 In., ilm-,- lii,.|,,| fl . m ,l i-, theroforo . ^ Tho numorioal (wflidionlH am ohtainnd UN liillmvM. Tht-iv j ,.,. ..... ,,.1, (iu-ior donvcHl from tho oxprctwi.m in [ ] un Mm lufMmn.l mi] ( , ,,r |,|,,, nprnlinii; vj,:, (;,. [11111J, wliioh contaiiiH llvu oijual HjinliolH, UHH furfur in t , ;!. ;i . |., ,",, . |>i n , n' (| . ||J|.H which contaiiifl bhiiHi uiinul HyinlxilH. il, IH I . . -| ( (! ; (U1 ,| , ...... , ,,;,', n ^.^ such as Li.1222] ooiitninin^ two equal symlmls, un.l i.h,-.-,, nj.iul ^ml,..!,, lit-' 'd.-iur woud bo i. 2.1. 2. 8, 12, and HO in ,.!. H i,nilnr ,,,.). lu m,v !,,, ,. ihn rU, . hand ado cil th u uquttfci,,,,, w , IIUts(l , i(l . a | h , (ill ,. Hl|( . h |ls , ,, , ' ivhj|1 , ( l ,, II , ai|ri ],, equal flymbos .miltiply by i; fur a fnntnr ], ll!f ( || , ,, vv | )irll ,,, ....... , symbols, m^ by ii( and nn. And in i\ w ,,,, win,,, u ,, tlH . .',, ,x,,, ,|, m ( 122 )(,,) or (gs)(l8)< , , vMtl }} (|tr(|| , N imin . ||IHI| ( ; ' I' - ' ( bo taken aooonnt of HU , h U.no thai, it n,mn,; n, wlmi in ,|,,. ,,. ltim , .MHV ,,, oooffioioiit obta,nod nbovo in tho fi,r ra.h ,,.,.,,, , tll , , ..... ,!,:, l]l(lu , - above to bo multiplunl by thn nn.nbor of ,h, ,,,,, ,., ,1, L n cxumplo, talcing in ordor tho s,voml h-rnm of U,,, ,M,,,SM ( ,,, fll , ,.,,-, ,,' Actor iB-0. and tho Hovoml ooollini.nl.s a,v ' ' '' "' (fIIIIIM " h '1204. 1 20. A. II O.J. 1.1. ^.., nL: the whole InorenHocI by imUyuJlo; vii fi- l "'" 1 ""' i " tl >'" ttlllitl1 '"**^.l u,,-! [ 3 *lPl[n.uiia2] 1 [nia],r mn i the sums of tho oooffioionlH aro and wo have ~ ' ' '" 2 ' "'* a ' " (i . +24 n-*ipfrLivlv. 1 + (-l) ' ^. i" 5" r r partitioned in tho forms 1.4,23 Mm I m> n " f W " P '" whioh lhtt i bo theorem is Uwa a theorom in tho ft^UoJ. If ^ {^y. nd tho iMHnenUonod " ..... 264 HI-ICON 1 1 MKamnt UN 'nu; i.tn" i * numbor of tho propni 1 solutions, 1 IIH tJiiw inmll i lu il> himiiiM ,; ;<>'-,M t . lM im > n, h CMC tho oxpruHHinn of tho Sn|i]ilt'tiiiml. ; l.hr r\|n.-> .inn M. ..hiuin. .1 . ;( !i m .,-ia. 5 ., , bo tujcounliod for m\<\\\y omm^li, and Mm l<ini\vlf<I^i- ! tli.< ulil vu.-, ,( rlu-m v,iil bo a convonionli IHISJH lor ull.drior ii in'c for pniufn in a lin> th papor in tho dtitiijtton ,/iVW//,v, June July I Hill. ivt.>nr.l (.. ,., .; Kn y M, ;IJI . FJI in oxtiindtid to unicnmil (inrvrw in a pup,.r ,,t ih,- w.i.i.- <,<-n. ..,, M.ut-li |Miitj nun on a lit^aUon (lu not lio tho MVHO of u oiil-vu of K ivt'ii .lulirirnry 1) r..i r .t<|.>i.'.| ih .,> ,,,,, , ,,l a (Hirvu uf tho onln- w will, H W ..|)( H( jj) rt ' M)l |.|, ,,.,,,,1^ n..^^"'.^ coimlniiiliUH (:, ?A j) im > |imii..r|.i..i,,il to ,- (l ii, t , )l( | ttll ,| ;,..,.,, I|l . (ill ,, l> ti , _... i i\ - * ' 1, but , pp 4M (1% ,,,!, my |M| H.r--.in i|, ml .StoiV//. No. Ul., (Vi. l.sii.'i. |;IH||. HATIHFY (HV.UN CONDITIONS. 2G5 JJii, Siippusi! Unit Mm nHTifMpowlintf points aro P, P' and imagine that when P givmi tin- c.niTtinpiiMiliiig puinU P' an- tlm inliwioc.binnH of him givon curve by a curve (l.hi' i>i|iinl>itni ut 1 Mu! I'lirvn H will of (ionrw! contain lihu eoowUimtus of P as TH, fur nUii-nyim' llir position of .!" would not clopond upon Llmt of P). I l.liitt. if Mil* nwvn H Ims with !.lui givon uni'v*) & intciwocbions at tho point P, I hen in I. he n t vt.i'in tit' (mints ( P, /") Uun immbnv of nnil.iul point* is win 'lint 1 in puH'i HI lit r il' (.In- c-urvi* ( H ) ilntiH nol; pawn Mivnugli thu point P, then the iniiiiliiM 1 nl' uiiilrd jininlH in a -|. ft', UH In l;hn IMSU of a iniicnii'Hal cui'vo, (I have in Mii< pup'T nl' April IHlilj H.IIDVP niH>rri!(l to, provuil MUM Muiorani in tho particular case wlii'i'i 1 tin 1 /, iiili'i'm'i-l.iiinM IL(. MKI jmlni. /' tnko {ilmm in conKoipionon of the curve f 1 ) liiivin.iv H /.'-l.npli' pninl nl /', liul, have unl gone, intn thu inoro diflioulfc invsbigabion Hi] 1 Mu- nisi* wlicri' llir k inl.ui'si'rlioiiH jirisd wholly or in part from a oontaot of tlm i-iirvi' H, ni 1 any hrum-h nr hnuii'ln's l-hovnf, widh 1,1 HI givtiu nuvvo at -/'.) !Hi, ||. i |.<i In- iiltMi-rvnl thai. Um gintiMiil noLion of a nnitiMl poiuli is as follow*; tnkiii^ Mir ptiiiit /' n!, niniliiin on thn jfiviiii itnrvu, tlm (surv H hus at thin point b iiiliT.'ii'i'liiins \vilh l.hi- ^ivi'ii rurvn ; l.lm niiuaiiniig iniovHodliionH aro liliu cori'CHpoiiding p u iiilM /"; if Ihr a tfivi'ii psit.hm of /' onn or inoro tif tlm points P (some to miiiftili* wil-li /', llint. in, il' I'm 1 ihi' given jiosiMtiu uf P tho ourvo f) has at bhis point niniv limn / inh-iMi'i-tioiiN with tint givnii urvn f tluni tliu point in quoHtiim in it nnili'il ]'iitl, il infill HI IliHl. Mi^lil. uppi'iu 1 thrtt. if for n tfivmi position of P n iiuinbor a, Jt, . . nr / uf tin' pninl* /'' Mhnnl.l nuiiii !,u roinnidu with -/', tlmn that tlio point in question Nhimltl ivcLuii, fur a. .'1, ... nr ,/ (ns Mm nine niiy !') unitdil points: but thin is not w.. Thin !N pnliii|s HI, niHily HIKIM in tho mwt of a iinimnwiil ourvo; taking tho t .,,iml.j,iii i.r i'.,n-i-s|...n.lrnn- hi In. (& t \) a (0\ l) (t '-0 ( fchun wo luw fl-3-a' united points rurrrHpi.n.linx I" llio vului-K nf I) which wilisfy tins uiiniiliion (0, l)*(9, l^'^O; if thw cipmUitn hiw it /-Uiplc i'..,,|. lh--\. llw point /' which lumwui-K to this valuo X of the pani- nirti-r IK iv.-knn'.'.l im / unitr.l pmntK. Jint HLurtin^ from tlm ouuatioii (0, !)(0', !)' 0, if mi wriiiiiK in MIIH fijinilinn fl=\, tho ivnulting o<p.uition (X, \.) a (0' t iy0 has a root (;, X. il. K.II..WH ilu.l tin. tH|iiutinu (0, \Y(0, 1)"' hn a root = \, and that Ih,. p,.ml whirh IK-IIIIIKH lit U*i. viilmi tfX w a unitod point; if on writing m tho iini. a. Ih- rrK.ill.iHK ^jmition (X, 1)- <fl'. !.)' ImH u j- tuple root fl' = X, ^ HI./. /* Lhl, HH- .-.piation (fl, l)(tf, :i)"'-0 hoH a j-tuplu root 0**K> nor oon- itiiil. ihi- ]iuiiiL nnnwrnng to flX in anywiHO rockoiiB us j uiutod pomta. 26G SECOND MEMOIR ON THE [-107 gives a point which reckons as j united points. But if 6= A, gives fchu j-fol<l rnul ^ = \, this shows that the line 6-K has with the curve j intoraoetioiiH at fcho pninl, tf = ^ = X; not, that the line = 0' has with the curve j intoraoctionfi tit fcho point, in question. 08. Reverting to the notion of a united point as a point P which in fmoh tlmt mm or more of the corresponding points P 1 come to coincide with P\ in tho oiwtj whom P is at a node of the given curve, it is necessary to explain fclwfc tho point P nnisl- be considered as belonging to one or the other of fcho two branches through fchu nod.', and that the point P is not to be considered as a united point unless wo hnvn tin the same branch of the curve one or more of the corresponding points V oominL' Lu coincide with the point P. If, to fix the ideas, k = l, that is, "if'tho cui-vo H simply pass through the point P, then if P be at a node the curve pooum through tli'c node and has therefore at this point two intersections with tho givcm curve ; hub iJu- second intersection belongs to the other branch, and the node is "not a united point,- m order to make it so, it is necessary that the curve 8 hould at tho undo tuiicih the branch to winch the point P is considered to belong. The thing app.a.-H vni-y clearly m the case of a unicursal curve; we have here two values 01 x fl ,, b * r bianch of the curve; and m the equation of correspondence (6, l)*(0', :!)'(), tf-A, we have an equation (X, i).^. i)^ satisfied by ^' = V bub iirt by 0'~~X and ^^^i 1X= ! " thus , nofc " twted b ^ *' val - ^^at Jl a node qitd node is not a united point, V- V v i = x fo r> and the equati<m o , .mited point, the cp 1; r ^ t "T- "" the ' ., u to be rej e od , he > ' along with any other spec also tiomJ " P""" 1 W "^ illchl ' lu thu " !" and instead ^^^ "^ ^ " Sup, ,lu- -- write -_' + Sup p. = g/,/). oxamples in whfah 1;ho , Mll , vo JF ^ WitK * "^ "' " joi-nng them pass e s Lugh" a I ed 1 H, ^^/T P into 8Uflh thftt the linu of contact of the tangents througho tW ?.' t " ^ P UltS wU1 be tho P" 1 ^" be equal to the o.ass of the c. The el' IT Ti t *" I " litod ''' "' !11 the given curve a single intersection * P h h '" h " e ftP ^ih haa with corresponding to a given position of P are' tlT /C = L The l )ointa -?" ^ith the eurve, that is, W e have .*-?? ^-"T"* 8 " l ~ 1 ^^ of O.P , and m hke manner = m _ j. E(loh of tho 407] CURVES WHICH SATISFY GIVEN CONDITIONS. 267 cusps is (specially) a united point, and counts once, whence the Supplement is =tc- Hence, writing n for the class, we have n+2(m- 1) + n - 27), or writing for 2_D its value = m a -3Hi + 2-2S-2, we have 31 = m 2 - m -2S- SK, which is right, 101. Investigation of the number of inflexions. Taking the point P' to be a tangential of P (that is, an intersection of the curve by the tangent at P), the united points are the inflexions; and the number of the united points is equal to the number of the inflexions. The curve is the tangent at P having with the given curve two intersections at this point; that is, &=2; P 1 is any one of the TO -2 tangentials of P, that is, a'=w-2; and P is the point of contact of any one of the 11 - 2 tangents from P' to the curve, that is, a=-2. Each cusp is (specially) a united point, and counts once, whence the Supplement is =*. Hence, writing t for the number of inflexions, we have t-(m-2)-(tt- 2) + /e=4D; or substituting for 2D its value expressed in the form - 2m + 2 -f K, we have t = 3 3? + K, which is right. 102. For the purpose of the next example it is necessary to present the funda- mental equation under a more general form. The curve may intersect the given curve in a system of points P', each p times, a system of points Q 1 , each q times, &c. in such manner that the points (P t P 1 ), the points (P, Q'\ too. are pairs of points corresponding to each other according to distinct laws; and we shall then have tho numbers (a, a, a 1 ), (b, /3, $'}> &c., corresponding to these pairs respectively, viz. (P, P') arc points having an (a, a.') correspondence, and tho number of united points is = a; (P, Q 1 ) are points having a (/3, /3') correspondence, and the number of united points is -b, and so on. The theorem then is p ( a _ a _ ') + q (b - j3 - /9') + &c. 4- Supp. = 2M), being in fact the most general form of the theorem for the correspondence of two points on a curve, and that which will be used in all the investigations which follow. 103. Investigation of the number of double tangents. Take P' an intersection .of the curve with a tangent from P to the curve (or, what is the same thin# P, P' cotangentials of any point of the curve) : the united points are here the points of contact of the several double tangents of the curve ; or if T he the number of double tangents, then the number of united points is =2r. The curve is the system of the -2 tangents from P to the curve; each tangent has with tho curve a single intersection at P, that is, & = -2; each tangent besides meets the curve in the point of contact Q' twice, and in (m - 3) points P' ; hence if (a, a, a') refer to the points (P, Q^), and (2r, , /3') to the points (P, P'), we have From the foregoing example the value of a-a-a' is .*-*. In the case where 342 2G8 SECOND MEMOIR ON THE 4Q7 tlie point P is at a cusp, then the n-2 tangents becomo bho -,'! tungtmtH from the cusp, and the tangent at the cusp; hence the curve tmsots tho givon urvu in 2 ( - 3) + 3, = 2w - 3 points, that is, (>i-2) + (-- 1) points; this dooH noli pmvti (&,, Xo. 90), but the fact is, that the cusp counts in the Supplement (w - 1) tiniGH, unil tint expression of the Supplement is = (n - 1) /c. It is clear that wo huvu /3 = fi'~ (. ~ B)( m _ ;}), so that the equation is 8D that is - 2 ( - 2) (m - 3) or substituting for 3D its value --2*+2+ and reducing, thin is 2r = n a + SHI - 10a - 3, which is right. 104 As another example, suppose that the point P a K ivon ,,, , (ih ..... ,[ * and the pomt Q on a g, curve of tho order ' havo an (,, ') uom-H,,,,,,,!,^,. -Hi let ,t be reqmrecl to Hnd the ola SS of the ourvo envelope,! by h, lin, 'J ' arintar pomt 0, join 0, and let this ,neet tho cnrvo. in /"'>' i ^ a given line is ='+' + "' s - r ( " 10 Iill0a P( 2 whioh '"" by 103 to m . arbitrary. g .veu Olll ,e, (3^)( 2) ancl (3Z) , cond, 10 , 1S , and besides have with the rive, (- the case may be) two eontaets tClst p, 2, and then 6nally (5), (4, 1) f , have w,th the given eurve a oontao 'of t d also of the fa. o^,.., flye contaot ftm , " iOS whioh satis fy """T f th S8 OIld 1> 1 '' ' md on with the condition. numban! of tho >"' which contact of t ho TouHh 407] CURVES WHICH SATISFY GIVEN CONDITIONS. 269 10G. As regards the case (42) (1), taking P an arbitrary point of the given curve 7)i, and for the curve H the system of the conies (42) (T) which pass through the givun point P and besides satisfy the four conditions, then the curve has with tho given curve (42) (I) intersections at P, and the points P' are tho remaining (2w 1)(42)(T) intersections: in the case of a united point (P, P'), some one of the Hystem of conies becomes a conic (42) (1); and the number of the united points is consequently equal to that of the conies (42) (1); we have thus the equation ((4Z) (1) - 2 (2m - 1) (4,2) (l)J + Supp. (4,2) (1) = (42) (1) . 2D. 107. It is in the present case easy to find cl priori the expression for the Supplement. 1. Tho system of conies (42) contains 2(42 )-(*#/) point-pairs^); each of these, regarded as a line, meets the given curve in in points, and each of these points is (specially) a united point (P, P'); this gives in the Supplement the term m{2(4-)-(4<#/)j. 2. The number of the conies (42) which can be drawn through a Gimp of tho given curve is =(4?-); and tne CU8 P is lies pect of eaca ^ tnese (ionics a united point; we have thus the term K (4t2>),_ and the Supplement is thus = m{2(420~(4#/)}-|-(42.). We have moreover (42) (1) = (42-), 2D = H-2m + 2 + ; and HubBtitnting tliese values, we find which in right. 108 It is clear that if, instead of finding as above the expression of the Supplement, the value of (4^(1), (^-) +mW), hod been taken as known, then the equation would have let! to Supp- W (1) = m I 2 (*% ') - (*% />! + * (4 ^ ^ ; and this, as in fact already remarked, is the course of treatment employed in the remaining eases. It is to bo observed also that the equation may for shortness be in tho form _ 2 (2m _ + Supp. (I) = (1)22); viz. tho (42) is to be understood as accompanying and forming part of e; and the like in other cases. 109. Wo have the series of equations (42) ((1) -(1) (2m -!)-(!) (2* -1)1 = ( 1 ) LL) + Supp. (1) . The oxp,ossicm . point-pai, b ** as equivalent to ana standing ta that of a o*u BOO First Memoir, No, 30. SECOND MEMOIR ON THE [407 (32) 2)- (2) (2m -2) -(I, 1)| 4 Sl 'PP- (2) = 2(2)27;; (32) 2 ((2) -ft l}-(2)(2,-2)} + {2(1, !)-(!, l)(2, tt -3)-ft I) (2m -3)} + Supp. (1,1) =(Ij 1)2/); W f(3)-(3)(2m-3)-(T, 2)j + SPP.(3) = ;i(IJ)2/); (22) 2{(3)-(2,l)-(2 l l)J + 1(2, l)-(2, 1)(2 W *-4)_(T | 1, 1)2) 22) 3 {(3) -ft 2)-(3)(2-3)) + !(1. 2)-(T 1 2)(2 M -4)-(T l 2)(2fl l -4)J + Supp, (1, 2) ' -(1, + Supp. (l t i, i) -, , (2) [(4)-(4)(2*-4)-(T l 3)J + Supp. (4) = 4 (4) 27; ; (Z) 2 {(*) -(3,1) -(2, 2)) + ((3. l)-(3, l)(2 B -5)-(l, 1, 2)} + S PP- (3, 1) 2 M3 , I),. )-(2, + Supp. (2, 2) + Supp. (2, l, _ + Supp. (i, 3) /; (1,8) 971 407] CURVES WHICH SATISFY GflVEN CONDITIONS. (Z) 3 {(3, 1)-(1, 1, 2) -(3, 1) (2m -6)} + 2 {2 (2, 2)-(l, 1, 2) -(2, 2) (2m -5)} + {2(1, 1, 2) -(I, 1, 2)(2w-6)-(l, 1, 2) (2m -8)} + Supp. (I, 1, 2) = ^' l ' 2)2Z>: (Z) 2 {(2, 1, 1) - (I, 1, 1, 1) 3 - (2, 1, 1) (2m - 6)] + {4(1.,l,l 1 l)-(I 1 l,M)(2-7)-(T>l,l ) l)(2 7)} + Supp. (1, 1,1,1) -^' L ' *' ; " * 110. I content myself with giving the expressions of only the following supplements. Supp. (4Z)(1) =w[2(.)-(/)] + <0- Supp. (3) (2) = *w. [2 ( : ) - { / )] + i* (' /) Supp. (3^) (1, 1) = ( 2 w wl ~ 3 " 3 - - 2m -I- Supp. (22) (3) =-im[S (.'.)-(: Supp. ( 2) (5) - + 6 (2 + 20, where a, 6 are the i-opreaQnbativoa of tho coudition Z. It may be added that we have in general Supp. (2)(4X) =Supp, (4Z-) + & Su PP' (4Z/), where (4Z) stands for any one of the symbols (3), (3, 1) ....(I, 1, 1, 1). 111. The expression of Supp. W (I) ^ ^ ee " oicptainod ^-A, No. 108 Tlmb of Supp (82) (2) may also be explained. 1. The point-pairs of the system of comes W ^reg^tng each "point-pair a lino, are a eat of linos eave toping a aarvo; the ass T his eurve is equal to the number of the ifc* wlneh pass through au SSnJ point, that is, as\t first sight would appear. *T the ^;/^^ the system (3^0, or to a<8*:)-(B*./>' it is, however, neceeaary to ndm i that number of distinct lines, and therefore the elass of the curve, IB eme-ha f of Ihu lTp(8*0-(-/)]; -Hch being H0 ,the number of the point- p an,(^) wboh, ^dcd as lines touch the given curve (of the order m and class W ) is 4[2(S^:)-(3^/}]. The pi f contact of any one of these Hues with the given curve ,s (specially) a umted point, and we have thus the term * ft pS(8Z:)-(W - /)] of the Supplement r The number of the conies (32) which toueh the given eurvo at a given oup thereof, or, say, the conies (32) (M), is -*(8*./), -d the ousp is in of these conies a united point; we have thus the remaning term 4* Supplement- 272 SECOND MEMOIR ON THE [407 Article No, 11 2 fco 135.-^ Mmi fo contact with a (jiven Curve. ^ qURti nS ' WMCh : firat '-"' " ->' I call their First equation: {(5) -(6) (2m -5) -(I, 4)} + Sl W -^ on = o (5) W. Second equation : 26)-(4 l l)-(2, 3)} + ((4, l)-(5, l)(3m-0)-(l, 1, 3)}' + Supp. (4, J.) ^ =4(4, 1)27J, Third equation : 3 {(6) -2 (3, 2) -2(8, 2)} + {(3, 2)-(3 f 2) (2t- G) - 2 (1, 2, 2)} + Supp, (3, 2) -3(3, 2)21). Fourth equation: 2 {(4, 1)-2<3 J 3, l)-(2, 2, 1)} + {(3. 1,1) -(3, 1, l)(2 t - 7) -(1,1,1,2)} + Supp. (3, 1, l) ' -3(3, 1, 1)2^. Fifth equation: 4 [(fi)-& 3) -(4, 1)} + R3, 2) -(2, 3) (2m -6} -(I, 1, 3)} + Supp, (2, 3) - ' =2(2, 3)2D. Sixth equation: 3{(*. l)-2(3, 2, l)-2, (3, 1, l)} + 2[(3 l 2)-(2. 2, l)-(2, 2, 1)} + (2(2, 2,1) -ft 2,1) (2m -7) -2 (1,1,1, 2)} + Supp. (2, 2, 1) Q ^ -2(2,2,1)25. Seventh equation: 2 {(3, 1.1) -3(^1, l, 1)- 3 (2, 1,1,1)} + {(2, 1, 1, 1)-< 2| 1, 1, 1) (2m -8) -4(1, 1, 1, 1, 1)} + Supp, (2, 1, l, l) * =2(2, 1, 1, 1)27). 407] CURVES WHICH SATISFY GIVEN CONDITIONS. Eighth equation : + {(4, 1) - (T, 4) (2m - 5) - (I, 4) (2m - 5)J + Snpp. (1, 4) = (1. 4) 2.D. Ninth equation : 4 {(4, !)-(!, 1,3)- (4, 1) (2m -6)] + 2 [(3, 2)-(l, 1, 3) -(2, 3) (2m -6)} + {2(3, 1, !)-(!, 1, 3)(2m-7)-(l, 1, 3) (2m -7)} + Supp. (I, 1, 3) = (T, 1. B)2D. Tenth equation ; 3 {(3, 2) -2(1, 2, 2)-(3 f 2) (2m -6)} + {(2, 2, 1) - (T, 2, 2) (2m - 7) - (T, 2, 2) (2m - 7)} + Supp. (1, 2, 2) =(1, 2, S)8D. Eleventh equation: 3 ((3, 1, !)-(!, 1, 1, 2) -ft 1, l)(2w-7)] + 2{2(2, 2, 1)-2(T, 1, 1, 2)-(2 ] 2, l)(2m-7)f + {3(2, 1, 1, 1)-(T, 1, 1, 2) (2m -8) -(I, 1, 1 ? 2)(2m-8)) + Supp. (1, 1., 1, 2) =(1, 1, 1, 2)2.0. Twelfth equation : 2 {(2, 1, ], 1)-4(1, 1, 1, 1, l)-(2, 1, 1, l)(2m-8)} -1- {5(1, 1, 1, 1, !)-(!, 1, 1, 1, l)(2m-0)-(T, 1, 1, 1, l)(2m-9)J + Supp. (I, 1, 1, 1, 1) = (I, 1, 1, 1, I) 2D. 113, I alter the forms of theao equations by substituting for 2Z) its value- = n 2m+2 + /c, and by writing for the expressions with (1) their values, (1, 4) = (-4) -5 (5), &c., and except in the terms {Supp, (6) (5)}, &c,, by writing for K its value Sn + a, The resulting equations, if the Supplements were known, would serve to determine the values of (6), (4, 1), &c. j but I assume instead that the last -mentioned expressions are known (First Memoir, No. 50), and use the equations to determine the Supple- ments, or, what comes to the same thing, the values of the terms in { } which contain these Supplements, We have thus the twelve reduced equations, with resulting values of the supplements, C. vi. 35 274 SECOND MEMOIR ON THE [407 114. First equation : (5) + {Supp. (5)~x- (that is, we have 15m 3m 8m + + 10w + 4- Supp. and so in the subsequent cases, the equation gives the vahio of bha town in which contains the Supplement). 115, Second equation: 2(5) , 1) + (Supp. (4, l)-*(4, 1)} + (4, -(1, 3) -6m a - - 30m- 30ft + a ( .1,8) ft 3 + 104m + 104ft + a ( 6m + Oft - 66) + 18m + 9n+ a( 3m - 0) i a - 36m- 30n -fa (-3m -8ft -hi 8) 'i a 56m 53n + a ( i$m JJj;, -|- 30). I stop for a moment to notice a very convenient verification of fcho term in f }; putting therein a = 3H, the term is ' Jl 18m + 9n + (9m?i- ," 1 ","- 1 ' --* -8. ^ who, any hfchor tormH outor = m " wnl=B '= 8 ' te - ""> iuo IB -is-e iH above by sirap , y ,_ term is of the proper form P S + question is material in the seoll of the numerical verification ' ' 116. Third equation: ,_ oeffloiot .aloulablo an ,7 ^ * " = wh(m "" C mplete reducti " '> tho form it, ' the P oint he & the mko 3(5) (3,2) {Supp. (3, 2)-* (3, 2)} , 2) - 45j?t- + 120m + + 15m - 36m- (- 4m Verification is 15 + 3 (1 . 2 - 7) = Q, -|{)7 riHLVKM WHICH MATISKY UIVKX CONIH'ITONH. 1 IV, Knurl ll rijimtinil ; MM. I, 11 I |Snp|i. (II. I. I) *( -'I I. hi I (M, I. h(4m I :l/i Zn) Vni)ti'iili"M ||H, Kiilli 275 - Km" (It -I- i^ll' 1 w I a I, fan 1 'itn< A i 4 i-aM-Hv-i V -7)a.-.:.o.-aM.|.a(i.4-.(V-H)a.|.as ?* i* " ,f HHJ - -!( hi. SiMh :u4, n , 4 W i* - HOifiH -- a4 lt -I- !H2t -h !1 1 2 ! 4- 240/u -h 240/1 I * .torn' 4- lOHmw 4- 4Mn - IliMlw - MOu 01 (1) ( CD (I) (W 352 276 SECOND MEMOIR ON THE + 18m+18-198) - 8- SB- 156) + * ( - lfi - lfi + 654) + tf ( 30) 6) + 50m + W n - 276) 120. Seventh equation = = Verification is ; 2(3,1,1) (2, 1, 1, 1) {Supp. (2, 1, 1, 1) K (2 1 (i) (2) + (2, 1, 3, l)(2t + jj a) W + W*+toM (3) - (i, i, i, i) -*- fl*-^^!^ {9) (1) <*) (3) <*) w - 3m . - Kteft, - 20 re - W + 109ms + 282m , t + mn , _ 86Sm _ 8flgB 6rf + 30^ + 8 (h,. + 6^ - 174 m . - Si8m)l _ U4tf + 1320m - l*- 5m=- fa 3_ 20m ._ 5mn+ y n , + 8Qm+ 8to -w-v^-i 8B ^- |fl . + mH 37))w+ na _ 150m _ 76a + 2 + l i+ Sn!ft , + K> + i|lraS+ 8tol + lfare ,_ 382m _ 40g/j (1) (2) (3) W W f C ( . + ^, + ^ . / ^+l 2 , m+3 ^ 69m _ 69?i+582)+aa( ~ 9) ^ a ^ a "-2flmn,-^ w + afam + aiia B -.060) + -(_^ m-f)i + 28) 19m + fw- 54) 407] CURVES WHICH SATISFY GIVEN CONDITIONS. 121, Eighth equation : 277 = = 5(5) + (4, 1) + fSupp. (I, 4)-* (I, 4) 4) - 76m- 75)i + ( 45) - 8m 3 - 20 wn - 8n? + 104w + 104n + a (6m + 6?i - 66) + 1m 4)i + a ( 1) 10m a (- 6m + 10) + on (6) Verification is 1-4 + 3.1=0. 122. Ninth equation : = = 4(4, 1) + 2(3, 2) + 2(8, 1, 1) + jSupp. (I, 1, 3)-* (I, 1, 3) U(m-2)(2(.l ) 8) -(/I, 3)) , 3) (4 (4, l) + 2(2, 3)) 3m 3 - 32m 3 - - 20mn a - 3i s + 109m 3 32?i 3 ( 109n s 16m" - SOwin + 3m s 8mn s + 24m + 24?t - 264) 278 SECOND MEMOIR ON THE 123. Tenth equation: = = [407 3 (3, 2) + (2, 2, 1) + fSupp. (T, 2, 2)-* (I, 2, 2) U(ro-2){2(.2, 2)-(/2, 2)) + 3(3, 2) 24m 3 + 360m + 860w <i) 24n 9 468m 40Hw ca> 64m : (- Urn - 12 - 234) + tf ( (~ 8m- 8n (3) W (5} +( 20w (B) Verification is + 33 + 3 (- 3 . 2 - 13) + 9 . 2 = 0. 124, Eleventh equation 3(3, 1,1) 4(2, 2, 1) 3(2,1,1,1) = = *)( 1,1,2) + (3(5, 1.1) + 2 (2,2,!)) - ff*'- (1) (.5) (a) 407] CD <*) CURVES WHICH SATISFY GIVEN CONDITIONS + 848m+ ^tp-w 3 - 1302m ~1302 + a( 96m 3 + 21Gmn + 9Gn a - 1872w-1872n + ( 1044m - 522n a + 3960m + 3960?i + a ,[3 2?)l 19Gm a -|- 56ft fl - 392m- 392i + a - ?" - 2m - mn" (7) 279 (8) (3) (-0 (6) (S) (7) (1) (2) (3) w w to) (7) - ^m- ^w+ 873)-!- a 2 ( - ^) - 32m - 32 + 1308) + a' J ( 2m + 2?t - 48) + 358m + 368n- 2880) + a"(-fm-$n+ 84) Vorifioation is 4 (- - ft - 6 - 3) + 2 (V - ^ + 18) - 2 - 191 125. Twelfth equation: 2(2, 1, 1,1) + 6'(1. I. 1. l > a ) + f Supp. (1,1. 1,1, !)-*(!. = = + )(>!, 1, 2)i(2, 1, 1, 1) (3) (6) (Q) 280 (i) HTCflOND MKMOMl ON T1IK -j- 12 W i !l -|. V" (fl) + MwM- tym'H -I 111 H' :l/Sm u - fi2wm I "]()// .(- a ( n + 8 (f A -I- i + 1. -I- A) H -h (- ii - .1 -|- ;j) 4 -I- (- H - (jf.) 2 - M) -|, II (( !] - , j) 4 -,. | ;t . a J . (( . 12fi. Ib will b nbfioiT(l thai, in bhn ni^lali HII.I lulhm-iuK i|tUit.i.H ( vi/. tln rojii tho expression of Uio Supplmn.mL tumtiuuH Mm Hymlml (1). I ] m v- inHiitli.. ilong wilh tlio Hiipplomunt within l;hu ( ], tl,,, tonns (i!|) |a(.4)-t/+)I *r v too ara -(m-U) int., n.uuhnv ,,f p,,!,,!^ (4), to,.: tl,i 'w f.,r n.un, ,(',,, )m | v t Hiinplifioa tho calculation, l.th fnu thn Hyuiuuarical form inul,,' wliit.h l,h, . ion pi^onfc bljomHulvcH in thu H.v.val .quulionH, ,! !.,,,. t | 1M ( , Kmni. in .juoHUoi,. (te, tormH bcnug imai. ,n,iUi^ w f R miwbor () y Auth^ fchoory known in torn w O f tho CJapi,:,^. It in u. 1 ....ticn) Mn (, r r V tion, to U Uie nyntom to wldoh tlio (Japitul^ l,ul<mg, w<, ,|h,,i,,M. h v zt r ; m ; 1 ^11 T,^/-* ^ u. " upp. (2, 1, I, 1), bho Capitals belong to tho Ryatom (1, 1, i. i). 407] CURVES WHICH SATISFY GIVEN CONDITIONS. 281 CO (2) (4) 0) 127. Referring to Nos. 41 to 47 of the First Memoir, for convenience I collect the capitate whieh belong to a single curve, giving the values in terms of m, n> a as follows. (1, 1, 1, 1) '11 - 8 (n - 4) (m - 4) - %m a n 8)i a + m + (-fm 2 + fm- 6 + f) - 2m 3 - fm tt ji + 4mu 2 + 10m 2 - 14m - 16?i 2 - 8m + a( 8 + 2m s - ^m^ - fmn. 9 + a(-|i 2 +*m ' -In); - 18m- m + 6). (2, 1, 1) (3) (8) (2) (1) (2) (5) ff = [= t . i (i - II) (w - 4) supra] ; J = + a( + ( + 2m - n+ 6); + a( -3m- +9); mw" + 8m 3 - m-4n 9 -32m+ + a ( - 3m + 12) ; mu - 3m - 4?i + 12) ; 3). (3) (2) CO (2,2) L =(n-8) Jbfi(*-l) JV* ~K 0. VI. _ 8n a 4m-^ + a( -I); i); i). + ( 36 282 (2) (2) (5) (4) (3, 1) P = Q [= (4) [= tcsuprti], SECOND MEMOIR ON THE + ( -3); + ( - 3); + a (m - 3). 3m+ct (first equation). a(2m-G) + On ( 8 econd equation). 6JiT= -9) [407 2) 15m ff(?i-7) (third equation). (fourth equal 407] CURVES WHICH SATISFY GIVEN CONDITIONS. 283 7i s 4- 8m n - 3 (fifth equation). = 3?>m 2 4- 24??i 2 - 3mn - 12n a - 96m 4- 12?t + a ( - 9m + SG) 4- / 3m7i 2 4- 9?/m 4- 12 3 36?i 4- (fli 4-4J" = - 12m 3 + 36m +a( i. gjp Qti~ 4- 18ft + o. ( .w-3m-4w4 lii) 4m 12) 2?i 6) 12??i 2 4- Ginn G?(' J 60 i Qn + H (?j. in - 8m 27i + 30) (sixth equation). 7^ ^m-ft - 2m 3 - flm* + 4w-' + 10' - 1' to-16 2 - 8m +C i-fcft (U .[. 4^' _ yjftp + g??!, 3 ??t a ft - 9'wt a 72wi 3 4- ! )m 4- 20?i s 4- 160m S !0 (2) + 4D - Gi ;i + 42i a 72m (31 ioi" 1 fill ( 1) Tl' 9?" 4- X/ ~ ^ ... " a "" (i) (8) -|7i s 20 i 2 i 4 Gm4-6)t~-24) 4- 27m - 60) - 14m 4- 24) -n+ 8) ? 4- 19m 4- An 54) (seventh equation). 2P + J' a (eighth equation). (2, 3) = -4m-4?i- 6 1.2(5, 1)= 2m 4-27?.- 12 - 2?ft - 2rt - 18 + 3a (used = -2*71" +2w-16n+( ^ (ninth equation), 362 284 3Z + m ~w - o *(2, 2, 1) = -2F - ff 22)' J"' 0) (2) (it (5) (3) (7) (8) W (10) WJ SECOND MEMOIR ON THE ( 3a- 9) a(-6n- l) + s - 2) - 1) + S3B + (-S-13) + a (tenth equation). [407 4" fl 3n + a , -80 + m 2 -f m)l+18)ia _ Stt-I91n (i) (a) (3) (i) (5) (e) (7) (a) (8) (10) (ii) (m + ft-jy.) (eleventh equation). 407] I.l, 1)= CURVES WHICH SATISFY GIVEN CONDITIONS. 285 (0 2D & is) + 36m + ( + 18 + a( -10m- 10ft + 40) + 7m - 12) n a + S- 6) 4 -Hi 2 - mw + fw 3 - - m ~ 129. We have consequently, by means of the results just obtained, Supp. (5) = (5) (twelfth equation)' + N Supp. (4, 1) = K ft 1) 4-2J + J2 Supp. (3, 2) = (3, 2) + 67iT + i Supp. (3, 1, 1) = *(3, 1, 1) + }} + J5 + Supp, (2, 3) = (2, 8) + Q Supp. (2, 2, 1) = (I, 2, 1) + 3& + I + Supp. (2, 1, 1, 1)- *(2, 1, 1, 1) 3 J" 4- J* (first equation). (second equation), (thh'd equation). (fourth cauation), (fifth (sixtl (sevi SECOND MEMOIR ON THE r , Q (I, 4) = , I>4 _ - i + |0 ( eighth equatioil) (I, 1,3) - *(1.1.8) + 2.8 *a , . . , (ninth equation). (I, 2, 2) = 2| 2 ) - /, ,, /T , (tenth equation). . (1,1,1,2)= .(1,1,1,2) + , (2, 2,1) BJ) +wyw , Observe that. (eleventh equation) ff-jup.o, 3^/ +8 ^^ +7/+s// to o^ the fom of te _ (twelfth equation). and therefore a l so ^ 3) = (W ' 3) + ^ 3) - (SI, 3)J, * 3) = K (aZ, 3) + * [(2, 3) - (2S ( 3)J, the second term IY2 3)-(Is\ q^7 u The remark app.ies to all the ^veTn.^^ te S f Zeuth -' s Capitals. tan 8 much as (M) = (JS 1 ^ T^' nly as re S arda *> t four of them, which are th us expressible by means of M!l ^ rigi ' W ' te S K " ( 5 ' L ^ fornn,!. (First nJoi,. No,. /9 HT? 8 '- '* ^ the - si *'- of the = A: = Deferring to (first equation). (second equation). 407] K (3, 2) (3, 1, CURVES WHICH SATISFY GIVEN CONDITIONS. = K (- 9 -1- ) = (3 (n- 3) + K - 1 + 1) = 3i -I- 2M + 287 to = K (w a + 2mw + i?i a - J .f m - tyn + 27 - fa) ~H-\- 21 + D 1 + J' \ix. K~ I H =^i 3 - iw +4 K- 1 . 27 = 2mn - 6m - 8 + 24 (third equation), (fourth equation). (2, 3) (2JS, 3) *(2, 2, 1) (2ia, 2, , 2, 3) /cTi i i n j- 71' = K {&K\.) -L, J-t *) "r Jy 4) - (1^4)4-0 (2, S) + 2(5, 1)^ * (1*1,1,3) + * (2^,3) +*(-3) , 3) , 1, S) + S*<a, (1 2, 2) * (F*!, 2, 2) + * (3 (n - 3) + - 1) ( - ' ' ' + (2*1, 2, !) + *( -3) 1, 1, 2) + 2* (2*1, 1, 2) (fifth equation). ( sixth eq ation )- (seventh equation). (eighth equation). (ninth equation). (tenth equation). ,. , equatzon). goo SECOND MEMOIR ON THE r ^ "(1, 1, 1- 1, I) + *(2 f 1, 1, 1) ' lleferrmg to 181. Henoc, substituting in the ex p, eS siou S of the sevo.al Supplements, we h ave Supp. (5) = Q + N (nrst equation). Supp. (4, 1) - (second equation). . (3, 2) = 8i + 2 jtf +0 /4.T , (third equation), f f a (fourth equation). (fifth equation), ,. , (sixth equation), . (5, 1, 1, !) = , fe lflfl) + Jy + ^ + 4(7 + 4J? + JX (seventh equation). Supp. (1,4) . {1S I 4) + + (m-f)(4J\T4.0) "^ , . ]|pl (eighth equation). - (2, 3) = ^2^ 3) + Q . 2,1) = -2P- 0-- (ninth equation). (tenth 107 I (MIHVIW WHICH HATIHl'Y (MVMN CONDITIONH. 289 >. (I, I, I, 1!) " (!*!, I, I, 2)-l-2/c(yi, 1, 2) i (M ... y,)(;tA; r ;t7<'.|. (itf-i-a/H- .//-i- a/-i- r ,/) - aw-.. a/- 1 - <; - ,\./H- y/'-i- ft/- jf >> 27/ -^ (nlovtmlih oquatinn). Hupp, (I, I, I, I, I)- *(U-l t I, i, I, I).|.SK(5J1, ], 1, 1)-|-./)' Hm-4j)(vl.|.'J/H.4r;.1"!U)) ..jUl--j7f--j|U~2/>--7>'. (twolfbh equation). Ilia, limn* Ihmlly, nii'ivly rnlli-rtiiiK l-hn tnrnm, wo hivvo liho following oxprcHBionfi til' (iln Sllpplt'liM'llhl ill til'' IWi'lVn Pipmtinim I't'SJll't'.l.iviily, Stipp, t,M ' A f i " ('"'Mb uqiialiiou), Hupp. (4, I) - '.(./ I a/M-.r (widow! (!(]iuition) ( , (II. H) it A' | 4A-|-^/-|.MiV.|.:iO (Uiil (Hiuation). , Cl ( L U -' " I /'' f /' t'^' I '//-l-aM-:U-l^>'-l-2./' O'nui'lilt oquation). .Sup)., tl HI A-rJ*l. -'I) M<> C Iiftl1 <I"".on). , |'A '*;, I) - *(;!*'!. 1!. IH-:UH'M'4./>|-!U' (Hixtli oipmbion). , ttf. 1, 1, Ct ..... tv!vl, 1, I, l)'h/M"KM'4/H<2/J' (mwrnth o(|imtion). i. ll. +) - *tUI. 4) I (4m-.7)A'4.(^- 1)0 (^^ liquation). , {I, I. M) - MU'I, I, :J)-M!tal, rt) t U!'" -(ij/ I 'Kaw-roy-l'(">w-10)./H'('tirt I (Sim ^.S)A'.Htm./,t(m-MJ)J/-|.( t 2//^0)A r -l-('m-!)}0 (Umfcli o Hupp (I, 1 ( t. S) - *U*, 1. I. UH>'M^'<- 1. 8) i , i a,,, .. Ti) /I -I- (Hi* - ) A' -I- 1'*''* ^ !I ) ^'-t' (' ;t M lfl > f/ IH ( m - 1 ) // < t - (!i - 1 ) t H- ( w - 1 r ') '' 'I' !1// (ulovonth Hupp, o, i, i, i, n- *(i ), (twolfth oquation). m 1 rocnll Urn ^imirk. .iHto. Nu. ISO. thai lu uuth ^imUon tho Capital bolong Iho .plorn ubtaiiHxl by .HmmwhiiiK Ui bnitinl number by un.ty and ro.nov.ng iho ; (4) fur Uw llwi wiuuiimi. (. 1} for Urn cil, > "- a vi. 290 SECOND MEMOIR ON THE forms of the hibit how the Supplement arises, whether from prope? , 6XI 3 , act or from point-pairs (comcMent];"^ ,)"];?" r'T^' " tOUOUng a < " CU3 P> terns Ime-pair-points). Thua fo instarl , "" noludm g <> f in those pl, n ati n s ,K rat P Mem^C o 4^rT ( ^ + ' **"* ' described a, "inflexion tangent teLinaI l^ number of the Kne-pair-pcinl del ib d a "oli, < P ," or in what is here the a onl ! T ^^ te ilMt ^ each Hne-pair each inUoxion tangent anTeaeh ^T/ W ' hare as o of the firat equation, ,vhen g the ^^ point 1= T '7*' ReVerti " t0 the 8 the curve e is the conic (6), hay:n e with * P ? genm ' al f the ^ Beting it in the 2m _/; it a |' 1 r VG * lnteraect; ^t P, and - 8 becomea the coincid nt 1^-paT 1 r ^ " " '" M ' number of intersections at P is therefore -f 1 > taDgent talren twioe . united point. Similarly, when the ' t ~- ' \ ^ m& xim is tllerefora (pe oomcident line-pa u . formed by the tanC L n ? * T' "* U1Te 9 beco - s -P therefore =6, and the ousp i Z s f, ^ T^ * 6 mnlbar f ^eotions at tota number of special united foin s - K if T"^ & Um f ed " int: ^ thus the result, Sivpp. (g).y + 0. * ~ + ' : "P 8 * Wlth th ^ Agoing A posteriori 13*. Or to take another example; f or the fifth equation we have Supp. (2, 3) = (21, B ) + Q. " " d "* ^nt terminated puipoae, ,ve have each double talent as a .1 ! IT a WPto for the present of its points of contact, and also a a coincident " ^ " "^ * ^ e * pomts of colltact , Eeve the Ze- on T;r m reSpe0t ' the other of IB a pomt in general on the given curve th!T n T^"' when the P int ^ toudnng the curve at P, and "hav ng Teddet \ h'i, " ""f Sy8tem f W ^ ^ S1 f a ^ eaoh =nic the number of inte L f p" . nte0t f thfl thM "toi rp- ras s r p is t r 2(5 - 3) - - th - ; (2i; s )ii the t tow mber ^ pomfB P. Suppose that the point P ; talrm f f if. 2 (2 ' 3 > "iterseotions are the angent; of the (2, 3) conica, 1 (I ^1" ^, t f 6 ,, p0lM f oontaot of a double formed by the _double tangent taken LTe and .iv?, HT ^ C in ident ""-P* the remammg (2,3)-l conics are pl . OB er '!!,? - , thel ' efore * ^'ersections at P, at P, or the total number cf inteZt loT f^^' ^ ""ersections s a gam of 2 intersections. As remarked Wo 96 t l ' ] + 2 intersections; or there that the pomt in question is to be consWeS' TV J^' f **>** *& I do not know how to decide d pnori whlh r i^. H (8Pe ia%) 2 mited P '^ points or as 1 united point, but it is &?* b V, e arfed being 2 united 1 umted pomt; and as the points in lesL a + Tf ed M ^ ( 8 P edal W *f double tangents, we have thus the numbef T O f t , ^ P intS f onta t & the point P is at a cusp, all the <f T? P UEited P<)ints - A ? ain ' whe " (2, 3) comes remain proper conics &,,)-& 3) OQ1 407] CURVES WHIOH SATISFY SIVEN CONDITIONS. First Memoir, No. 73), but each of these fend conic touching the cuspidal * with the given carve at the cusp not 2 but 3 intersects, so that the to al of intersection at P is 8 (Si. 3), =3(2, 3), and there is a gam of (2, 3)-(2l, <) intersections. Kach cusp counts (specially) as (tol, 3) united pomts and togothei the cusps count as .(Si. 3) united points; we have thus the total number K (2*1, 8) + of special united points, agreeing with the expression, Supp. (2, 3) = *(-!, V + V- 135 As appears from the preceding example, or generally from the remark, ante, No. f I tve not at present an^ d ^ method of ^determining the proper _ca 1 multipliers of the Capitals contained in the expresses of he severa Supplemcn s these conies becomes a coincident line-pair ; this regarded the given curve (-) ordinary interseckons ( a number, . 4 at most the contacts which the line may have with the curve) ; for each th P taken as a position of P, one of the conice which make up the curve becomes Sd^t Unfair, and kcre are in respect of this conic two cUo at instead of one intersection only. We have thus m respect of .the paiUcu a> " ttvrs ,rr thi-, bu" being unable to <&.. the results th any degree of com pleteness, I abstain from a further discussion of them. 292 [408 408. ADDITION TO MEMOIR ON THE EESULTANT OF A SYSTEM OF TWO EQUATIONS. f *? f y<d *** f m ' TO ' - , pp. 173-180. Received August 6,-Kead November 21, 1867.] mtkout tables fo, options in he fo is the standard form m nuanZ Thl V I rt 1 "' ' oaloulried. Hssultant of ooeffic: te . ^ r meam whereof the present tables were Table (3, 2). Eesultont of ( 6 _ ^ ( P , , 4 ; 408] ADDITION TO MEMOIR ON THE RESULTANT 03? A SYSTEM &0. 293 Table (4>, 2). Resultant of (ft, 6, c, d, e^oi, Table (3, 3)*. Resultant of (a, 6, o, (Zftflj, 2/) 3 , (p, g, r, ajflj, y) 3 . * N.B. la the corresponding table o the memoir, there is an error in the signs of the last two terms ; they should be 294 ADDITION TO MEMOIR ON THE Table (4, 3). Resultant of [408 408] RESULTANT OP A SYSTEM OP TWO EQUATIONS, 295 Table (4, -4). Kesultanfc of (a, b, c, d, e$>, y)\ (p, q, r, s, tjaf, y)'. 296 ADDITION TO UKUOtB ON THE IlESULTANT OP A SYSTEM, & C . [408 Table (4, 4) continued: 408] 297 n. vi. 38 298 ADDITION TO MEMOIR ON THE Table (4, 4) concluded ; [408 408] RESULTANT 01? A SYSTEM OF TWO EQUATIONS. 299 300 409. January 9, 1868.] [IT is remarked, Proc. B, Soc. vol xvrr n <m H, t ^ , I had in foot i n regard to the 1- ' P .' 31 *' that the ab title is a misnomer: to the ro- f' ed "* th twofoltl relati < ^ t ?f 8efeld ^ "" W0uld ha " On the of ec l ual ts of a binary quartio or f of systems of equalities "P> of the . re e 1 uati < )na kno ! "- 7 =i ^ """ the1 ' f the two " llrtioM U " less - to the root-systems the rootle conditions for the existence O f quintic,"] In oomideiing the conditions for the between the roots of an equation > * imposition of relations A "ret o'n ^-0, or it i, say Wol d> o^tc b" as regards onefold relations, the theory of -F=0 i, a Ration co~d e d Jf relation satisfied if, and notTttfi er , , , la relations is satisfied. The 1 ke not ot f "' "" vk, the eompound relation I a eTation , the other of the two eomponeut a n s t t T fro,n any further di sous8 ion P of h e tt orv of r ' PU T rP Sdy "^ at P resenfc for the existence of given system, rf 1 posito. I say that the conditions furnish instances of and, ompo Z- in fT f' betWe n the r 0ts of an and its first.deri.ed functio? n , ' J SY T?" ^ "" ^^ &st-deriyed functions in regard tc x ' * ^ Same tUn & that Of course, relrtion it ia a -, as 409] ON THE CONDITIONS FOR THE EXISTENCE &C. 301 unless, there is satisfied either the relation for the existence of three equal roots, or else the relation for the existence of two pairs of equal roots; or the relation for the existence of the quadric factor is compounded of the last-mentioned two relations. The relation for the quadric factor, for any value whatever of n, is at once seen to he expressible by means of an oblong matrix, giving rise to a aeries of determinants which are each to be put = ; the relation for three equal roots and that for two pairs of equal roots, in the particular cases n = 4 and n = 6, are given in my " Memoir on the Conditions for the existence of given Systems of Equalities between the roots of an Equation," Phil. Trcms. vol. OXLVII. (1857), pp. 727731, [150] ; and I propose in the present Memoir to exhibit, for the cases in question ?i=4 and n = 5, the connexion between the compound relation for the quadric factor with the component relations for the three equal roots and for the two pairs of equal roots respectively. Article Nos. 1 to 8, the Quartic. 1, For the quartic function (a, &, c, d, eJiB, y}\ the condition for three equal roots, or, say, for a root system 31, is that the quadrin- variant and the cubinvariant each of them vanish, viz, we must have = ace- ad* - - c 3 = 0. 2, The condition for two pairs of equal roots, or for a root system 22, is that the cubico variant vanishes identically, viz. representing this by we must have (A, B, 60, 10D, BE, A = a?d - + 26 - Oao 8 0= abe - Sacd + 2b 2 d E = - ade+ F = - ae* - Zlde + 9o 9 e (? = - 6e" + Bode - 2d 3 = 0, C6 a o = 0, ==0, =0, = 0, Qcd* = 0, = 0. 3. But the condition for the common quadric factor is -0, o 36, So, d 6, So, 3d, e ttt 36, So, d &, 3c, 3d, e and the determinants formed out of this matrix must therefore vanish for (/, J) = 0, and also for (A, B, 0, D, E, F, (?) = 0, that is, the determinants in question must be syzygetically related to the functions (/, /), and also to the functions (A, B, 0, D, E, F, ff). 302 ON THE CONDITIONS FOR THE KXI8TJ3NOK 4. The values of the determinants are 1234 = 3 x 1235 = 3 x 1245 = 1345-3 x 2345^3 3 <v 4- 1 oVe - 1 a e a - 1 fl&e 9 - 1 aW - 3 affe - 1 nice + 4 firf a + 1 2f+9 orfa H- *! a(l j (i ~ \ alcd + 14 rrc 3 - 9 if I 3 3 flcrf 3 - 9 Wo -i- 1 u(J tf "- O bet fa -i- 14 &d - 8 4V + 6 b^cd + 2 fid 3 + 8 /l/i/y-l i O iVi/tt "I" ^3 t\ t\ -. ') oV J -i- ti 5. The sjrzygetic relation with (I, /) i s g i von b y means of bl . 36 , 3c , d or, as this may be written, "here //T u the Hessian Qf 6 - That is, we have o mit ft, firet 409] O3? EQUAL ROOTS OE A BINARY QUAE.TIO OR QUINTIC. to each other they will also be = ,= The equations may then he written ao - 6 a , . ad - bo, ae + 26cZ - 3c 5 , be - cd, ce-d* = Q, a , 26 , 6c , %d , e and the ten equations of this system reduce themselves (as it is very easy to show) to the seven equations f A T? /** n T? ii 1 n\ n LA, Jj, u ; JJ, Mi, J! , Cr) = U, which, as above mentioned, are the conditions for the root system 22. 8. It may be added that we have A Jl C D E F G .1234 = c ~4& + 3a .1235 = c -SI) + a = d -3o + a 1245 = ~ e -i- 'id -3 C = - o + Gc - = - d + 3c - b . 1345 = ~ e + 3d ~ c = - e + 3c - i .2345 = -3e 4- 4d C where it is to be noticed that the four equations having the left-hand side ~ 0, give B : G : D : E ;, F proportional to the determinants of the matrix d, -3c, . , a -e, . > 60, . , a d t 3c, 1) ~e , , , + So, - b the determinants in question contain each the factor o, and omitting this factor, the system shows that , 0, D, E, F are proportional to their before- mentioned actual values, Article Nos, 9 to 16, the Quintio. 9, For the quintio function (a, b, G, d, 0,/$<B, 2/)Y the condition of a root system 41 is that the covariant, [#=] No. 14, shall vanish, A = 2 (ae - 4>bd + 3c a ) = 0, B= af-Sle -f 2cc2 =0, viz.. we must have 10. The condition of a root system 32 is that the following covariant, viz. , = ] 3 (No, 18) a (No, 14) - 26 (No. 15) a , 304 409] ON THE CONDITIONS FOR TEE EXISTENCE &O. 305 12. The conditions for the common [cubic] factor are a, 4&, Co, 4d, e =0, a, 4&, Gc, 4dl, e b, 4o, Qd, 4e, / b, 4c, Qd, 4<e, f the several determinants whereof are given in Table No. 27 of my "Third Memoir on Qualities," Philosophical Transactions, vol. OXLVI. (1866), pp, 627 647, [144]. 13, These determinants must therefore vanish, for (A, B, (7) = 0, and also for (SI, S3, ... S, 9W) = 0, that is, they must he syzygetically connected with (A, B t G}, and also with (9f, S3, ... 8, 9ft). The relation to (A, B } 0) is in fact given in the Table appended to Table No. 27, viz. this is (? x +J9x 4- A x 1234 = H- Ga 3 - 12 oft 4- 16 ac - 10 i a 1236 = + 6 ab - 2 ao - 10 6 a 4- 6 flrf 1236 = ~ 2 ao -I- 8 6 a 4- G ad - 18 6e - 2 rf/* 4- 8 e a 1245 - 4- 18 ac - 6 ad - 30 be 4- 8 e 4- 10 &2 1246 = + 126c -1- 4 as - 4 bd - 24 c 3 + 4 ie 4- 8 erf 1345 = -1- 24 ad - 8 aa - 40 bd -i- 4 a/4- 20 60 1256 = - 1 00 + 4 bd + 3 o 3 4 1 a/ 1 4- 6 6*3 - 18 cd _ 1 bf + 4 ce 4- 3 rf 3 2346 = + 20 ae -i- 40 bd - 30 c 3 - 80 be 4- 20 erf 4 20 bf 4- 40 ce 30 rf 3 1346 = -1- 4 rtfi -I- 8 fo* + 6 o 2 - 36 cd 4 46/4- 8 CO + Grf 3 2346 = -1- 4 rt/n- 20 fto ~ 8 &/ - 4 oe 4 24 o/ 1306 = H- 4 60 4- 8 erf 4 4 bf - 4 cc - 24 t a 4 12 do 2356 4- 8 bf -\- 10 flc - G </ - 30 rfe + l& df 1456 = 4- G ao 4 G of - 18 cfo - 2 df+ 8 e 3 245G = + G of - 2 rf/ - 10 e a 4- 6,/ 8456 = + lGdf-lQ G * -i2*y 4 G/ 2 14, Between the expressions 91, 35, &c., and 1234, 1235, &C., there exist relations the form of which is indicated by the following Table : 306 991-l 9QSS 99fit 91-81 QffiTJ 9*Sl) SWll 982T to & ON THE CONDITIONS FOR THE EXISTENCE "a 13 X X Q MS 13 s o 13 V. o ("1 M) 1*1 ^ a S5f tn ^ "" O v_-> %.s + ' S* o ^3 - 409] 01? EQUAL ROOTS OF A BINARY QtJARTIC OR QTJINTIO. 307 15. Assuming the existence of these relations, we have for the determination of the numerical coefficients in each relation a set of linear equations, which are shown by the following Tables, viz. referring to the Table headed c9l, b, a(S, a . 1234, [first of the seven tables infrcl] if the multipliers of the several terras respectively be A, B, G, X, then the Table denotes the system of linear equations A + 3.B + 330 4- 0^ = 0, 3 A +0-B -1020 -16^ = 0, fee., that is, nine equations to bo satisfied by the ratios of the coefficients A, S, G t X s and which are in fact satisfied by the values at the foot of the 'Table, viz. A : B : G : X = + 66 : -11 : +1 : + 6. There would be in all fourteen Tables, but as those for the second seven would be at once dcdncible by symmetry from the first seven, I have only written down the seven Tables; the solutions for the first and second Tables were obtained without difficulty, but that for the third Table was so laborious to calculate, and contains such extraordinarily high numbers, that I did not proceed with the calculation, and it is accordingly only the first, second, and third Tables which have at the foot of them respectively the solutions of the linear equations. 16. The results given by these three Tables are, of course., G6c8t- 11 #8+ !< + 6(1,1234 = 0, 330 cm + 110 o - 65 &< + 9 a5> - 105 a . 1235 = 0, + 260478676 eSt - 617359490 $3 + 144200810 cS + 9666911 6JD + 9090786 a - 721004050 o . 1234 + 90914176 &.1235 -160758675 a, 1245 + 11559296 a. 1236 = 0. It is to be noticed that the nine coefficients of this last equation were obtained from, and that they actually satisfy, a system of fourteen linear equations; so that the correctness of the result is hereby verified. 392 308 ON THE CONDITIONS FOR THE EXISTENCE 17. The seven Tables are First Table. . 1234 a'bf + 3 + 33 a?ce + 3 - 102 J aW -216 + 3 a-be* + 21 + 135 + 1 a-bcd -12 - 144 + 120 V -16 + 480 + 9 ab'd + 30 -150 + 8 atfc- + 50 + 240 -300 - 6 b 4 c -25 -150 06 - 11 + 1 Second Table. + G ffl. 123ft 6. 1234 + 10 - 390 + 156 100 - 600 + 1600 + 126 -1000 Third Table, + 33 -102 -216 + 135 + 120 + 480 +2664785761-617359490^: 65) ~_ - 90 - 195 + 10 + 360 - 390 -1500 + 900 + 1800 + 155 .+ 225 + 100 - 600 + 1600 -1500 + 125 -1000 +9656911 + 9090785 fl.1234 - 16 + 36 + 16 -162 + 96 + 80 - 60 . . ~. -721004050 + 6.1235 .184ff l . 1*3 - =* flj + 10 - 4 -i- G . yj + 24 -20 , ',; -00 t- Iri + 00 + 4 ,, i ft -84 + 90 - 10 -24 -80 + 64 + 60 -40 +90914175 -1G07B8G76 4119S9-: 409] OF EQUAL K.OOTS 03? A BINARY QUARTIC OR QUINTIC. 309 Fourth Table, ft a5 tZ.1234 o. 1235 ft. 1236 6.1245 a, 1246 a. 1345 v + 3 -i- 3 - 114 + 4 a z ldf -12 + 33 - 90 - 264 + 6 - 6 - 4 - 24 W -I- 21 - 195 - 990 + 16 - 4 + 64 V/ -10 + 10 + 468 _ 4 -24 + 24 a a orfe - 144 - 102 - 390 + 1320 - 16 + 24 + 24 -208 # 2 c 3 -216 + 1080 + 36 + 144 aftv + 60 + 155 + 360 + 900 -i- 4 -22 + 6 + 24 rt&Ve + 30 + 135 -1500 - 2700 + 16 - 6 -26 -20 - 40 abcPe + 240 + 100 -i- 900 + 900 _84 + 16 -96 + GO alotP + 120 - 600 + 1800 - 600 -152 -24 + 96 - 40 ao s d -i- 480 + 1600 + 96 + 64 Wf -24 + 225 + 16 b a ce - 150 + 125 + 60 -10 + 90 &W - 150 -1500 + 80 -80 &W -300 -1000 - 60 -40 /JO Fifth Table. a c.1234 (2.123B c.1236 0,1245 i. 1246 5.1346 a. 1256 a, 2346 a. 1346 + 3 - 19 + 1 + 21 + 33 - 114 - 608 + 4 - 2 + 16 - 144 + 10 - 90 + 537 - 4 + 6 - 6 -16 + 20 -36 -102 - 195 - 245 - 16 + 16 + 16 - 80 -16 - 216 ~ 390 + 1740 + 36 + 24 + 16 + 60 + 36 + 30 + 106 - 264 - 245 + 4 4 - 24 -16 - 80 - 16 + 135 - 990 -1700 + 16 - 4 + 64 + 240 -1- 240 + 360 + 468 + 1740 -22 + 6 -24 + 24 + 60 -j- 36 + 120 + 100 - 1500 + 1320 -2000 -162 -84 - 6 -26 + 24 - 208 - 860 - 20 - 600 + 1080 + 600 -24 + 144 + 9GO + 4-80 + 900 + 600 + 96 + 16 -96 + 960 + 1600 + 1800 - 400 + 64 + 96 320 - 150 + 225 + 900 + 16 + 24 ~ 150 + 125 -2700 + 80 + 60 -20 - 40 -300 -i- 900 - 60 -10 + 90 + 60 - 1000 -1500 - 600 -40 -80 - 40 cfibcf tflcdf ttbcde abd* <tc s e 310 ON THE CONDITIONS 3TOR THE EXISTENCE h*22&& 409] OF EQUAL BOOTS 03T A BINARY QUARTIO OB. QUINTIO. 311 And the remaining seven Tables might of course be deduced from these by writing (/, e, d, G, b, a) instead of (a, &, c, d, e, /), and making the corresponding alterations in the top line of each Table. 18. The equations 91 = 0, 95 = 0,...., 9)1 = consequently establish between the fifteen functions 1234, 1235,... 31-56 a system of fourteen equations, viz. the first and last three of these are 1234 = 0, 1235 = 0, -160758675.1245 + 11559295.1236 = 0, + 11659295.1456 -160758675.2356 = 0, 2456 = 0, 3456 = 0. To complete the proof that in virtue of the equations ?l = 0, S3 0, ,., 931 = all the fifteen functions 1234, 1235, ,,. 3456 vanish, it is necessary to make use of the identical relations subsisting between these quantities 1234, &c. ; thus we have a, . 1345 + 4& . 1245 + 6c . 1235 + 4d , 1234 = 0, 1 . 1345 + 4c . 1245 4- Gd, 1235 + 4e . 1234 = 0, which, in virtue of the above equations 1234 and 1235 = 0, become a. 1345 + 46. 1245 = 0, I, 1345 + 4o.l245 = 0, giving (unless indeed ao ~ & a = 0) 1245 == 0, 1346 = ; the equation 1245 - then reduces the third of the above equations to 1236 = 0, and so on until it is shown that the fifteen quantities all vanish. 410. A THIED MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS [From , the Philo^Moal Tn^tions of the Royal Society of London, vol. OLIX. (for the year 1869), pp. 111_ 126 . Recelved May g0j _ I others! CI ' ^7 V S SUp P lemeQt ^ to my" Second Memoir on Skew Surface, e th t ; ' P ' -' ' an relates als ' eS K ' er ' " ' Uartic sorolls ' " Pointed out to in th8 Fesent Ifcmoirj and are Quart* Scroll. Xinth Species, S(l,), aM a triple directrix line. 64. Consider a line the intersection of two planes, and let the eauation of tho r nr n ; he or ^ 3 ' that f the sec nd p' e -tain L:r y paiameter 6, the equations of the two planes may be taken to be (P, 1, r, s%0, iy = 0, (, Jfl, 1) = o, Hebe, die 410] A THIRD MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. 313 where (p, q, r, s, u, v) are any linear functions whatever of the coordinates (#, y, s t w). Hence eliminating we have as the equation of the scroll generated by the line in question (p> !?> n s'ti.v* -) 3== o, viz. this is a quartic scroll having the line u=0, v=Q for a triple line; that is, the line in question is a triple directrix line. 55. Taking & = (), y = for the equations of the directrix line, or writing u=a! t v-y, and moreover expressing (p, q, r, s) as linear functions of the coordinates (x, y, z, w), the equation of the scroll takes the form and we may, by changing the values of & and iv t make the term in (,-, i/) 1 to be where the arbitrary constants a, , 7, S may be so determined as to reduce this to a monomial fee 1 , /cafy, or ]ca?y\ 5G. The coefficient k may vanish, and the equation of the scroll then is z (* ]5>, y) 3 + w (*'>, y)' = 0, or, what is the same thing, it is (#5>, y)*(z, w) = Q, viz. the scroll has in this particular case the simple directrix line 20, w-0, thus- reducing itself to the third species, $(1 3 , 1, 4<), with a triple directrw line and a single dvreotirito line. It is proper to exclude this, and consider the ninth species, S(1 3 ), a& having a triple directrix line, but no simple directrix line. 57. The scroll $(1 3 ) may be considered as a scroll 8(m, n, p) generated by a line which meets each of three given directrices; viz, these may be taken to be the directrix line, and any two plane sections of the scroll. The section by any plane is a quartic curve having a triple point at the intersection with the directrix line; moreover the sections by any two planes meet in four points, the intersections of the scroll by the line of intersection of the two planes. Conversely, taking any line two quartics related as above (that is, each quartic has a triple point at ita i section with the line, and the two quartics meet in four points lying in a line] lines which meet the three curves generate a quartio scroll $(1 8 ). This appears the formula 8(m, n, p) %nmp <wn fin yp (Second Memoir, No, 5); we have in the present case in = 1, n 4, p = 4, a = 4, # = 3, 7 = 3> and the order of the scroll is 32-4 12-12, =4, that is, the scroll is a qi scroll; there is no difficulty in seeing that through each point of the line there C. VI. *Q 314 A THIRD MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. |/' three generating lines, but through each point of either of the plane quavtir.s ,jiilJ single generating line; that is, that the line is a triple directrix line, bul> m*)'* the plane quarries a simple directrix curve. 58. We may instead of the section by any plane, consider the section by a |'^ through a generating line, or by a plane through two of the three iranoral.!,,,, I*' which meet at any point of the directrix line; if (to consider only the most iiJ case) each of the planes be thus a plane through two generating lines, the ,.*'- ^ by either of these planes is made up of the two generating lines, and of a rrH passing through the directrix hue; the directrices are thus the line and two ,>< each of them meeting the line; vo have therefore in the foregoing formula m*=l, n = 2 t p = 2, a = 0, /9 = 1, 7 = 1, and the order of the scroll is 8-2-2, =4 as before. Quartic Scroll, Tenth Species, Qfy wit k a ***** skew mhio met twicB % ^ generating line( 1 }. ^ ^ bteHleotion of '" P 1 ^ and let tho oqmti.m ,,f ,m a vamble fo ?, '-F, 1)' = 0, line the skew cubic determined hy = 0. ^::r scroll as described above. ' d ^ th 8Cro11 is ^"^quently a qn ftl . t i , In, ^ rii 2 indicting that it 1. me twice by flBflh ia a nod(l1 (") "ne on tho aorolJ, tbo o 410] A THIRD MliJMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. 315 60. The coordinates (a;, y,, z, w) may be fixed in such manner that the equations of the skew cubic shall be j, y, z y, z, w or, what is the same thing, each of the equations $</ f>(t = $ ) rq' r'q = 0, pr' p'r = is then the equation of a quadric surface passing through the skew cubic, or, what is the same thing, each of the functions prfp'ty, 1'$' ~- r '<l> 2 }r '~P >r ^ s a linear function of yw z", zy xw, wz y n -\ and the equation of the scroll is given as a quadric equation in the last-mentioned quantities. It will bo convenient to represent the equation in the" form (H F, G, 1$ A F, Crftmu z 2 zti anu xz~y^ = Q, or, writing for shortness which letters (p, q, r) are used henceforward in this signification only, the equation will be via. this is a quadric equation in (p, q, ?), with arbitrary coefficients. 61. Comparing with the result, Second Memoir, Nos. 47 to 50, we see that in the particular case where the coefficients (A, B, G, F, G, H) satisfy the relation AF + BG+GJ{=Q ) we have the eighth species, 8(1, &), with a directrix line and a directrix skew cubic met twice by each generating line. We exclude this particular case, and in the tenth species consider the relation AF + BG + GR=Q as not satisfied, and therefore the scroll as not having a directrix line. 62. I consider how the scroll may bo obtained as a scroll 8(m*,ii} generated by a line meeting a curve of the order in twice and a curve of the order n once. The first curve will be the skew cubic, that is m-3; the second curve may be any plane section of the scroll; such a section will be a quartic curve k"""" +.IIVOA nnr^ at each intersection of its plane with the skew cubic. ^ cubic, and a plane quartic meeting the skew cubic in tnree poiuw, BIMIU ui buem a, node on the quartic, then the scroll generated by the lines which meet the skew cubic twice and the quartic once will be a quartic scroll. In fact (see First Memoir, No. 10, [339], and Second Memoir, No. 5) the order of the scroll is given by the formula 8(irP, n)-9i([m] B + Jf)- reduction, = 16 - reduction. And in the present case the reduction arises (Second Memoir, No. 4) from the cones having their vertices at the intersections of the skew cubic and the quartic, and passing through the skew cubic. Each cone is of the order 2, and each intersection quA double point on the quartic gives a reduction 2 x order of cone, = 4 ; that is, the reduction arising from the three intersections is =12; or the order of the scroll is 16-12, =4. . 402 1 THIRD MEMOIR ON SKEW SUBFACES, OTHERWISE SCEOLLS. [410 6 lnr e t hZ;h illStead V 1 " ^"A * [ ^ in gene '' a1 ' *** ^ <^n and of 1 Z I*" 61 ' g \ 6; the SGCti0n iS hm ' e made u ? Of the *** and of a p ane cubic passing through each of the two points of intersection of - r;~r, *""""" <" "* " >- " - >. *. {instead of bdonrimr f n ,!,, ? . , '' e P"rtilar case where the line P1 Per lnVOl " tl0n) meet3 a ^ *e lo ., , , , '*; ;i r d!iiates " involution is rap esses that the hue shall belong to an (4,S, 0,f, G, or Mo irs: S'^i s ^ ^ - ^ -^Y : T tt ' -V* : ^ -a'^ : a* -& , p ^ , ^ _ ., , , , ^ "^ : ^-^ : *-> W-*o : W '. w : AB'^L- and denoting ather of these seta of equal ratios by : & : fl ; / then ( fll ft. Cl /, fl , , () Bntis/y identicall the z . ' a ' h > the line. "Jf^-hCft-u, and are said to ba the six coordinates of 410] A THIRD MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS, 317 We have to determine the locus of the line (a, 6, c, f, g, h) the coordinates whereof satisfy the relation and which besides meets the skew cubic yw~^ = 0, yz w 0, fcs ?y 3 0. The equations of the skew cubic are satisfiec by writing therein x ; y \ z : w = I : : f 3 ; t 3 ; and hence taking 6, <jy for the parameters of the points of intersection of the line (a, &, o, /, (/, h) with the skew cubic, we have 1, 0, 6*, 0>; 1, <j>, # # as the coordinates of two points on the line in question ; whence forming the expressions of the six coordinates of the line, and omitting the common factor </> 6 t these are (", 6, . o, /, </, ft) = 0$, -(8 + 0), 1, 6* + fy + # ty (6 + </.), ^^, and hence the condition of involution gives between the parameters 0, < the equation (A, B, G, F, <?, Moreover the coordinates of any point on the line in question are given by m : y \ z : w = l + m : 10 -\-imft : ffi + nuj) 2 : W 3 -t-i( a ; and writing as above p, q, r = yw z i ) yz xw, ccz f, we thence find, omitting the common factor (0 d>) a , ^j : q : r = ^ : - (0 + </>) : 1 ; and eliminating ^^), ^ + <^, we at once obtain (A, B t 0, F, G, fl^pr, qr, r*, f-pr, -pq, ^ a )- 0, oi'j what is the same thing, (S t F t 0, if, jl-J'.-GSi), ff ,r)'0 as the equation of the scroll generated by the line in involution which meets the given skew cubic twice, Reciprocal of the Qmrtic Scroll 67, I propose to reciprocate in regard to the quadrio surface B + 2/ a + s s + i a = the foregoing scroll 6 6 (J2-, ff, 0, ^A-F,- ff Ip, ffl r)* = 0. If the coordinates (a, 6, o, / g, h} of a line satisfy the condition of involution (A t B, G, ff, G, ff-ya, 6, c, /, </, A)0, THIBD ME C01lditioil a,l l c, f, g , h) = . Tim reciprocal of the before-mentioned skew cubic r vol. vii. (18BC), pp. 87-82, [872] " " equation of the osculating plane is f Pneter whereof is i, the 1, 2, ff 1 , t or, what is the same thing, the equation IF =0, s Hence fo, the H,e ,Meh is the intersectimi rf F)=O oraitbing . the common factor we have thus between the parameter, ft ^ the relati(m 410] A THIRD MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. 319 let p be the third root, then we have y 6 + $ + p = T , w 6<}>.p = -^ and thence (X, -F, Z, Substituting for + <j> and ^ their values in terms of p, we find Fp [7 a - ZX - pXY] - GpX (7- p,Y) 4- or, what is the same thing, from which and the equation (x, -F, ^ we havo fco eliminate p, 70. "Writing for shortness ( ., H t -G, A^X, Y,Z, (-JJ, (-^,-5, -0, .$ )==S, and therefore .y+ jQF+-y# + 8F0 : the two equations are SF = 0, Writing the first equation in the form multiplying by -p, and reducing by the other equation, /3(p"X - pF) -paX ~ 7 F = 0, or, as this may be written, 320 A THIBD MEMOIR ON SKOT SURFACES, OTHERWISE SCROLLS. [ 4TO From this and the preceding equation we deduce the values of p'A'-pF-^ ttd pX-Y\ viz. writing for shortness P p ^ awi we find p'X-/> or, what is the same thing, whence also and thence and we have therefore E F , r (Mepeiident of p) a 1. I fab any one of these equation, for instanoe ^-3 Tr = I/jr or, what is the same thing, t s m tt to , obtain f the r we h at f ' m the values f A 7, S, if only + 5ff + Off aot = 0, 410] A THIRD MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. 321 and substituting these values, the equation in question becomes qi' (- Ba -t- A/3 - HS) -r fl ( Get-Ay -ffS) This becomes .Ar (q/9 + ivy) = Tlr (- pa) = - -Bqra + ff (pr - q a ) a + F (pr - q 2 ) a + If {- qrS 4- (pr - q a ) 7} viz. the whole equation divides by a; and, omitting this factor, the equation is .Apr -I- .% + Gi a + #(q a - pr) - #pq + f/p s = 3 or, what is the same thing, it is (H t ff, O.B.A-F, -<?$p, q, r^ = 0, where I recall that we have p, q, r~/3S~7 a , fty ctS, y j9 a , a. j3, 7, S being linear functions of the current coordinates (X, Y, Z, W), viz. we have =( , , H, -G 3 A$X t Y, Z> W), ' p = (-H, ., F, J5J ), 72. It thus appears that when AF -\-BG- + OH is not =0, the reciprocal of the scroll (H, F, 0,3, A- F,- G^p, Q t r^ - has an ecr nation of the verv same form. 322 A THIRD MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. [410 equation contains the constant factor AF + BQ + CH, so that throwing thia out, tlm reduced equation will be only of the third degree in the coefficients. 74. The transformation is a very troublesome one, but I will indicate tho stops by which I succeeded in accomplishing it. Each of the functions (p, q, r) in a mimlrir function of (A', 7, Z, IF), say, P = (, b, c, d,f, c/, h, I, m, ftpT, Y, 2, W}*, q = . . 5 )-, '=(">- - I ); we have to form the value of viz. representing this for shortness by Ca> 6, G, d, f, g, h, I, m, if ',.. t" .. me cootncient of Ji* is ' " (B. F, 0, B.A-F, -fftot, a", ., M ', , ), that of X'T is (H, F, C, S, A -f, of the t, ms ,Y, ^. 7 , fe in viz. this fa P '' = lfe '' ' <l ' f < ff ' >'' l - m ' "W. Y, Z, HO I A THIRD MNMOIll ON HKKW HUUIJ'ACIIW, OTU13KW1WK .SOHOLLS. 323 Eitiil Mm viilmvi of lltn (inulliimmdH <*, />, ... wluuh outor into iho Inrnnilu. 1 am ^ivmi by mirnmi itl 1 Lliii I'tillowinj,; viilucH nf p, i|, r; vi, UICBO urn A*- !' # IP r# &Y A'r A' ir ,nr #ir P . . / yi //. ... /' ij , -. (u<\ - <? t ... tw, /iV' T , /u/; AM, v^, - ,mrQ;A r t r, ^ H ; X J , I- - ti* -van -i-ii/'Vr' - af/^ .|-2r/.A' Hl t IHl, <!(!, JUl, -Ml, -Aft, Alt, A*. AH, A(! f . Ml// -1-7-'^ -|-7'7/ >\- lift -///'' -M'V' 1 Q - .-,. p* (,'// VTi, AM un iiiMliinrp til' lltn culniiluMnn nf a nin^lo innn, l.liu t'-n^llii-iotil nf A'- 1 (//. A 1 , (/, /.', A -*!<\ '- vi'/. thin in // (/i // - (/'y ill,. wlin) h-nti in UMIN (AF+IHl \-Vll) 11*. vi/. i-lmni is tin* liiilm- f \F-}' Jltt-\-<<l/i tt.H llli'lllinllf'il illtnVC, 7(1, Tlii-nwiiiK mil lln* fin-fur in in.-hi.k.n, AF-\> IHt '\->Wl, lh" iM|ulifin ttl' llu> l HtT'lll 1M 1'nltltll It" 111 1 I- A' B l' .- lXX , ^.VMI' t - \\AU1I - >f.A"*r* . 2/v/+^<// tf Iltf* - 1 ' 412 324 A THIRD MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. [410 + XY 3 -2FGH + XY*Z AFE - BGH - SF*H + 2W + AT'IF" 2ABH- UFG - BFH + 206H + 8F*Q + XYZ* - AFG- 2BFff+BG*-OGH + XYZW AF- 8AF*-UBG + ACH - 2B*H + J3FG + oOFIf ~ 2CG* + XYW> 2A*B - 3^F+ 2AGG + &Q + BOH - QG'FG X&W 2ABF-2AOG- 2BCB- 8BF*+OfQ X2W* A*G+AB t XW S 3ABG-& BFH-F'G + Y*W AF* + Ml - 2CFH - F* ABF- + YZ* + YZ*W ACF+ 2&F- BQG - 4 YZW* + YW* A& GF* But the option may be TO , itton n -r (- a^S + 2af Z ( a 7 S - 23'g that of VlPiB^-arof^^ 6 c 3 efficienfc of -J'ff, that of 7& h-aiml^r* ^ ^~ gain ' the coefflci ^ of 410] A THIRD MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. 325 or, what is the same thing, as may be verified by actual substitution of the values of the coordinates. 78. By what precedes, substituting for p, q, r their values in terms of a, & -y, 8, it appears that we have the remarkable identity (H, ff t C> B, A-F,- -t- 7 (- a/3S 4- + 2 ( a.yS - 1 4- T7( aS 3 - 79. In the case above considered of the tenth species, $(3 2 ), for which not = 0, the three forma of the reciprocal equation are of course absolutely equivalent to each other. The first form has the advantage of putting in evidence the fact that the reciprocal scroll is also of the tenth species; the other two forms do not, at least obviously, put in evidence any special property of frho reciprocal scroll. Reciprocals of Eighth Species, 8(1, S 3 ), and Ninth Species, 80, If AF+BG+ (7// = 0, then the equation (ff.^a, JJ, 4-JF, is a scroll of the eighth species, $(1, 3 a ). The first form of the reciprocal equation becomes identically = 0, on account of the evanescent factor AW -\-BG-\- OH, but the second and third forms continue to subsist, and either of them may be taken as the equation of the reciprocal scroll. Taking the third form, and calling to mind the significations of (a, /9, <y, S), viz, H t -G, , W), ff, a, . it is to be observed that = 0, /3 0, iy = 0, 5 = are the equations of four planes passing through a common line, viz. the line whose coordinates are (A, J3, 0, ff> G, H) t and the equation thus puts in evidence that this line is a triple line on the reciprocal 32G A THIRD MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. [410 scroll; that is, the reciprocal scroll is a scroll of the ninth species, 5 (1 s ). Or stating the theorem more completely: For the scroll, eighth species, S(l, 3 3 ), (H,F, C,E,A~F,~G\^^ r)s = 0, generated by a line meeting the line (F, G, M, A, B, G), and tho skew oubio p = q-0, r=0 twice, the reciprocal scroll is of the ninth species, X (- =5 + 3807 - 2/3 3 ) + Z ( 7 S + T^( S^ having for its triple line the reciprocal line (A, B, G, F, (9, //). 81. It should of course be possible, starting from the equation (*P r , 7)<+^(*;pr ) 7)3+ IF (^ Y) r)>o <> "> - Hence writing for a moment Illl .l A Tllllth .MKMiUH ON HKKW mNlKAOIW, OTHKIUVJHH HOUOLLy. 327 thai in y i r n ii' ( r IK - %*) -i- AM* < A'#.- ro -i- a I'^'A' IK, II, liy I hr <<i|iuui<ui nl' I, In; Miii'ull ; llllil \Vtl l.llllH HIT I hul. ||IM I'ljimiinll (if |Jli! IVI'lpl'lK'ul HLil'llll 1M ( //HI - y) (.I*; //-) ( i/o .r^if t,-. 0, "i" ' ;11 .V M* ' ' 1"' (l - v 'w- '' ' H ' L "Will ' s '(l. "I J ) Kdnnmlii'il by a lino nnuM,iiig Llui lino ", '" t), ami I In* cnliiiT ritmi p 0, n^ I), i" i() Uvinn, Tho <!i|imtitm M nliviuiiHly iiii'linlnl in tin- ^'i-ui'Vnl '-ijiiiil imi Nil. lu'lui'iiiu^' ID (In* ^fin-nil I-H-H- nl' tlm n-rull, diV/ArfA w/wxjiVw, (S'tl, H a ), il. IH |ti'n)ir-r in hliuw ^riiiiit'tfii'iiilly hn\v it i iluil iln- ivrijimnil in n ftornll, /i/'/i^t N/JIK;M,W, S'U 1 ), Ciiiihitli-r in dif M-n.ll iS'{l, H') uy 1'lmm |,hrmij.jli 1,1m tliri'ttirix lino; t.hiH rniiiuiiirt l-tu'i'f ^.'iM'UUinti liitiM nl' Iliti wrull, vi/. Lilian iii'd Mm Hitli' ol' l>hn trian^lt) litiu'f| Ity llir llti'i-f [inintH j' iiih'ivu'iHi'Mi mf Uii; |ilarti' \vi(li l-hn Hkmv t.niltio: huniMt in iliD i'iTi|ir..i'ii| lit^inv \vi* hnvi* a ilirritlrK Mm* niu-li ||i,(, til. I'm;!) |iuint. nt' il. l.linvd ai<- llii'fi* i,;rn<')nliutj lini'n; llnil. in, wn Imvn u ni p nll N(l*) wil,li u Iripln tlii'tirdnx linn, I '"iivi'ivu'ly, iti in linj< wirli [lip wmll ,S'{ |'), rurh pliuii' lln-migli llin l.rijilu (liviail^rix linn itjt'i'f i (lit* siri'iill in tlti^ HIM' llir.'i- linn's ami in a Him(lu ijnin*vuling linn; wlii'min (In-n* i 1 * in !li< ri'i'ipnu'jtl nt'|-tl| H -aiti|iln tlin'rlrix linn; but. in unlcr l,ti wlinw l.lial- i) ii n )-.i-i,.ll ,s'(| ( ;i j ). wi liavn yrl i.u H!MW Ihut Un-i'i* in, us n ntnlul ilinmlrix, u rtkiw niliii* nifl (Avii'r liy twli ^rii'Malin^ linn; thin iuiplii'M tlnif,, iiMiiprutiully, in l,fin wi-i'cll .V(t 3 ) nii'li grin'mliiit; lint* IM lite mtriwiHitni nf (AS'tt OHtinUrin^ (>liuu'H of u M^i'sv ftiliit! (liMtj^i-nt jiliiiM-H nf a t|itiuliti (UIHC), wh mittli fitiiuo cotihiiuiug l,v" liiK-n M|' tin' wntl! a jfi'nini'trirjtl proprfty whh'h IM far from nbvimiH; iu ill-- Ncritll, niititt hjtirii'x, XflC), wlu'n; the i ..... : ...... ..... ' , flu* pr<.|iniy iJnit. I'iM'li jiohrnniiig Hint in u linn t' !i!mbi IM ih' |)riipiMly ili'tt fiivli line is jtten UMI .MI-.-., ( ,, W *. ...... ......... .. ........ n i'H trf a nkiw (riihit 1 i MI*, wluit SM iht* Hnmn lliiiiXi iw liingont ]ilam!H of u tjuartic Ainimus, J/y IM, IHOO. iho loa-goiiitf Moinuir ww* wriltou I runiMvistl from ProfcBwor Cromoiia a I altar (foiiod Milan, Nnvoinbor 2U, 180K. in which (btwiduH tins ninth and ItmtH pooioa 'roiiidn i> (l above!) ho refbm t two oilier tfpuaiu* of ijumtio Morollu. Ho tromnrkfl that 328 A THIRD MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. [410 * Deficiency No. of species, Nodal curve, Bitangent torse Corresponding to niy species, 1 r 3 2 3 10 2 -?/ 2 + J?j ^s + 5, 7 3 -fl, 3 JTa+JP, - (say, 12) 4 R 5,+^ ^i 3 -(say, 11) p = Q 6 ^sf l ^a^ 2 5 7 r a W 8 8 ^! 3 ^ 9 9 -^l 3 7?j' 3 3 10 J?,' ^ 6 p-l 11 A+A' J?, + A' 1 12 ^ # 4 where T 3 denotes a skew cubic, 2, a torse conic, K, a qlla drio cone, A, -R, S diff ent or three u: QH ^ n ^ / line" R* R* ' Xl Mi I have not yet examined the two new viz these aw , ' f '- " twice moT, mn j *i mentioned m this enumeration: 411] 411. A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. [From the Philosophical Transactions of the Royal Society of London, vol. OLIX. (for the year 1869), pp. 201229. Received November 12, 18GS, Head January 14, 18GD.] THE present Memoir contains some extensions of Dr Salmon's theory of Reciprocal Surfaces. I wish to put the formula on record, in order to be able to refer to them in a "Memoir on Cubic Surfaces," [412], but without at present attempting to com- pletely develope the theory. Article Nos. 1 to 5. Extension of SALMON'S Fundamental Equations. I, The notation made use of is that of Salmon's Geometry, [2nd Ed.] pp. 450459, [but reproduced in the later editions, see Ed. 4. (1882), pp. 580592], with the additions presently referred to; the significations of all the symbols are explained by way of recapitulation at the end of the Memoir. I remark that my chief addition to Salmon's theory consists in a modification of his fundamental formulas (A) and (B) ; these in their original form are a (n 2) K -H p + 2<r, b (n - 2) = p + 2/3 + 87 -f 3t) o (n - 2) 2<r -i- 4/3 + 7 6 (n ~ 2) (n - 3) = 4/c + [ab] -\ c(?i-2)0-3) = 6A + [c]H where [ab] = ab - 2p ( [ac] ~ac - So-, [bo] be. - 3/3 - 2' a vi. 330 A MBMOm ON THE THEOBY OF BEOIPBOOAL SURFACES. [ 4U * to node, these will be but these present themselves cip rMa l equations. respectively, fonnl betje *"* 6', cuicnodes, -S, binodes, j , pinch-points, %, close-points, 0, off-points, t", cnictropes, f , bih^opes, j' , pinch-planes, %', close-planes, 0', off-planes; [06] = ab - 2p -j, [ac] = ac So- - Y A> i abore ; e r reci P ro cal term *>, in th, M ^eir vahes, the which replace the original formula (A) and (B). 1 This addition to the theory is in fnnt a' from tlie t af UB - L e i 411] A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. 331 5. For convenience I annex the remaining equations ; viz. these are a'= n(n 1) 26 -3c, / = 3ji (n - 2) - 66 - Bo, the equations (jr=&s-&-2/c-3y-6i, 7- = c a - c - 2/t - 3/3, (q, r in place of Salmon's R, S respectively); the equation. a = a j and the corresponding equations, interchanging the accented and unaccented letters, in all 23 equations between the 42 quantities n, a, 8, K ; b , k , t, q, p , j ', c, h, ?, or, 6, % ; /3, 7, ; S, (7, ', a', S', *'; 6', k', if, g', p', j'; c', h', r', o-', 0', K ' ; (3', y, ' ; ', 0'. Article Nos, 6 to 12. Developments, 6, We have (a- b- c)(-2) =(/c~ J /3~5)-6 J Q-4 7 -3, - 8& - ISA - G (60 - 3J3 - 27 - i) ; and substituting these values of S, in the formula ' = a (a - 1) - 2S - 3, and for a its value, = n (w 1) 26 3c, we find n' - ?i O - l) fl - n (76 4- 12o) -h 46 a -h S6 + 9c a + 16o - Sk - ISA + 18/3 + 15W 4- 1 a* _ fl* 332 A MEMOIE OK THE THEOBY OF BEOIPHOOAL STOFACK . [4U Writing in the first of these '~2 = (_!)_ other ^ by mMM of the vaiues rf ft . '-=: -20-42 + *- ff _2- (Salmon's equations (D)). W hich to the q , , . 8. - * and reducing the eq ,, ati on, -, which, writing thereia a = n(n-l\-9h * i (n 1) - 26 - 3c , and * = 3n (n - 2) - 6& - So, becomes a' = 4n ( - 2) - 86 - lie - 2/ - 3 X ' - 2C" - 45'. ' * ^ -iproea! of ,. of the e generated by he tanj 11 T* ^ f th S P ilMcie torse ". what i, the same th Sg t i tfe n, T ? ' f"' 6 SeCti " f the pla D e section; that i s , it s ea'n I 7 , ' ! Pm tle ^ mes which to oh the spinode curve and the plane ZtiL m fl ? f. f P iutS f ^<* of the The spinode curve is in fact fo ' "', J' ^ " ^ <nd " f the s P inode interseotiou of the surface by the He ^"Jl * f"?'' Y"" 1 ' ^"g" 1 ^" a curve of the order -~ ^^'^ " ~ t * order a redaction + U . a, S in g Hessian surface ns the fee in the , "' ! ," """ S8me thing ' that cuspidal curve taken 1! time s -a e ,1 w h oh T 1 i"? *, aken " tim6S ' md in the also as appears post, No 'T , , amved at b y othe1 ' me , and p - . pmoh-pomt, a close-point, a cnicnode arrl 7 I i ' ' ' must signify that the surface d 1 ff 60 * which are not regarded as belon"L t !h " itaelf that for tL reciprocal of'T Lh reciprocal of a binode it is a lL ? * thls curve ls a Pineh-pcint it is a ,i ue c^nti 2 CCd f ti '^ fc , hue counting 3 times. fche reci Pcal of a close-point, a reol P ro ls of a Geometrically thi s " ertain at Ce S ^ est ic, and for the 411] A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. 333 9. It is clear that p' will in like manner denote the order of the node-couple curve. 10. I express in terms of n, 6, c, h r k, & 7, j, 8, x , O t S such quantities and combinations of quantities as can be so expressed. We have rt = ft' = n( 1) 26 3o, K' = S (n - 2) - 66 - So, 4* = 127i 4- c (fin - 6) - Gc 3 - % + 30 - 2& 24( - (- Sft + 8) 6 4- (15n - 18) c + 86 a - 18c a - 2 (8ft - ISA) + 20/3 - 167 + 4>j 4-904- 6# q = & a - b - 2& - 87 - 6, (i auprk), r = & - c - 2A - 3/3, 2o- = c (n - 2) - (4/3 + 7) - 0, 8p = (IGn - 24) & 4- (- 15m + 18) o - 86 s + 18c 3 4- 2 (8fc - 18A) - 9 (4/3 + 7) - 4j - 90 - 6^, 8 = Sn (n - 1) (n - 2) + 6 (- 32 + 56) + c (- 17n + 40) + 86 a - 18c a - 2 (8A - ISA) + 1^ (4/3 + 7) + 4j 4- 170 + 6^ 4- 85, 2S - (n - 1) ( - 2) (n - 3) + 6 (- fai 2 + 20/t - 24) 4- c (- 6w a + Ion - 18) 4- 126o + 18c a + (8/c - 18/t) - 9 (4/3 + 7) - 0(9 + 20, 8ft' = 8n (51 - l) a + (- 32 4- 40) & + (- 21 4- 30) c + S6 a - 18c a - 2 (8/c - 18/0 + 21 ( 4 ^ + 7) - 12 J + 21 - !8% - 160- 24J5, c' = 4re (n, - 1) (n - 2) 4- (- 16i + 28) b 4- (- 10a + 26) c 4 4& a - 9c" - (8/c - ISA) + 10 (4/9 + 7) - 4j + 100 - 6^; ~ 6C - 8B, f = - a + ' (n' - 1) - So', c' supra), W - 9-S' = - lift ( - 2) 4- a (n' - 2) 4 226 4- 30c, (', a supril), -0'==c'(n / ~2), c' supra), 4/c' - 3 (i 1 4- 3/3' + 27') - 2p' - / = (- 4 y + G) 6' + 26' a , (w' f 6' suprii), 6/1 - 2 (*' + 3/3' 4- 27') - So-' - tf = (- 4n' 4- 6) c' 4- 3o' a , (ri, o' supri) ; [or in place of either of these, 8/c' - ISA' - 4/>' + Do-' - 2?" 4- 3 X ' (2f - 3c') ((n' - 2) (n' - 3) - a}, &', c', a supra)}, p' + 2/9' + 87' 4- 3i' = &' (n' - 2), I)' supra), 3 r ' + 2i' 4- ^ - off 7 - & - 40' = c', (c' supra), (twenty-three equations, being a transformation of the original system of twenty-three equations). 334 A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. [411 11. Forming the combinations 4i + 6r, 2U-Sq + l8r (the last of which introduces on the opposite side the term +m), we obtain Alices te + Or = c (5n - 12) - 5 y ~ IS/3 + 30 - ^ equations which are used post. No. 53. ,* ' lt" e ? r;? ttt f diff T !t points f on face by any piaue oint of fche line f - B z R, ,, -, ,? on * T face by any piaue * hrough the Hue tai s tfe i; *Z' A, '? ,? e r ? oint of fche line ne and the same ta g^ pi-. hen the sect um of the .uvfaoo by the tangent plane oontams the line at least twice li tl VT 3 '' le lme " fo '' Mi ' if thMe times the line is far turned a Z,';'; P "T* " le t0ml " " nl " U " e ^ like m ' ^ * ra '' oscl ' kr a ^ tl ' trned a Z,';'; P "T* " le t0ml " " nl " vm be oo,* ra '' oscl ' kr a S^ plane. The* epithets, S erola r> torsal and oscular, will be convenient m the sequel. Article Kos. 13 to 39. Explanation of the New Singularities. I proceed to the explanation of the new singularities. 13. The cnicnode, or singularity 0=1, is an ordinal , conica i . ; . , the tangent plane we have a proper quadricone. = ' s cnot is P conteot ; " say rather the oauta.e !s . touch.ng a surface, not at a single point, but along a conic. the te 3 a--V U Tr aVlg "'Y 110 * 0-1, and the reciprocal surface having ctr:tS~ d SocTZetr? 6 of the cnicnode sk direoti ns f touching it at x points ^ The , k^ nf t " lWS ^ f "" onicta) P e conio ' twice and in a S T , the C ' 11Ctl ' pe meets the surface in * nic twee and in a les dual curve which touches the conic at each of the six iioints It would appear that fee six contacts are part of the notion ,f the cnictrope " that ae idu7cuLT e !' ave ; surfao with a conio of i 3lme contaot . i> * on o rix t me" v at al l f lnteree tl0n the P 1 * f -c does not touch the asmte "" ' Collic and here the conic = 0, F= 8lven as ^e s of intersection of the Uvo couos 411] A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. 335 17. For a surface having the cnic trope G'=I, the Hessian surface passes through the conic, which is thus thrown off from the spiuode curve ; or there is a reduction = 2 in the order of the curve, which agrees with a foregoing result. 18. The binode, or singularity B= 1, is a biplanai- node, where instead of the proper quadricone we have two planes; these may be called the biplanes, and their line of intersection, the edge of the binode, The biplanes form a plane-pair. 19. The bitrope, or reciprocal singularity B' l, is the plane of point-pair contact; but this needs explanation, 20. Consider a surface having a binode, and the reciprocal surface having a bitrope. We have the bitrope, a plane the reciprocal of the binode ; in this plane a line, the reciprocal of the edge ; in the line two points, or say a point-pair, the reciprocal of the biplanes : these points may be called the bipoints. There are in each biplane three directions of closest contact; the reciprocals of these are in the bitrope three directions through each of the two points. The section, of the reciprocal surface by the bitrope is made up of the line counting three times (or fche line is oscular), and of a curve passing in the three directions (having therefore a triple point) through each of the two bipoints, The bitrope contains thus an oscular line ; but it is part of the notion that there are on this line two points each a triple point on fche residual curve of intersection. 21. We may however have on a surface an oscular line without upon it two or any triple points of the residual curve of intersection. Such a surface is Mtv -\- Ny 3 = Q ', the intersections of the line aa = 0, y = with the curve = 0, JF = will be all of them ordinary points, The reciprocal surface will have a binode, but there will be some special circumstance doing away with the existence of the directions of closest contact in the two biplanes respectively, I do not at present pursue the question, 22. For a surface having a bitrope B' = 1, it appears from what precedes, that the oscular line must count 4 times in the intersection of the surface with the Hessian ; for only in this way can the reduction & in the order of the spinode curve arise, 23- The pinch-point, or singularity j = I, is in fact mentioned in Salmon; it is a point on the nodal curve such that the two tangent planes coincide, or say it is a cuspidal point on the nodal curve, If, to fix the ideas, we take the nodal curve to be a complete intersection P = Q, (3 = 0, then the equation of the surface is (A t B, <7JP, Q) a =0 (A t B, G functions of the coordinates); we have a surface A0-J3-0, which may be called the critic surface, intersecting the nodal curve in the points P - 0, Q = 0, ^6'-J5 a = 0, which are the pinch-points thereof; or if there be a cuspidal curve, then such of these points as are not situate on the cuspidal curve are the pinch- points : see my paper " On a Singularity of Surfaces/' Quart. Math. Joiwn. vol, IX. (1868) pp. 332338, [402]. The single tangent plane at the pinch-point meets the surface (see p. 338) in a curve having at the pinch-point a triple point, = cusp + 2 nodes, viz. there is a cuspidal branch the tangent to which coincides with that df the nodal curve; and there is a simple branch the tangent to which may be called the cotangent 335 A MEMOIR OS THE THEORY OF RECIPROCAL SURFACES. particuiav case he ode, ,0 CllS1>klaI bl ch >' and the cider n - 2, the tangent to which is the cotangent, ' Mdl1 "' 1 ' 1 si [411 he of the let Plane fa " H . tl, ai ' H P , is in fact a torsal plane touching twi e aud in a "H'X'l e reaSOn that the am by the , Secti " h at ^ * ident nocle - oou P le P' a wit1 ' two coincident nodes. The plane " conple aue in a HM * p"-** thi g h ^ -^ to,, inng o the , - " --P1 fig-, the reciprocal of the pnoh-plaue la tln.s a point of the nodal curve, and is a pinch-point' the ' ' ? "".P^-point is the reciprocal of the point P - th tang "to :;; d e i ;: t reeipi r :f the H r " that is - f the tai ^ nt <* p * * , and the cotangent at the pinch-point is the reciprocal of the torsal line. 2a. There is in this theory the difficulty that for a surface of the order n the l plane meets the residual eurve of intersection in - 2 points 7 a d if oo - virtue of h only ono of the (-2) pomts of intersection of the torsal line with the residual lane onm, a po.nt of the node-couple curve of the reciprocal sul I, ft f a proof. of A rn ft /- rf / 7, / v 01 p t a, di /, g h> t> m> B) i', Oil + 2l^r fu aw ' See Salmon, p. 2ig, wlMM it ia only Btaw[ ^ th() HflgBian ^^ ^^ functions of the Hoe. 411] A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES, 337 and representing this for a moment by A, H, Cf, L = 0, H s B, F, M G, F> G, N L, M, N, D then in the developed equation D (ABO- AF* - BG 2 - Off* + 2FGH) ~(BG~F\ GA-G\ AB-H\ GH-AF, HF-BQ, FG - GH^L, M, JV) = 0, observing that 0, F> M, N, D are of the first order in x, y, the only terms of the first order are contained in B (- DQ Z - G# + 2NGL) ; and since G, D, N are of the first order, we obtain all the terms of the first order by reducing B, G, L to the values 2i//-, G, D ; viz. the terms of the first order are = - 2^ (C*d + D*c - Hence the complete equation is of the form - 2i/r (G*d + D*c - ZCDn) a; + (so, y}* = 0, or, what is the same thing, <]> + f*p ~ ; the Hessian has therefore along the line <c = 0, y = Q the same tangent plane a = as the surface; or it touches the surface along this line ; that is, the line counts twice in the intersection of the two surfaces, 28. If instead of the right line we have a plane carve, say if the equation be axj) + P 3 ^ = 0, then the value of the Hessian is a;<3> + P 1 !' = (viz. the second term divides by P only, not by P 3 ), so that, as before mentioned in regard to a conic of contact, the surface and the Hessian merely cut (but do not touch) along the curve {o~0, P = 0, To show this in the most simple manner take the equation to be a' + |P 2 = 0; let A', B', G', D' be the first derived functions of fa and (A, S, G, D), (, &, o, d, f, g, A, I, m t n) the first and second derived functions of P ; then if in the equation of tho Hessian we write for greater simplicity = 0, the equation is ' + Ph + AB t C " + P(I + AG, 1 7 + PZ +^ID PI +& , P/ + 8G, Pm + 5.0 Pf+BG, Po + 0* , P?t + OD Pm + BD, Pft + OD, Prf + ^ a = 0. D' + PI + AD, The equation contains for example the term - (D 1 + PI + AD)* {P a (bo -/*) + P (bG* - 2/BO)}, dividing as it should do by P, but not dividing by P a ; and considering the portion hereof - D"*P (W + oB 3 - 2/BG), there ai-e no other terms in D' B P which can destroy this, and to make the whole equation divide by P s ; which proves the required negative, 0. VI. 43 338 A MEMOIR ON THE THEORY OP RECIPBOCAL SUltPAOJilS. [411 29. For the off-point or singularity = 1; this is a point on tho cuspidal oumi at which the second derived functions all of them vanish. In further explanation hwnf consider a surface U=Q, and the second polar of an arbitrary point (a, # y, S) ; vi/. this is (a9. + ^9y + 79 / + S3 w ) s yO l or say for shortness A a /"=0, whom tho ttonniciuhl.* of the powers and products of (a, ft y, 5) are of course the second dorivod functions of //; this equation, when reduced by means of the equations of tho cuspidal oiirvo. may <uM|nirn a factor A, thus assuming the form A(aP + /9Q+ 7 ^-|-S6') 3 =0, and if HO tho inJiormxitimiH of the cuspidal curve with the second polar (-20- + 0, if, as for simplicity in Hitpjm.s.nt, there is no nodal curve) will he made up of tho intersection* of tho cuspidal cmrvo with the surface A = 0, and of those with the surface P + /9Q + r -|, &Sf o i-nctti twice; the latter of these, depending on the coordinates (a, ft y, S) of bho arbitrary points, are the points a each twice; the former of thorn, or mtorsootioMH of tho rnspidid curve with the surface A=0, are the points tf, or off-points of tho cuspidal !.,. 1 there _ is a nodal curve, the only difference is that tho off-point* aro H.ioK uf thr above points as do not He on the nodal curve. instance of the mor in which this nia^nlariU' iuy a+ ^ 3 = ' Where ** dofi" of th fi,nti,L m - * if * e th rdor f HW " rfMu - W-; and we e t - f Uu- CUSP"'"' ~ " " " " " 8l - ll 'hicc equation of th and if fe an ; ri Sco"Z's tho = fo = A ^T " ( "' '"' :'*il), then \ has the cuspidal conic = o, Sx' + an -n ,, has tho off-point s OT = , j, = , W + l-n. (l S C ( 0rain 8'/" ldor tho fmm M + OQ'-O) (.- 0, y . 0, w = 0) eaoh twice ; 9 ^ ' * ' 8 ' thB l )oilrta (0. -0. U 0). But siting the same equation in tho form whore (4 ' O*- &! + j!/' - ^ _ ,), = _ 411] A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. 339 it appears that there are also the three cuspidal conies y 3 2a? = 0, xfl ssw = 0. Reducing by means of these two equations, the equation of the second polar is at first obtained in the form = ; but further reducing by the same equations and writing for this purpose y = WJB (<u 3 = 2), the equation becomes (4, (Ja;, 0flj a 5fl) a (3o) s Ay-6Aa!), 2a;A - sAw - wA-s 8 )" = 0, that is a; 3 [20 (3o> 2 A?/ - 6Aa;) + 3 (2#Aic - sAw ~ wAs)]' - 0, and we have thus the off-points e?~Q t if- 2a 3 = 0, a?-ffiu-0, in fact the before- mentioned two points each G times ; and the complete value of 9 is & (4 -h 12 =) 16 ; viz, the off-points are the points (03 = 0, j/=0, = 0), (# = 0, y = (\ w-0) each 8 times. On account of this union of points the singularity is really one of a higher order, but equivalent to = 16. I am not at present able to explain the off-plane or reciprocal singularity 0' =\.. 33, As to the close-point or singularity % = 1. I remark that at an ordinary point of the cuspidal curve the section by the tangent plane touches, at the point of contact, the cuspidal curve : the point of contact is on the curve of section a singular point [in the nature of a triple point, viz. taking the point of contact as origin, the form of the branch in the vicinity thereof is y a -.-'== 0, whore y = is the equation of tlio tangent to the cuspidal curve], such that the point of contact counts 4 times in the. intersection of the cuspidal curve with the curve of section. At a close-point the form of the curve of section is altered ; viz, the point of contact is hero in. tho nature of a quadruple point with two distinct branches, one of thorn a triple branch of the form y s = a?, but such that the tangent thereof, y = 0, is not the tangent of the cuspidal curve; the other of them a simple branc!^ the tangent of which is also distinct from the tangent of the cuspidal branch : the poii-t of contact counts 3 -H times, thnt is 4 times, as before, in the intersection of the cuspidal curve and the curve of section. The tangent to the simple branch may conveniently be termed the cotangent at the close-point ; that of the other branch the cotriple tangent. 34). We may look at the question differently thus : to fix tho ideas, let tho cuspidal curve be a complete intersection P = 0, Q = ; the equation of the surface is (A, B, G^P, Q) S =*Q, where AC-B 2 = Q, in virtue of the equations P = 0, Q = of the cuspidal curve, that is, AC-B* is =MP + NQ suppose. We have (as in the investiga- tion regarding the pinch-point) a critic surface .4 (7 - I? a = 0, this meets the surface in the cuspidal curve and in a residual curve of intersection ; the residual, curve by its intersection with the cuspidal curve determines the close-points ; the tangent at the close-point is I believe the tangent of the residual curve. Analytically the close-points are given by the equations P = 0, Q = 0, (A, B, G^N, -Jlf) a = 0. It is proper to remark that if besides the cuspidal curve there be a nodal curve, only such of the points so determined as do not lie on the nodal curve are the close-points. 432 340 A MEMOIR ON THE THEORY OP RECIPEOOAL SURFACES. ("4 1 I 35. I take as an example a surface which is substantially the same as one which presents itself in the Memoir on Cubic Surfaces, viz, the surface (1, w t anj$M*-iey t *) g ^0, having the cuspidal conic w n -~xy = Q, 2 = 0. Since in the present case' AO-l&Ll\ w <> have JI/-1, iV = 0, and the close-points are given by P=Q, Q = Q, = 0; that; is, fchi-v are the points (2 = 0, w=Q, = 0) and (2=0, w=0, y = 0). 36. I first however consider an ordinary point on the cuspidal curve, or conic wa- wy - ; the coordinates of any point on the conic are given by IB : y ; g : w = 1 Q* - () (f where $ is an arbitrary parameter; we at once find fl+ y - Qtg -\- 2 W ) =0 for |'] M ! equation of the tangent plane of the surface or cuspidal tangent piano at fcho f " ng to M the mtersection of this i *i. and thence and the equation thus is or, what is the same thing, viz. this is . or reducing, it is . ; b 8 p - ? ' * - rhe the conic, and in the vicinity of tH ' ^ , Maumed I 3oin(i 0, b-ch of the for m . ?**- - - - 411] A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. 341 = 0, that is in the line <e = 0,w = three times, and in the line ce~Q, w+22=0 (that the section consists of right lines is of course a speciality, and it is clear that considering in a move general surface the section as denned by an equation in (w, z. ?/), the line w = represents the tangent to a triple branch w 3 =^-j-&c., and the Hue ?y + 22=0 the tangent to a simple branch); these lines are each of them, it mil be observed, distinct from the tangent to the cuspidal conic, which is cs = Q, 2 = 0. And similarly the tangent plane at the other of the two points is 7/ = 0, meeting the surface in the cuvve y = 0, w 3 (w -h 2#) = 0, that is in the line y = 0, w = three times, and in the line y = 0, w -h 22 = 0. 38. The close-plane or reciprocal singularity %' = 1 is (like the pinch-plane) a torsal plane, meeting the surface in a line twice and in a residual curve; the distinction is that the line and curve have an intersection P lying on the spinode curve ; the close-plane ia thus a spinode plane; it meets the consecutive spinode plane in a line p passing through P, and which is not the tangent of the residual curve. In the reciprocal figure, the reciprocal of the close-plane is on the cuspidal curve, and is a close-point; the reciprocal of the point P is the cuspidal tangent plane; that of the line p the tangent of the cuspidal curve; that of the tangent of the residual curve the cotriple tangent; tliat of the torsal line the cotangent. 39. The torsal line of a close-plane is not a mere torsal line ; in fact by what precedes it appears that the surface and the Hessian intersect in this line, counting not twice but three times, and it is thus that the reduction in the order of the spinode curve caused by the close-plane is =3. Article Nos. 40 and 41. Application to a Glass of Surfaces, 40, Consider the surface FP* + GR-Q S = 0, where / p, g, r, q being the degrees of the several functions, and n the order of the surface, we have of course /+2p~0 + 2?' + 30. There is here a nodal curve, the complete intersection of the two surfaces P = 0, R = ; henco 6=pr, k = %pr (p -!)(? -1), =i& (b -p -r + 1); t = 0', whence (q) = pr(p + r- 2). There is also a cuspidal curve the complete intersection of the two surfaces P = 0, Q ~ ; hence o=pq, h =* ^pq (p - 1) (q - 1), o(o -p -q + 1); whence (r)=pq(p + q- 2): I have written for distinction (g), (r), to denote the q t r of the fundamental equations. The two curves intersect in the pqr points P = 0, Q = 0, R = 0, which are not stationary points on either curve ; that is, /9 = 0, 7 = 0, i =pqr. There are on the nodal cnrve the j = (f+g)pr pinch-points .F= 0, .P = 0, _ft = 0, and = 0, P = f R = Q. There are on the cuspidal curve d=fpq off-points ^=0, P = 0, Q = 0; and there the gpq singular points G = 0, P = 0, Q = 0. I Hnd that these last and also the points each three times, must be considered as close-points, that is, that we have # = (g + 3f)pq. 342 A MEMOIR ON THE THEORY OF RECIPROCAL SUBFACKS. ["411 41. We ought then to have b (n ~ 2) = pt c (n - 2) = 2<r + $ ; the first two of which give p, and then, substituting thoir values The first of these is 2 = 2^ + 2 ,. + S?+/ + , /j . ^ +/) + (2) , + ;j and the second is *-* + 8 + - that the equations are satisfied. Article No. 42. The Flecnoclal Curve /: SW-tsr^'=."-r5j? Article Nos. 43 to 47 Stirf 3c . 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IIHVP Mm.-* t.wn *!<< i-udi ul' / liiu-i, mit-li thai ulmif* flm linen M|' t-iu-li jH'l, III- Miirliii'M is liuicliiul Ity nn (inui^iiuu'V) ini'iiilinn jilium; VJK. lln-a; an 1 ill.' riirulnr plitnrM .1- !/// (), .r ..///:, jmnninj.f llir.iiirjli elm nxjs. I MI nillmnl. !iln|ipini( I" nlimv i) fhal. llii'H.' 'Jw lim-M urn lint-L imi j' Inn y' 4 Elm! i-i. Mn-y I'Hi'h rcttnftt |.!n> itnmr -M' Mit< n|>iiM>il>- nuvc liy M('). Tlic iiUl\inii>< i-n' .'h' J ' lim tiiS MA- pjivli-n wliii'h ^iiimMhih* flit- Npinmli- ciirvn uii Ilir fiininuv. I- !. Atnl WP ran Miuu vnily Mm). I In- rmupli'in iuti'mi-i'iinn n|' MIP fiiirl'ai'n ilin lli'iiniini in fiiml' 1 up in mvMiiluttn' \villi Mn f t'>n.'i(niii!;- Mtriirv ; vi/ t hd>T ul' MM i I'ai'i' - 'J,ni t ( )nlT i'l' I lt-ciiui -M*^/;j '), whi'iii-n ni'diT ill inli'n-ii'rliuii lli/H . . \fltn Nntliil I'uiA'p, .J(/M a -wi|-|-ft i-in-l.-i. S titii.-i .H/n ? . SF | ItiiS <!itHpii(ul i-ui'v.'. A- ciivl-(. II fiit ..... i . \>\>K t-in'h'rt nl nintur!, m j - m - *JJi - - ?U, l!//i' 'Jj -Iri < liv I'iili'H 'Ji , .'I liiiii-i r Urn l-"i. \\ r n limy Ity n ^imilur iTii^niiiuj -Jinw Mint lln< MiiiTit-r amt Mm ll*'i-mn|it H Mt'i'l, in flu' imtlat i-tirvi' ItiKdi U:J liiiu-s. nut! in lli>' ciuipi'lul nnv.' tukni *,!'(' ami nin^'iph'iiMy Mmf, Mn- unltT M|' Mi- r.-.iilnul iiil'-i^rniiMii i,r Ht-rntnJii| t-nrv.^ in .(! In--- Ull '^/t -'J7r. Ti flli'i't. (JiiM, nliHcrv Mini, jil any pMiut, \vht'v.-r nf H .jiiiuhiir nmla*-"' ilic ijintfriil, plitim iiH'utM lln* Niiriiu-,' in M jtuii' ui' liui-ii, linn i-t, in u cuiv-* liuvini; at Mir pinl i.t t it< tnuli 1 with an iallrxir.ii MII each lnMiii-li, m 1 miy, H U.-H-'i-HMit,'. rjuHs,{iim in ptiuii' lixiii'c a t'-mir having its *'*! n MH tht< uxi* .!' iMiufi.in ami itt '"- : riiltiiit Ilmi'f"\vilh, iiml Mm trmmf huviji)r with tin* caiv.) .(' tin? niil>'i o -I. IN ittiitn al, any juiinl /'; iln jmiui 7* gi-m-ml*^ H rircl*-, MIK-!' Hun. Hrdp thu MiiHiit't- in imfiiltitnl liy a f|iuulrii' MirtJiro it' iv*'"l"ii-" \\w inuntlian iwM'iiiiiiM Imv.,- a f'HU'j>niniir matact; tlm ri Mio surJUi! a lU?llmMiinln t'ircli'; uutt I a^miin Miat it nrolo. HUIICO iT itiu niiinlufi' nf tin 1 poiatw r In- fl, w.* luivr *n the mirlm-** f HuHcianodo circloti, **%& HrciiiHlw riruliw, lliitl w, a nVi'autl" t-arv.' nf tfiu unlur ift I wish lo Hhw ihnt wet lmv j? wfiin.<-ni 1 Observe iliai Uus trm in m cant te t;tii riil of in a difff-rcni umiin#r, by nny Bticrallioit **f H itucE 11 io which Urn prw^nt ii nHmufi- n***! 344 A MEMOIR ON THE THEOKY OF RECIPROCAL SUUFAOJM. 4*3. The problem is as follows: given a curve of tho order m with K onsps; it is required to find the number of the conies, contm on a ami an axis coincident in direction with this line, which have with 1;]io n 4-pomtic intersection, or contact of the third order. This may be w .lvu f fomnhe contained in my "Memoir on the Curves which mbisfy given m /. vol. CLVIII. (1868J, pp. 75-144; see p. 88; [400]. Taking a - = for the given line, the conic (a, 6, c, f t CL AT),* -jyj- . s cent, oa the given line and an axis coincident thomw th tf ! y 1L' ^"-tmg these two conditions by 2Z ( it is easy to see that wo Imvn )=2, (2,T.//)=2, $ given mvini <! b will " . " ' f/i | ] H run! !j m . nim' mns ' !,.. ' " * thon , where *, ft % S denote {2,Y.-.), (2Z-/) rST./A t9Y im - - , TO ^.o 1. 2, 2, 1 respecti ely, a, d whl "' ( ^ ^ "" "' " W '"'^^ >*' respectively. ^' wil01 ? ft ' / 9 - 7 denote (M/;), (^./j, a w (loni(IH ih ^ contact of the third order with a l e ' I 7 S do oto ^ liU<,!i uf u <. ' > ( the order, the class 4Z\ " , (l ? S ' *i or a( V with a W ivun CM,,,,, -2S-3, aa3 + #) ) th.m wo Imvo (3: )-4OT Order of fl ecilo d e 8Urf 0-Ior of intersection curve, Hfli Cuspidal curve of contact of contact . w 411] A MEMOIR ON THE THEORY Off HEOIPUOCAL SURFACES. Article Nos. 48 and 49. The Flecnodal Torse. 4)8. Starting from 22&' -I- 27c' = 6 (66' + 8c') - 7 (26' -\- 3c') = G (3n' a - 6n' - ) - 7 (it' 3 - ' - S) 345 that is 11*- W- 226'- 27 c' = &n' - 7S + 6*. I find tt'(ll?j/-24)-22&'-27c' = w (?i ~ 1) (llw ~ 24) + b (- 69n + 96) + o (- 04 + 156) + 266 3 + 87c a - G2&- H4A+ 141/3 -|- D4y + 77*+ 3y' + % - 15(9 - 45^ - IOC- 9.8. 49. For a sin-face of the order ?i without singularities this equation is n f (lift' - 24) - 226' - 27c' - (n - 1) (11 - 24) ; to explain the moaning of it, I say that tho reciprocal of a floonodo is a flccnodal piano, and vice versd; the reciprocal of the rtecnodal torse of tho surface n (viz. the toise generated by the floonodal planes of the surface) is thus tho flecnoclal curve of the reciprocal surface n'; and the class of the torse must therefore bo equal to the order of the curve. The flecnoclal torse is generated by tho tangouts of tho surface n along the curve of intersection with a surface of tho order lln-24; the numbor of tangent planes which pass through an arbitrary point, or class of the torso, is at once found to be (n-l)(ll ft -24); for the reciprocal surface the order of tho flccnodal curve w by what precedes '<11' -24) - 22i>' - 27c'; and the equation thus expresses that tho order of the curve is equal to the class of tho torso, Article No. 50. The general Surface of the Order n without Singularities. 50. In the general surface of the order n without singularities, wo have n = n, K = n (n - 1) (n - 2), & =0, h =0, 4. , . A 2=0, ' ' 44 ' m WP U (i A MEMOIR ON THE THEORY OJ? RECIPROCAL SURFACES. [411 c =0, A =0, * =0, o- =0, C'-O, /?-0, d =o, 7 =0, i =0, '* n {*-!)', '- (-!), *'3(B~2) 1 /j'= /i(ji- /=0, c' -in (n - r 2 ^-0, A "j *-* u, -2) (3ft -4), 7' = 4^(71 -2) (-., t' =0. ^ ' 41 A MKMOIIt ON 'I'lIK 'I'lll'XWS" ()]' ItKril'ltOdA!. HlMtKAtHW. rl-inli! Now. ill lii (!!'. liu't'stt'iiuiuni of FnrntHlu fm- ('I'. fil. Tlu< valim //' a{H-ii)(llH..ai) I'm* n by Sulimm liy intloju'iidriil, j;mnii'Irinil rniiM \villi,.iii , vi/. he ul iiuinlmi' !' inf,i ( rsi-r|.innn nl' f.hn Npiimtli* iMtrv.t (nnlrr - -I-// (:!}) ly |.|m llr nl' rim tinlor Lin :H. fiy. Tin) vttliin M|' /-J' nilt.'il. Itn uhliiiiutliK' in tint cicic nl' 11 siirlMm with Minifiilui illrM, anil 1 liitvii linen Inil lu ruiirliul^ Mini, we Imvc I'liliriu- 1'llUt'Uutl (>",,/, (f, ( y, f? ( //, (',,/", /'', x> fr . 'O. ImL I liiivn luil, ynf, cuiiipli'lt'ly iti'du'iiiiiii'il I hi* rui-flii-ii-nl-i ol' lltn MIMMIV ttniMliun, T 'iil ItirnitiliL in I In- i-imti nf 11 hiirKfi ...... f I In 1 nnl.-i' i* williniii itii X> (; > M* '''/ "' A:'- ( '" W tllt ' 11 lt! (V, //, */', o', -r', ft', y, I' hiivin^ tint vuliit-H in tltt* tl'ivx"i"K Tultlc). ll wiw liy ii^ i'ur # tin t'x|in'nniuii f flin Wti lurm Iml with iuiliMnritiiiitaii iMiriliripru^ niitl tldtonniniiijf (luwi in Hiwh winn Iliut tlu ivriprtHtjil Mi|uiiMn itlxmlt! In- an iil that tint liiri%'(iin^ runnnlu (ur fl' wiw firrivnl lit, fill, 1 IIHNH1UU (;U ~ 7/1 -MA/ (,/J - A'} H- ^r -h linciu 1 funotion (i, j, 0, ^, (7 t /{ t where it if* to bo runmrkocl that, in virtuo of Ihw m|u the coofltaumbi of thu form tiro ronlly nrbitrnry: I i which lad mo to write JDmUQ, 303. 348 A MEMOIR ON THE THEORY OF RECIPROCAL SURF AC EH. 54. Forming the reciprocal equation 0= 2n'(n'-2)(ll'-24) -(?/?' -Hy'-If + linear function (^ j', ff t ^ G", ]]', i t j, 0, x , 0, .//), and substituting herein the values which belong to the surface of tho onlor n singularities, we should have identically = 2 ( - 1)= ( n - 2) (V - 12) [^l - 2) [D tL ( n _ or dividing the whole by ( n -2), this 0= 2- 9 + 2 (3 - 4) o o 'J A MKMtMIt, ON TIIK TIIKOllY (IK ltK< 'N'HlMIA I, Ml) IlKACIW. vix. M 1(! nqimf.imiH am n-iul vcrl.imlly tlmviiwnnk Tin- I'usl,, HIT ..... I, ami Miinl <' 4 iml,inns, " ml ''' ln H1 "" "'' '-'I* 1 I'unHli MIII! liff,h, nil |;i vu tlm M umi! mlal.Hm, Kl*.! -.'I/I /..(); MICTM urn (iuiim'ipn'iil.ly, iunhinivn ul' (Jiui, live iii<lit]Huiiliinl, mltil.iohH. My i-nnibininif (Jin cqimliioiiH HO IIH l.o Minipliiy Ihr nunibn'H, 1 liinl llu-Mc in Im ivl >li^\>(! -m) r(), 7 .-i /M.^A; .|- UA'-I. ;w-. 47*1 o, Jii/l --/M.,M/> |. N/y -- i;.:^ MI 5(5, I fniiiul, us pivHrnMv t.i.'til,inu,'.l, A 111), // iM-'. (! (..!.; vuliir-^ wli (an |,]uiy slmnld tin) MH- m-rntiil i-i{iiiLtinii ; ami MMMI ii^iiniiii^ D IK) :unl wn htivo A':-^. f/ ..'HM, 7/..., a .|, Hi / ,... HIS; HIM) ).).,. InrnniUi | H MiiH'iir Iniu'liun (/, j, D, y, (', H. *",/", /)', y*. r", lit.- piiiiui.HH mil, (iiitiblinif l-hc ili<h>niiiiuvti ..... if the ciii-l1iiMt>nls ui' Mu< lin.-iU' li ill,. ^.ui. m | iht<..r.t m rlmt. il' tin-,-,' in 'it ' 'liWH / which IH ..|.upln MI. /i. /^Mipl- .,n . ;[ 7 4n|.|.* <tu ,.. Ih-h fit., numb, , ,.|* Mm poiul.H ui' inti'i-Ht'nt.iiiii nf tin- tlivcn BiirftMtt-H in 'p - HI {/7/A ') vat- -f - n Apply l.luM tn thu t-a.sn ..!' M Miirliin- nl' lli,- nnl.-r M with a nmlnl riinv ,,f ninl clnsn r/, inirm-Min^ ibt' ,l!c^nii tiiul D.-rnnilui Hiirliin-H, vv Jmvn Onli-r. PiiH-.iiiK tltmiixli (ft, y), li '] H .(, H 2-t 1 1 number of inUM'Ki.TLiuim in ihnt JM 350 A MEMOIR ON THE THEORY OJ? RECIPROCAL SUKPAHI'M. [4 1 1 and the value of /3' is one half of .this, = 2 ( - 2) (1 1,, - 24) - (110 - 272) 4 -I- g. I have not succeeded in applying the like considerations to the cunpidnl oui'vo. 08. As regards the general theorem, know (Salmon, ,,. 274) that if two Hurli,* of tho orders ft partially mtersect in a curve of the ordor m nml ohm r besides , a cm , ve of tho m , der m , ota . . of the order ,; the surfaces vt "1 ' ^ Jmglm tl '''" *., residual eurve rf the ote /:1 ,/" , T ""' " *""'"' '""' '" " points, and thence the Lf JL. JT in fll "' IU1 " ^ '" " ( "" ..... """> M a si,np,e eurve on each of t L /T "tl? M '" 'T '"" Hi """ -iO-r] points, W hence ' " """' ..... " ""^ "' - in a residual curve of the order ^ fl "'M"" "' tll( ' flllm> "' ^ '''. ,,-^) points ;w henee the fee 8 : f ;r!; ^ '"? " 11!Uta t!l " "'" P " to curve being a _ t k ^ ^ "^ moo, m p(la - mafi) , f ^ , ..... *B enrve fl times and in a residua cu of tl ,' '^ C " "" >'' "'' ' ..... "' surfaces meet in mve ' tl10 I' p - 0m, whoimu !;!, |,|, Ivn ^(^-/3m)- a [ m(( , + /3/ ,_ 2/3 , pointoj and equating the two values, we have Lastly, if the curve m be nd in a re of . a of thl , 00 the value of ^' by - (life- 303) A MKMMIH ON TIIM TllKOIt.Y OK ItKOI J'itOOA L HHIt I-'ACKS. IJIi | mill (nliHcrvinjr Unit, l-lir Tallin uf SiiitfiiluriU'H in my Memoir tin <!iil>i<- Surliinw writ iihluini'il wilJioul, (Jin nitl of lln< fitrmulit mm- in iiiuMl.inu) I unili'iivuiir hy iiii'iuis f Mm rcHiiltH Mmivin riniLiLiiiril In litnl Mir viilnt'M id' Mm miluimvM roHHrirnlM ft, if, a; \, /i, n, A', ,</', .'<'', A.', /i', i'. (H). Km- a rnliin mirJiu-c n<--'.\, unil I'nr n nuliin titn-I'm-.n willmul. Mimjulru' linrs (in litnl, liir nil Miti I'uscH cxi-upt, |,h' milm; Nrrolln XXII inn! X X 1 11 ), tlin liinuiilii in Tahli', and rHrnvil |,n hy |,]u< utlixi'il ruiimn nmiihci*, Ilir r,.|,.. A. ;U-. i/-- Hi,/, ft-l.-2A --V, iM am (I) (ID (!!!) (IV) (VJ) (VIII) (IX) (XIII) (XVI) (XVII) (I ^ 5-1. - h - '.'</ > V (I i,". -".| ,;.... ,'|J/', vhii'li uvc nil HiUJHlini if nnly A 2-I-. // h Mil-': :,:;((; // .|- if' ra 1 M. 01. .If WP upply tit Mm name Min-fimcH lint rn^ipi'iu'itl i*i|iiuiinit I'nr /#, in-, what, IM I.|HI MUIHI thing, iipply ihu uriginal rt|inttitin to thi- n-i-ipi-tMinl mii'liir'-s, IIM ^ivcit hy , wi* IKIVH Mm uppMV ami Itiwrr liuU't-M uf I In* Tttliln uf Si r Hnvti'fi uf t'liuiiMuiiN, vi/. thin in HA' + 10i* I- -7 0*.117Kfi~n75(i-2V-X, GUI 7 Ot r iH4 A'-~f/' 0^ afi4- U522-..W-3X. OM. 3024- 8144-2^-101* Om 14.M8- 1385 -2fc'-/ - 518- 504~4A'-6X. 0^ 383- 322- A'2/ 0- 54 -So -%/, (1) - r ', (li) IMS, (1U) !>, (IV) 45. (VI) 12. fVirn 120. (IX) 47, (XUT) 1-K (XVI) 01, (XVH) H (XXI) 352 A MEMOIR ON THE THEORY OP .RECIPROCAL all satisfied if only II hence if - ^ c/' + ty = 88, 9 +</' 18, * =- 1. I remark however that the cubic scroll XXII or XXIII gives = 54 -(330 -272) -2 (M- V), ^' a ' "" IWH tho f -section numerical contration. G3. Combining the two sets of results, we find A = 24, ff =</> (0 =(C "" f iul,,- n(; (,h, h' = 5, and the formula thus IS - 272)6 A MIUI01K (IN TIIK TIIKiWV OP HKdil'lUKJAL M f.<> (Inl.iinmiH! I Iii'iii, I'm- rhc iv/i>^m ! Imt ,',, ,,n ,1 , j ... ' , , ""; llm1 ' l " nl! "I I'linin wii luivii ' . #' ' *V ' " : " I1H '" f| l IN a v.'i;y ivniarhiibln rrliU.tnn, "I wlimh 1 d.i init jii'ivrivr any .? ftn'tiri rcii.sou. I. inn a r>:t () ,,... ' h ;, ; u - wriUoII in lh) , vain.- ..!'//, nn.l >n nil 2 Nr.. tin (u (IS. IttWl i!mi WM lmv ( - l,,l,w,vn V- /'../'I ', A. i', rr. ./', ,,'./; -;'. A'. ,.'. (r ', ", Vl/. Illl'f!!' illV " ; (', "' 11(11 I) 'J/. He, A 1 ' "- tilt (;i - i!) llfi Ni: t (n . (ii - 8 fl) ii) 7 f. fl. C, VI, 45 354 A MEMOIR ON THE THEORY OP BEOIPHOOAL f a' (' - 2) (' - 3) = 2 (S' - C") + 3 (oV - So-' - tf) + 2 ('/,' _ g^' _ /), together with the equations for /? and /9'. 66. The symbols signify as follows; viz. n, order of the surface. a, order of the tangent cone drawn from any point to tho ma'Auai, S, number of nodal edges of the cone. *, number of its cuspidal edges. b, order of nodal curve. k, number of its apparent double points. t , number of its triple points. q> its class. G, order of cuspidal curve. A, number of its apparent double points. r, its class. ere cuspidal oiim is mot number of close-points. f atatiolmiy ou , nnmbe, of i n ts ra eotio lls , statiollaly points number of cnicnodes. class of the sm-faoe. of f 41 ' A MUMCIIH UN Til 1-3 TI1I*WV O|-' HKOI I'ltOOAl, HUltRAOKH. 3 fir, tf', lllltlllllT nf il,H illlli'XMHIN. //, nliiHH (if )im|i'"i'nup]i' j.nrs //, limilliiT (if i(.fi nppiivt'lil. /,', tminlii'i' M|' JIM Uiplr piru f/, I|H unli'i 1 , //, dl'tliT n|' Ilinll'-l'mipli' ('', I'lllHH ti|' H|JtlHH|M fi', nuitilirr nf iti* ii ll. ul. lr |p|anr.,. 0'. imitiltrr ul' M|I'-|I|JMII-I, ami Hpinniln ti.rnrH, MUHinnurv |<'M'nii,ln |,,n-rtO. /.f, iniitiltiT ..I 1 riminion 'l iJlP HjiilMllIi' (pi 7', uuiiiW of rnMininii piitii.-H, siitlimmrv plmu N , ( |" ij..t|<'M'nii|, i". iiiuiiliiT !' t -,i)iiiMfii pliuii^ lint, Nliitiiiimrv plum-M nf .. idler //', imiiili.T iij' ItitrnpfM .(' Mtii'1'in'i'. ('', iniiiiliri' dl' \l* i-nirfrnos. (IK. It. is \ wri \\ y n.-r,K,m-v l .v.Nili Mml lt H|l j ..... | (1 plan,. \ tt H tannin, planr m.'.-hn K Mi,. HUfin.-i. ,u a nnv,- Imvi.,^ m, ||,,, ,, t i Hl . nf , ...... art , u H,,!,,,.,!,, Ml . IM|M|1 . ,,), Iho |,ii,,..|.. nii-v... An,! Hiniliirlv i* u..i!,,.ifnii|,l,, plui..< in u tlniili plum-, nr plum* iiM-nliux ...... irliicn in a nirv,- !mvm K two ..... |,. K; |,|, ( . Hlw lnp nf tl.p ]ilnii,.!i IN thn M.U.-Mnpl,. t( ,, w . ( lm ,| ,], | I(I . MH , (f lh(1 |luj||tH |(( . (1((I||m . ti ..... ni]( ^ 'oupln inirvr; i,lm olh.-r I.-n.iM mud.' .,* tt f ,, n , nil Mxlui.,,,1 j,, th,, ,,..,(; A,..' AlllU'iHtX, Aayttrt M, IH(HI. AH in Uu* (.hiirv ' 1'iirvrn. . in ihut nf .S.iHlu-.^, l.lt.-iv r<- rtti-trtiii I'm Lha tiwUT, 1'lr.KH, &,-. uitil Hin^iilni-iiit'K wliicli hnvw thn wunic vallum in Mm orh Uw ivfipnwnl xiiroM n^piu-tivcly ; for c-uiivoniiMir-n I r.-p^-Hmt. any mu-h Mr nuwiM f ilu. Hynili.,1 v vi& ^ ,,./,,..., ^ v , K , 1|u(|tH ( , |a| . (||(1 nnii . linl| , ((( JH otjiutt l tlm Huind liiiicrii,u (N'. ', &',.) nf tint uuiuMitod li-tttm Hy wh.it WO IltlVb mV. (ll , d j t jr( I|WWH|Vep t ., Mr Ujlll ahy fniu . |it(I|i (jf th(( nnu( ,. ( , nt(l whioh IH -0, or which in .tjtml t Hynmun.ru-nl niuotinn nf miy nf fclu- m-cfi minoutiiitucl Ictiow, ..r lu n funciinn of a. i S ; for itiHUuioo, frutri tlm mpu No. .1 wo Irnvg :i''-HN-i', nml thuno 3 -o*9u'-.#- '. tlmt iXfln-c 45- 356 A MEMOIR ON THE THKOKY OF JlKOiJ.MtOCAL HUUIMOMH. and from one of the equations of No. 11 we have n 2(7 4#-|-/e - o--- li/'-^y;. :n-\')i' <!, we have thus the system of eight equations, V a (n - 2) - K + B - p - 20- b (n - 2) - p - 2/9 - 3 7 - 34 o(n-2)-2ff-4- 7- , v Y , V . V .-. V 2^-2/3 + /3 + St H- j 3r+ c -So-- S-4 or if from these we eliminate , ^ o-, then the system ul' (ivo n<|iui|,i <.nN, V 2)- ing Memoir: "H ,,mi,l,,,. ....... , fl , m a to and thence of l+4W ., _ M) 6 + (12 preceding , cttei ,, W() fll)ta , , and mldin^ lit'ivl.u rh wo linvr or, wluit, iw I)MI name tiling, il' fur -q,...^ . 10,,. ) -|. it . ,, , . o ( ,. Wn ,. i ,s/; aq/ .. , ;IO|V ... as,, -i - -Wj n.s A - iH win, M,. viu,. ni'/-r, NH, )^- V' 1 "- l&;~ia# ..... V? 4) 11:17 4- 1 1*2// I itit link nuuinpl In v.-nlj ihiN ,ti]itiai..ti, l.uf ] will purliully vnrily u n-Hi.lt ilcMluuiblu from it,; vix. if*' i tlm likr ium-lim, ,,r Llm a.rcntu*! k-ttum, tliun wo liuvo <r> *!/ n n' t II (4.^ - 4* 40) i + 8-y 4- 1 2<ty + 2247? + 306* - tf whcr 358 A MIOIB ON and IT is the Hke function of the accented letter, And this being so, TO should h ave - 48 C - 48.3 or, as this may be written, 13on - 48c- 48/9 - 13y + H = S. We have 26ft-lS C +0-i-7j- 8^ + ^-40 and multiplying by -4 and adding, the equation to be verified i 1)to C -8)-18(^ + r ) + n + + 28 But we have from the Memoir s which reduces the equation to n + ii-a^ 1 - 46^ + 110 -300 -648 = 2; or substituting for II its value, this is that is ( ^-^ an equation which is aatiafiecl if and ^7-20, or else 41 a 1 1 f~t i A MKAIulll UN UinW! 'Ynln (I,- /V,,Ym, v ,/ (( V,,/ VY,,i,./,W ,./*,/, /,>,// X nn ;, f]f (J/ - A||Hf/( J'-r I*!!". ,.,,. -II :(!(}. lt,r,ivv,| NHVVM.W IS. inns,: -H,, W 1 . n | ^ ;(j , , y U. lH(i.] .. ,,, ll( , , u Jt n,,.,,^,.,, Hupp.M.H^Iui'y l,u f.lml. y I il.-f.H|- N-liiiiHi. -(> ||, ( , UiMirilHUi,,,, t .fSiirfa.VH ,,r tit,- Thinl t)n!,. r iiil'u Kpi-nii<H i i,-t,-iviii-.. I., iln, ,,1,.,,,.,, ..... 4il ..... u.-.' ..I' Sin KM |,ir P.m,^ ,MH| Mi,, jvalilv i*f Mtnir .111.-, /'A,/. 7V,,.v. v,,l run. (IK (i ; !( , ,,,, |)i;i -J-H. Mm M,,. !,} ..... , (1 f ,,| ln " M , IIlltli| . is lll.Mrl.1. I ,b,|r K H 1( ! Mll,^-! !,, || lt . nlrilliuir (liu^.I) . IMJH-IH lihjj Hi! (| t ,' |Vuli|, V ut' llu- !*'->, Hll.-iitliii^ .ri.ly [. i!<r tlivimMh JitfM lim.iityhvM, .u- UH I j.ivf' f .|- I., ivrlimi il,) -v,-. t iy Iln... t -,.M,-H .IrpM.iir.i? MI. ill., miimv ,,1' i|,, Mi^nihirili.^. And I uU ..... I |<, (,!. n.-^h.Mt v.-iy iitii.'h ..it tir,'..iuir ..I* ll... li K lu . I,,- .,l.luiu.'d in M^VMI-.' t.. Lttn l.lm.ry U.'ri|up-ul SIII|,IO- M . 'IV .M..m..ir ,,.(rnv ( l In InniiNlirH in liirl, a slnro nf ittulorialH I' lN |,lti|.,.n,*. iiii^ii.irl, ,,,H l i;n,, (pllltiully nl" n,|i(|.|,.tl.ly ^t'Vi-|n|ll'll) tll<- t'l|im|.il>U |lmn'.i -...!. liimfr, ,,r I In- H ,-vriul r.M^ nf nil.i.- Mnrlm-ps, nr| whui in tin- niuitt* Iliing r<iiiulitim in ihiiiii.'1'.t.inliimtt'*. u|* i|| t . Ht'VMial MiflhiTs i.nliM : ........ 1 360 A MEMOIR ON CUBIC SUBFACSES. [412 WXZ + WXZ + (X + Z) ( F 3 - fa, 6, c , 7X a - # = 0, P + Y- (X + Z+ W) + WXZ+(a, 6, c, d][Z, F) 3 -0 IFJ^ + (Z + } f 7 - jf ) = 0, ) 6, c, d^X, 7) 3 =0, 0, - 0, Y 3 = 0, =o, IV =12-20,, VI =12 7? f VII =12-3 fl , VIII =12-30',,, IX = 1'? 213 x =12-^-0;, XI =12 -A, XII = 12-17,, XIII = 12-_0 3 -2a 3j XIV =12-B e -(7 a , XV =12-17 XVI =12-4<7 2 , XVII =12-23,-^ XVIII =12 -A -20,, YTY 1 9 n r< .A.J.A. = liJ 2j fl Ujjj XX =12-f7 g , XXI =12-35 8) XXII = 3, 5(1, 1), XXIII = 3,5 (171), 2. Where C' s denotes a conic-node diminishing the class by 2- .# 5 4 J? j9 a biplanar node diminishing (as the case may be) the class by 3/4, 5, or 0; and ,nu i "^P^^' node diminishing (as the case may be) the class by 6, 7, or 8. The affixed explanation, which I shall usually retain in connexion with the Roman number shows therefore in each case what the class is, and also the singularities which cause the reduction: thus XIII=12- 3 -2C 2 indicates that there is a, biplanar node, A dimmishmg the class by 3, and two conic-nodes, 6\, each diminishing the class by 2; and thlls that fche lass is 12 _ 3 _ 2 . 2j =5 _ Ag ^ case / xxn ^ XXII, these are surfaces having a nodal right line, and are consequently scrolls, each of the lass 3 vi, XXII is the scroll 5(1, 1) having a simple directrix right ,i no disinc from the nodal line, and XXIII is the scroll 5(0) having a S imple directrix righ, hne coincient_with the nodal line: see a, to this my "Second tt i Skew Surfaces, otherwise Scrolls," Phil. Trans, vol. CLIV. (1864), pp. 559-577, 3. The nature of the points tf a , tl S tt S t , S 6 , U 6> U r> U 6 r C'(=a 2 ) is a conic-node, where, instead of the tangent quadnc cone. on ^ a proper 412] A MEMOIR ON CUBIC SURFACES. 361 In 7-? 4 , the tangent plane is distinct from each of the biplanes; In B s , the tangent plane coincides with one of the biplanes; we have thus an ordinary biplane, and a torsal biplane : In B u , the tangent plane coinciding with one of the biplanes becomes oscular; we have thus an ordinary biplane, and an oscular biplane. U (= V a , U 7 or UH) is a uniplanar-node, where the qimdric cone becomes a coincident piano-pair ; say, the plane is the uniplane. It is to be observed that there is not in this case any edge. Tho uniplane meets the cubic surface in three lines, or say " rays," passing through tho uniplanar-node, viz. In f/n, the rays are three distinct lines ; In U 7t two of them coincide: In t/o, they all tbreo coincide. 4. To connect these singular points with the theory of the preceding Memoir, it in to bo observed that they are respectively equivalent to a certain number of the cmicnodoH (7(-(7 a ) and binodos B (- B s ), viz. we have G = G B[ = ' B, B, = 2(7, B, = G+ B, (B a = 3(7, (U t = 3(7, U 7 = 2(7+ B, U s = C + 27J. 5 I tako tho opportunity of remarking that although the expressions cnicnode and binodo properly refer to the simple singularities G and B, yet as 0,= 0, 0, is properly Hpokon of as a onionoOu, and we may (using the term binode as an abbreviation for biplanar-nodo) apeak of any of the singularities B 3 , B tt BB< as a binode. Thus the auriaco X = 12 -#,-(7, has a binodo 3< and a cnicnode a a ; although theoretically the binodo ., in equivalent to two cnicnodes, and the surface belongs to those with three cnicnod^ or for which (7 = 3, I use also tho expression unode for shortness, instead of uniplanar-node, to denote any of the singularities U, t U 7 , U 6 . 6 Tho foregoing equations (substantially the same as Schlafli's) are Canonical forms; tho reduction of the equation of any case of surface to the above form is not a way obvious. It would appear that each equation is from its simplify in the foi n be t adapted to the separate discussion of the surface to which it belongs; there is the Si^Lg. that Aquations do.t *^(^^*^ <^ the Hurfacea they ought to do so) lead the one to * . F2 _ ftZa _^ )==0 ^jL-i-w^rsrTKS.S'.^iK.: -^ form a tueory or tne qutii- the geometrical theory of the surfaces to means i"j n c*iu, -" _ . . they respectively belong, and the imperfection is not material ^ 0. VI. 362 A MEMOIR ON CUBIC SURFACES. [412 7. I have used the capital letters (X, Y t Z, W) in place of Sohltifli's (m t y, z t w) t reserving these in place of his (p, q, r, s) for plane-coordinates of the cubic surfaces, or (what is the same thing) point-coordinates of the reciprocal surfaces; but I havo in several cases interchanged the coordinates (X, 7, Z, W) so that thoy do not in this order ^ correspond to Schlafli's (at, y, z, w) : this lias been done so as to obtain a greater uniformity in the representation of the surfaces. To explain this, let A, ./?, 0, D be the vertices of the tetrahedron formed by the coordinate planes A = YZW, B=%WX. G=WXY, D = XYZ\ the coordinate planes have been chosen so that dotcrmmato vertices of the tetrahedron shall correspond to determinate singularities of the surface. 8. Consider first the surfaces which have no nodes .B or U. It is clear that; the nodes O a might have been taken at any vertices whatever of the tetrahedron they_are taken thus; there is always a node G, at D; when there is a second node 0,' this is at 0, the third one is at A, and the fourth at B. *' 9. Consider next the surfaces which have a binode , #, or # (j . t h is ia taken to be at D, and the biplanes to be X=0, <)(') (the edge being theruforu JJB) viz. m 5. or B ti where the distinction arises, Z = is the ordinary biplane ^-O the torsal or (as the case may be) oscular biplane. If there is a second nodo' th ls of necessity lies in an ordinary biplane; it may be and is takon to bo in tho biplane A =0, at C, I suppose for a moment that this i s a node G.. It is only when the hnode is B 3 or S t that there can be a third node, for it is only in those rihnfnode 61 '! r * "T" ^ ^ ^ ^ in ^ < the thud node, a 6' 3 , may be and is taken to be in the biplane Z=Q, at A 6ach is a ' ; rr. ^ "- - O to e _ There . f oot oT f (ex ? fc r the X=0 being the ordinary binlane 7-n tl , 7-", ' (f 1 ' a bmod8 ^ or S " ^ X=0, Fo, those oth W blod /T T f ^ 8e OIld Mm)de XII = 12-ff.) the iraiplane is T-n TP , ' and that (exoe P t i as represented b y to^^z ?%/?$* ^ ( /VH. 12 toing X = ^0, a ^ at , 2,f ^ + ^V-0 F-ol/ f & r common biplane), and a C7 a at X. (therefore ^ = the 3 In the case, however, of asineleJ). JTT-iQ n n. u- , tfe^in-ia-*,, thobipla^ are talcen to 412] A MEMOIR ON CUBIC SURFACES. 363 13. Ifc will be convenient (anticipating the results of the investigations contained in the present Memoir) to give at once the following Table of Singularities; the several symbols have of course the significations explained in the former Memoir. w a' 5' <J' <j' 8" o ; ' <JT <j' 1 ii i . . M 1 1 i, i " n ^ ef o ^ e? ft? co T ef b u 4 B? ft? tT ^ ft? f s & 3 ""* rH I 1 III! 1 i i i r-T CM 01 CM CM CM <M CM CM <M CM (M CQ H II r-l U 1 11 II II II 11 II II II II H II II I II II II II t-l II a u b , S Ei > H E H ^! M R M H H fc> p > t> M M M tl M H M t*i M H M M i M / ,, n H !) 8 8 3 3 3 8 3 n (1 l\ G 11 li (i G G C (i 4 a 1 2 1 8 2 4 1 u f) t) 7 fl 7 (J 8 7 fi 8 y 3 1 b k t a u 2 3 f. h r a- & X. I) y i 1 1 2 1 1 3 2 2 1 4 1 2 8 B I II III IV V VI VII VIII X XI XI IX XIII XIV XV XVI XVIII XIX XVII XX XXI XXII XXIII r 10 (I 9 u li y H il y 7 6 n fi fi 9 6 y 6 C 9 c 9 4 6 9 3 6 9 8 4 3 n' a' 8' ' 2 ' 21 ', d ', ' 2 1C (10 IB IB y 18 u y 7 12 b 7 1 i) 8 t) 9 1 3 3 1 1 1 8 1 3' G 1 k' . . , . 2 c' ' 2 ' 18 ' 8 18 90 24 10 72 42 12 88 17 10 24 24 G G y 12 32 2 5 o 2 2 h 1 ' 1 * 12 12 10 10 9 8 1 fi 16 2 2 tf K' ' fi 8( 18 18 8 1 P, y i' ' ' 1 ' 1 . , 1 3 B 364 A MEMOIR ON CUBIC SURFACES. [412 Article Nos. 14 to 19. Explanation in regard to the Determination of the Number of certain Singularities. 14. In the several cases I to XXI, we 'have a cubic surface (n = 3), with singular points G and but without singular Hues. The section by an arbitrary plane is thus a curve, order ?i = 3, that is, a cubic curve, without nodes or cusps, and therefore of the class tt' = G, having S'=0 double tangents and K '=Q inflexions. The tangent cone with an arbitrary point as vertex is a cone of the order a =6, having in the caso 1 = 12, S = nodal lines and *6 cuspidal lines, but with (in the several other cases) G nodal lines and B cuspidal lines (or rather singular lines tantamount to G doublo lines and B cuspidal lines): the class of the cone, or order of the reciprocal surface is thus '=6.5-2(0-!-<7}-3(G + j5)=12-2-3a. 15. In the general case 1=12, there are on the cubic surface 27 lines, lying by 3's in 45 planes; these 27 lines constitute the node-couplo curve of the order />' = 27, and the node-couple torse consists of the pencils of planes through these linos respectively being thus of the class ?' = &' = 27; the 45 planes are triplo tangent planes of the node-couple torse, which has thus *' = 45 triple tangent pianos. But in the other cases it is only certain of the 27 lines, say the "facultativo lines" (as will be explained), which constitute the node-couple curve of the order p' the pencils of planes through these lines constitute the node-couple torso of the class &' = // the t' planes, each containing three facultative lines, are the triple tangent pianos 'of the node-couple torse. Or if (as is somewhat more convenient) we refer the numbers r, ? Vi e ^i SUrfMfll then the lines ' *al8 of the fucultativo lines, constrtute the noda curve of the order V; and the points *, each containing three of these lines are the triple pomts of the nodal curve, Inasmuch as tho nodal curve foZa W me V he T mber *. of its apparont doilble >^ * fcy * fonnula 2fc =6-J -e^; and comparing with the formula a> = b'*-b'- 2//- V- fit' we have g + 3 7 ' = 0, that is, ? '= (,' the class of the nodal curve), and also y.Q. the OTbb > pn e CU1 ' Ve S tG C0m ' 3lot "^^ootion of the cubic smface by the Hessmn surface of the order 4, and it is thus of the order ~ ; , ther CaS6S the m P lete mterseotion connate of the , reoon connate o the ogether w:th certain right lines not belonging to the curve, and tho wiu be curvo are common tangent planes of the spinode tors and n planes of the spincde torse; or we have% - L - T ^^ """"^ instead of the 27 lines we must lakemilv the\~ t V' r * GS ' howew ' is not a double or a single ^^ ^^M^^ ^ f hioh J9 of the surface at the points of contact are th, V ^ ** ta ^ nt P Unea that is, the number of c ntacts give 7 O t ^ gte~ 412] A MEMOIR ON CUBIC SURFACES. 365 18. There are not, except as above, any common tangent planes of the two torses, that is, not only y'~Q as already mentioned, but also i' = Q. I do not at present account d priori for the values ff = lQ, S, and 16, which present themselves in the Table. The cubic surface cannot have a plane of conic contact, and we have thus in every case G' = ; but the value of B' is not in every case =0. 19. In what precedes we see how a discussion of the equation of the cubic surface should in the several cases respectively lead to the values b',t' t p', a-', I3',j' } %, B', and how in the reciprocal surface the nodal curve of the order b' is known by means of the facultative lines of the original cubic surface. The cuspidal curve <?' might alno bo obtained as the reciprocal of the spinode-torse ; but this would in general be a laborious process, and it is the less necessary, inasmuch as the equation of the reciprocal Hurface is in each ease obtained in a form putting in evidence the cuspidal curve. Article Nos, 20 to 28. The Lines and Planes of a Gubio Surface; Facultative Lines; Explanation of Ditif/mms. 20. In liho general surface I = 12, we have 27 lines and 45 triple- tangent planes, or say simply, planes: through each line pass 5 planes, in each plane lie 3 lines. For the mirftiooH II to XXI (the present considerations do not of course apply to the Scrolls) Hovoral of the linos come to coincide with each other, and several of the pianos also come to coincide with each other; but the number of the lines is always reckoned HB 27, and that of the planes as 45. If we attend to the distinct lines and Urn distinct pianos, each line has a multiplicity, and the sum of these is =27; and HO oach plane has a multiplicity, and the sum of these is = 45. Again, attending to a particular line in a particular plane, the line has a frequency 1, 2, or 3, that is, it represents 1, 2, or 3 of the 3 lines in the plane (this is in fact the distinction of a scrolar, toiml, or oscular lino); and similarly, the plane has a frequency 1, 2, 3, 4, or 5, according to the number which it represents of the 5 planes through the line. It requires only a little consideration to perceive that the multiplicity of the plane into its frequency in rogard to the line is equal to the multiplicity of the line into its frequency in regard to the plane. Observe, further, that if M be the multiplicity of tho plane, then, considering it in regard to the lines contained therein, we get the products (Jlf, M, M.}, (2Jf, M), or 3j, according as the three lines are or are not distinct, but that tho sum of the products is always = 2M, and that in regard to all the pianos the total sum is 3x45, =135. And so if Jf be the multiplicity of the lino, then, considering it in regard to the planes which pass through it, we get the products (M\ M', M', M', M'), (2Jlf, M', M', M'), ...(6Jf'), as the case may be but that the Hum of the products is fiJf , and that in regard to all the lines the sum is 5 x 27, = 135, as before. 21 Tho mode of coincidence of the lines and planes, and the several distinct linea and planes which are situate in or pass through the several distinct planes and lines respectively, are shown in the annexed diagrams I to XXIC): the multrphcrty i See the commencements of the several sections. 36G A MEMOIR ON CUBIC SURFACES. [412 of each line appears by the upper marginal line, and that of each piano by the left-hand marginal column (thus in diagram I, 27 x 1 = 27 and 45 x 1 = 45, 1 is the multiplicity of each line, and it is also the multiplicity of each plane); the frequencies of a line and plane in regard to each other appear by the dots in the square opposite to the line and plane in question, these being read, for the frequency of the line vertically, and for the frequency of the plane horizontally ; thus " ! indicates that the frequency of the line is =3, and the frequency of the plane is = 2. There should be and are in every line of the diagram 3 dots, and in every column of tho diagram o dots (a symbol ' '. being read as just explained, 2 dote in. the lino, 3 dots in the column). 22. For the surface 1=12, there is of course no distinction between tho linos, but these form only a single class, and the like for the pianos; but for tho other surfaces the lines and planes form separate classes, as shown in the diagrams by tho lower marginal explanation of the lines, and the right-hand marginal explanation of the planes, I use here and elsewhere "ray" to denote a line passing through a single node; "axis" to denote a line joining two nodes; "edge" (as above) to donofco the edge of a binode; any other line is a "mere line." An axis is always torsal or oscular; when it is torsal, the plane touching along the axis contains a third lino which is the " transversal" of such axis; but a transversal may be a mere lino, a ray, or an axis; in the case XVI = 12-4a 2 , each transversal is a transversal in regard to two axes. 23. In the general case 1 = 12, each of the 27 lines is, as already mentioned, part of the node-couple curve; and the node-couple curve is made up of tho 27 linen and is thus a curve of the order 27. In fact each plane through a lino moots tho cubic surface in this line, and in a conic; the line and conic meet in two points and the plane (that is in any plane) through the line is thus a double tangent plane touching the surface at the two points in question; the locus of tho points of contact, that is the line itself, is thus part of the node-couple curve. But in the other cases II to XXI, certain of the lines do not belong to the iiodo-oouplo curve (this will be examined in detail in the several cases respectively); but I wish to show here how in a general way a line passing through a node, say a nodal ray, fe not part of the node-couple curve. To fix the ideas, consider the surface II =12-0, there are here through G 2 six lines, or say rays: attending to any one of these, a plane through the ray meets the surface in the ray itself and in a conic; the ray and the come meet as before in two points, one of them being tho point C 3 : the 7 a 7 t6 tei> P int ' ** * d GS not M * face at 0.. node h Pen qUeSti n ' Wh6ther Ol ^ ht *> *t at a n d 1 rh C ' *>* "ra-7 through C a is not a tangent plane.) The plane through the ray is only a r pl r ; and the * * ^ Ai ^ - e" acoordin, 412 ] A MEMOIR ON CUBIC SURFACES. Article Nos. 24 to 26. Axis ; the different lands thereof. 24 A line joining two nodes is an axis; such a line is always a line, and it is a torsal or ocular line, of the surface. But _Bome furtherdistinctions are ^qu.sito using the expressions in their strict sense, cmcnode = 0, bmode = * an ax, < a CO-axis joining two cnicnodes, or it is a OB-axis jommg a omcnocle and a bmode or it is a%B axis joining two bmode.. A Off-axis is tonal, the transversal bomg a 14 L not a 4 through either of the encodes; a CB-axis is torsa^ the EanS b-g a ray of tL binode; a BB- axis is oscular The *^ course carried through as regards the higher biplanar nodes ft, ft, ft, ^ ^ uniplanar nodes 17,. 17,, 17.: t!n,s (ft -B) the edge of a bmode B not " s ^ nil but (ft = 20) the edge of a bmode ft is a OC-axis ; (B ? = .B+0) the edge ol a all, ov>< OH axis- (A = 30) the edge of a binode ft is a thriee-taken CO-ax I3 ; S?^. r of ^ ra^l regldod a 3 S a 0,-axis ; (J7.-H + .J, the ^b.e ray regarded as a twice-taken CT-axis, and the single ray as a CC-axm; (U, = ^ + G) the ray is regarded as a BB-axig + a twice-taken CB-axis, 25 It has been mentioned that the intersection of the surface with the tfarian consists of the spinecle curve, together with certain right linos; these hues are m act tlTaxes-vis. the examination of the several cases shows that m the complete ^1 each 00-axis presents itself twice, each OB-ia 8 fees, and each ^ax, ftoes. We thus sec that a 00-axis, or rather the tonnl plane along such ax.s, is he pLh-plane or singularity / = 1 ; the OB-axis, or rather the torsal pane a ong such a s,7e close-plane of singularity *-l; and the .BB-axis, or oscular^plane along s, eh axis the bitrope or singularity B" - 1 ; for a cubic surface TO tl. Bmgular hues Ihe cxlsson *> being ta fact \> = 12 - %"- * - '. There are however aome casen eS explanation f thus fer the case VIII = 12 -ft, where the edge IB by what Zedes a CT-axis, the complete intersection is made up of the edge 4 time and of !n cctic curve; the consideration of the reciprocal surface shows, however, tha the Sge aken once is really part of the spinode curve (vi, that tin. curve is modo up of the edge taken once and of the oetic curve, lts order being thus , =B) and he interpretation then of course is that the intersection u. made up of the edge taken 3 times (as for a OB-axis it ehould be) and of the spmode curve. 368 A MEMOIR ON CUBIC SURFACES. [412 Article Nos. 27 to 32, On the Dete-mination of the Reciprocal Equation. 27. Consider in general the cubic surface (*$X, Y, 2, W) S =Q, and in connexion therewith the equation Xx + Yy + Zz + If w = 0, which regarding therein A r , F, #, TT as current coordinates, and as, y, z, w as constants, is the equation of a piano. If from the two equations we eliminate one of the coordinates, for instance W, wo obtain ($Zw, Yw, Zto, -(X(s+Yy + Zz))* = Q t which, (X, Y, Z) being current coordinates, is obviously the equation of the oono, vortex (X = Q, F=0, #=0), which stands on the section of the cubic surface by the piano. Equating to zero the discriminant of this function in regard to (X, Y, Z) t we express that the cone has a nodal line 5 that is, that the section has a node, or, 'what in the same thing, that the plane atf + y7+.e# + wTF=:0 is a tangent plane of tho cubic surface; and we thus by the process in fact obtain the equation of tho cubic mirm.ce in the reciprocal or plane coordinates (, y, z, w ). Consider in the same aquation *, y> z, w as current coordinates, (X t Y, Z) as given parameters, the equation represents a system of three planes, viz. these are the planes aX + yY+*Z + wW' = whoro W has the three values given by the equation (^X, Y, Z, F') 3 = 0, ol . what / H tho samo thing, X, l,Z,W are the coordinates of any one of the three points of interaction of the cubic surface by the line | = f = |; (Z. F, *, W) belongs to a point on tho surface, and is the polar plane of this point in regard to a quadric surface Z +?> + #+ r-0- ' the equation (*jZw, 7w, Zw t is thus the equation of a syatem of 8 planes, the polar planes of three- point* of tho cubic surface (which three points lie on an arbitrary line through tho J point i-0 = = eqUatlUg t0 Z61 ' the disci ' imin ^ in regard to (X, Y t g)\ Q find tho - ^ ..L.tv.iu j.,, my m 1.1 iiu I A . t io I \ envelope f the system of three plane,, m say O f a plane, the po ,,r pi arbitrary point on the cubic surface,-or wo have the equation of the Z 6 ;- mg ,' " ", Wni the Snme tWng " S the D[ l uatio " of Wo wie reciprocal or plane coordinates (x it ?/A T 1 412] A MEMOIR ON CUBIC SURFACES. 369 29. But let the parameters, any 0, </>, be regarded as varying successively; if $ alone vary, wo have on the surface a curve , the equation whereof contains the parameter 8, and when varies this curve sweeps over the surface. The envelope in regard to c/> of the polar plane of a point of the surface is a torse, the reciprocal of the curve , and the envelope of the torse is the reciprocal surface. In particular the curve may "be the plane section by any plane through a fixed line, say, by the plane P 0Q = 0; the section is a cubic curve, the reciprocal is a sextic cone having its vertex in a fixed line (the reciprocal of the line P 0, Q = 0), and the reciprocal surface is thus obtained as the envelope of this cone; assuming that the equation of bho scxtic cone has been obtained, this is an equation of a certain order in the parameter 6 ; or writing 8 = P : Q, we obtain the equation of the reciprocal surface by equating to zero the discriminant of a Unary function of (P, Q). 80. With a variation, this process is a convenient one for obtaining the reciprocal of a cubic surface; we take the fixed line to be one of the lines on the cubic surface; the curve is then a conic, its reciprocal is a quadricone, and the envelope of this qimdricone is the required reciprocal surface, This is really what Schlafli does (but the process is not explained) in the several instances in which he obtains the equation of the reciprocal surface by means of a binary function. I remark that it would bo very instructive, for each case of surface, to take the variable plane Hucoonaivoly through the several kinds of lines on the particular surface; the equation of the reciprocal surface would thus be obtained under different forms, putting in evidence the relation to the reciprocal surface of the fixed line made use of. But thin is an investigation which I do not enter upon: I adopt in each case Schlafli's process, without explanation, and merely write down the ternary or (as the case may be) binary function by means of which the equation of the reciprocal surface is obtained. H('l't)H n H2. Tim clwrnimmuil, ul' a |itmti<!, whit'h r n /,-, ,. n|Miil,inn li,.in w iJ u, WO Hhlll H of M y ur a liinuiy !iinrli.,in'). vi.-, tlii<. i-> < ., iv,-n in ii )r . ).-, ' ) yH-iM,V. utnl wlii.-li i, I!UM,,U ,-nf -,, jv.lnn.il h. llu- pr..,,,!- m.l,-.-. Th-,, 1: fl t .- in nvid.^ii.-n (in- ni-.pi.hl mn,- .,) r),.- ),;,,, (In* I..HM i.lti.mi.-.| ,, ,\'' /' n ' |y (N II. 7' .11,, ,.,, ,, ,1... ,, !M ,, V j,. ,/ / ' 'I tin- |,v ..... (,,, .., (IM I)t .|....., l4 ,,.. 4 l,!, ,. HH ,nrvu m n ,:,(, ,,,,,,, , ,1,,. ,-,,,,!,,,, ,,,...., ,;..,, i (ll , (l . Ml(i , t| '" v,,,,| - ..... - ' -in, !!,,,, f|l ,,< .1... ,,!,| 1 ' l | "I* u iJn'iiwti mil, liii'tur. AH , notation ,,,H| l In Homo onv MUM, for v ft fllroot oll - mi,.. W 1 ,, XXIII ,' r.. I- - v """ "' '-' A MKMOIU ON UIMIK! f 371 "I" lln i cipiiilinii M|' tin- nripiwiil mi Hunt iJirnn miiilylioul uxpniHwiim liuvn no -Hi'iil. !ip|ili'nli"ii, Ami mi in HHUIM ul' |,lm ut>xt liillnwin^ minimus, no applinil'iun IH ' ' liWili' nl ill' 1 utiillyl irill i'\p)'rs>iinll!i ol' l.hn lilh'hl lUlll Ilk 1 I'ull I" wiutl llml il' a HUM lu< j;iwn u-i Uin inUii'mMitiwi nl' Uin two plains Uii'ii I In' i'K I'nni'iliimli'.'i nl' I he lint- im i, /i, r, f, if. It AD' AD, Itlt' li'D t CD' -H'D, IiC' fl'C, VA'-U'A, All'" A' It, ami llml. in l-'iim "I it-> n\\ cnimljtmli'i Uu< linn is i{i\c\\ JIM (Jut cuiniiion inrnr.si-ol.ioti M|' |.)ii' luiir phtiir:i ( . A. //. H O.Y, i'. /, ii'i^o, 'A. , f, / .*' ' / (I ',1. I, -!. . iiixl I lnil Un-i|'rM.'!iliiif; mi iisital in n'i;.ml In A' J I i' 1J I ^f ! IP - tl) (ho rnniiliiiiitrM of llti* H 4 rijir..fu| liii*- HI.- ( /', /, A. ii, ii, r} ; lliut- IN, ihi'i in |,hu rnininmi inliU'Hiuiliinn nl' t-lu 1 I '', ! /'. f O' 1 '. ,'/. -. '") ' '0. 1 (', , H, *I i ! * I A, - (i, . h s -/. //. A, .1 Jt, IN lit M'tint 1 rii^'^i ni>'i'< niiiV''iiifiit, in funkier 11 linn an ilcli-niiinril us Um iiiLui*- NMI'tinn nl |.,WM plilll- *f Kllll'T I llUll ll.V HH-Ull-l M|' its HIX I'tml'lHuall'S j l.llllH, I'nl' illHllllKri', In h|H'iik -( I lif lin.' .Y , )*' niilirr tliuu f iln i liim (0,0,0. 1,0,0); iitnl in wiiiit* nl* lit*' Mvlinn-. 1 liuvr pi'-r'-uvit hfil. In jj-ivn tht i'Xpivw*iniiH nl' Uin nix (iimnlini 1 ' 1 "- n|' tin* Ni-Vrliil llKt'w, A MI'IMllIII. ON rlMIH! it in HO (iomplimLuil dlml> il. can Ininily IIP onwiili'ivil in at- nil put I in;; in .'\i<l,-n,v Mio 1'iitaMimn of Mm MUCH ami plain's; Mml. "(' Dr Hurl (Sulnimi, " ' hi Mi Triplr Tniitfont Plant)* of a Kiu'l'm'n n!' Mm Tliinl Onli-r." MI ..... v.-limi,-, pp, ;!;,:! \\i\o) t ilnimndiiig on an unuiigiiiM'iil- of Uio l!7 Ihnvi iii'rni<liiii| I" u i-ulu' { :i rni'ti \vay, i;( a Hinuliu'l nh^iinl. IIIIK, anil will In* jn-i'in-nMy I'l'iirmiin'ftl, .'Hi. JUil; Mm nui9l. (innvunii'iil- HIM- is Si'hliilli's, (iiHrlinif In-nt u .Innl'Ir -.is,, r ; vi.-,. wo can (mid lliaf. in fid dilVmviil. wnyn) [ii'lcnl. nut n(' the ;!V lin- i iiv,. '.v.i'iirf ,-w\\ of nix liii(*n, Hiitih Mmt no l-wo linrn nl' Mm tiiinm Mynl'-iu iiih-i !'(, l-nl ih.it .--t.-li Hn,. dl' tlm ouu Hynliioii infiriw^ln all linl. dim HHTi;)iHinliuff linr ul" lit" .-lli'i '.y^t. m ; nr, Hay, if UKJ liniiH urn I , li , It , i . .'i , II r, *.!', ;i', -i', .v, i;\ Mmn MiiiHii ltav<! tlm Miirl.y iiidi'i.-iriiriMnn Any two HIIIIH Himh IIH 1, il' !!' in u |ilriiif vvhirli limy li< - ;i !!, d 1^', linrH I', U lin in a plaiii- wliirli nniy I'" rullr.l |'^ ; ih,-. lt . f WM piu Him 12; nnil any Miivn lim-M mn^li \\a III, III, ,'iii m ! tit |..i)ii>, Ki l2,U'l',5(i, \Vn liuvo UiiiH l-litt ttiiiifit .'lydft'iu "|" il^ %'} in,. > ..i.-l i ollUc.ti (ioniplobily oxpluiimil liy wlial. IIIIM ln'.-n t.Ulnl, tntl wluuli i .^liil tho iliagmin. , f ,iinil,iilv I In* n.'i KI*- ( m =t i.i,; in n \i\w , [.l^tf^, * in .i^'l in !<!! ut ;!7. Tlm. (liagmni oC Mm lines ami plum 412] A MEMOItt ON OUBTO SURFACES. 373 Linos. CJi hi", ill. C OJ OS tO W M K> M M M M M OV CS t)l CTJ Ol hi* O til hi- CW Q3 CJi K- W IO O> til if. M K) H O) til rfx W U) M 1OI li ~ll -1 Il ! X ~l --I Ifl' . 10 Hi' . r 14' . z ]fi' . y. 1(1' . y, 81' ' e 2H F . 84' . f ar' . Pi an' . V HI/ . na* . i) 114' . B fi' . 1 nil' . 'Ji -U' . r w . y -ill' . h , llfi' > < * m 1 'J' 1 ' . m K fit' . > > 1' r.a 1 . y r.:i' -IB x 1 ! i . i ; 04 ' . i fill' . i ill' . . PI IJM' . y III)' . r- (14' . ' n 06' . U 18.84.00 I i > il! 18.8B.40 . 12.00.46 . . 18 . 84 . 60 . 18.85,40 . 1' lit. 80. 46 . ' l7 l 14 , 28 , 60 * * K 14 . 26 . 80 i ' h 14 . 2(1 . Hfi > * i 16 . 28 . -Ill 1 ' <ii 16, 84. BO . riii 1.5. 20. 8-1 i > . TI 10.28.46 * * <i 10.8-i.86 * fii 10 . 86 . 84 46 46 . .L # jf t s o? * rjrrr** * * fi * ? ? Jf ^jLfjflf . JL. M74 A MKMimt, ON (IUIIK1 HlMtKACI-IM. !)H. It IIIIH ImiMi mniitiininl t<hul< Mm imnilirr nf i|onl'l<"M\<'ni \v't Mil, us Pollown: I , i!, ;t , 4, fi , li A:tmiui'il |Mimiti\'< I r *>' ;i' -I,' r/ ii' I i f r f i i i' i " ( f ' I, I', 2!l, 24, 2fi , 2ti liilti' iiiiaiii;''mi'iii i 1 ,'t l >. '' l:l M. 15 Hi *J ) M f I *' I * " | * ' * I ' " 1 , 2, II, .1(1, 411, 'l-.l l.iUf iiwm.n.'iuriii- 2:1. i;t, it:, 4, n, n wlmro, if wi! I tilt u any itnliiiini j^ ut' IAVH lintvc, \\<> huvr llit< r.nti|il<>t. pairH nf niiii-iut.ri'Hwtmj,; linn:i (rnrli linn nii ( nl-i 10 liiirn, ilu-ir tM<- fh.'h't'.ti" ; l ,V I III. ; " 1(1, whinll id ilui^ iinl, iiii'i<l, it nil I he iiiiiiiln<i' \' HUM iui>-iur<:f iuu j,,,n , i , i | 1( | fp i. 27, Hi r. i 2 Id). !l!l. Wn rim mil. nf llii> -tn jiliim^ Hpti'.'l, uitil llnU in UNI w,i\n. ,i (nli-Hi.il p.tti 1 , thivt IN, l,wo [I'lttdd u!' |tluin-n t Midi lliui flie |il!ini"( ..I din .in-- hiit'i, iisr.'i 'i".-riitjj t)tt-i" uf Mm olluu- l-riuil, KIVK !l "'' Mi*' V limvi. Aniil.vlii'iillv il ,V II. (' H, / -ti fiml //s.^0, I'V' 0, IT; (1 HIM tin* iN|iiuli'tii'i nf tlh' ;>i\ jilaii'"', tlicii ihc i ^inuinii ! llir liii'i! \H XY% \-klIV\V -il SCM in (M l)ii->t i^.f. ,\',., H, Tlu> t-rilii'dnil plrmn jmii-n mv : 12', 2H', :U' N... i ,:iri,2i!. 2:. Ki.ru, 1 2 M x 1 5 (i .'I I 2 5 I! 4 2 II I li ^ r, li 14 23 JHi Mf HI ;H H IB immaterial how tltu Lwo compiiiuitt tritub ISJI mitl 4-^ii an Arranif^l, wi bmin fttways thu snmo Irihwlral jitiir. A MMMdlll ON (UMIK! HlIJIKACIKS. 375 Hart, ai ^7 lini'n, rnliiriilly, l. lh-iv l.'i.ii'i'.'i il' ill.- :mmi> iilpliuli.'t. ili'iinl.i' linns in lilin Kiutiti piano, if only bho lottos V ih<' mum* "i lit.' !itilli\i"i Hut uiiiiii'i t-'tiw yl,, A. it A a \'w in u piano ^d^i^a; .. /v ''i I'*' " ' l I 1 '" 1 "' -li'W 1.1'll.i-rn !' ilillVnnil, nlpliabnts tlonotn HnoH whioh moot TMifliiiu l" th' Tiil'l' 1 V.. i V, fl, ttl(l , lv ,,, ,..., iu ,| t ,, ,,.),,, ,.!' Mi.' H,|ii,m. .Ir.mh-H a linn lying in bh KIHUO uiMt It..* In,. .I.'iil".l i'.V Mir 1HI.-.H ..I* rurh v.-r(.i(Mil pail- in th<( Hiuno H.pm.u UUIH ,t lirri in Mi iilintr .1,'"^,. *'liM.-. ^lt'*.7u ('i" 1 ' > l ' lln lH!l'm-nii!iiUiH!(l two iihvnuH -H, / liu.'l liiui MII,' wy i> vvhii-1. this may hn id.mtiliml with liho doublo-sixur i*( l4i ivjn.^MiiI, Mir ulu'V.' urnt!iK''i"'-nt. by I. X, 13 li 1*1*1 Mli t r i "* - V. fi . 'I' 1 " 1 n, 24, fit r, ui, o , nfltitjttid in (720xH(l) 25020 ways, viss. a -i fi '' lii rh VIK ukittg !JJ -J.J .! .-to/- . ,.,,Un, -r II,, 4,1 pluu,, uru ..bUM^l i.> -y Wr Int reforrocl to, A M mom UN riniH MI ni ,\t i /A' .) iiiY+n'/, I II' A'-O, HI \ ir ti, 'i. n'/, | H" , |. ,,#.,, jr. 7* f' / ;(/"") -I- ir.. 412] A MEMOIR ON CUBIC SURFACES. 377 X -I- - 1 r + nZ + W = 0, [56' = I] .- 7 - m (p - a) - ' [64'-n] Y + mY + + F = 0, [15 - 26 . 34 = a) nZ + F = 0, [16 - 24 . 35 = m] IX + - 1 - K - - -7 2 -- ..... x ^ + TV 0, [14 . 25 . 36 - n] u ^ ?/i m (^J - a) Z (?) - a) 7 1 z + F = [46' = inj ' _0 [16.25,34 = 1!] V/) _A [15. 24. 36 = mi] U, u r , 1 y ? (^ - ) 7 + IT t 14 ' 26 ' 35 =Sl ^ L -p "" .* ' n.., ' W " =0, [51' -p] = 0, [35' =q] IX -r - * - gm -a n ^ ,,_,_ 2 + ^+(p 0. VT. 378 A MEMOIR ON CUBIC 8UEFAOES. 'If, 1 l\. , 2\ W [I5.28.40-q,] , [or = p,j The coordinates of the 27 lines are then found to bo UK follows: 7 TO - - * V m/ \ n 0) 1 o w / i\/ i m-- [ -_ W \ n =("-i)('-J) *'- ~ T M "I - 1 n m) -4A, ?v nk \ m j \ "^("~")( n i) m 412] A MEMOIR ON CUBIC SURFACES. 379 (/} (?) (A) (12=0 (S'-ft,) ( 1 ="i) - m ( 2 =,) n -Z (23 =* Jj) - in ' < 8 '-*> 1 1 ( 1 ~ a 3 ) n m 1 1 ( 3 = 6 a ) n ^ I 1 (13 = a,) m 1 1 m (34 =04) n n 1 ; (24 = 64) 1 I (H = c 4 ) m 48 380 A MEMOIR ON CUBIC SURFACES. [412 412] A MEMOIR ON CUBIC SURFACES. 381 (/) ((/) (h) -n 1 m (56=a 6 ) 1 n - (*-*,) -m 1 (*'=.) o .-!* (35 = a fl ) (25 - 6.) m (p - a.) ? -(P + W n (p a.) (15 = OB) 2(p + j8) ?) + j8 (16 - a,) 2 ( p + ^3) w j (y) a) (6=6,) rtl / 1) 1 rt J \/ / ,-^ (5'^ 7 ) 3J(p-fl f B} (36 = a.) ^J a -(>.- p a (26 = 6 B ) 2*1 (7? /}) ^j a -(,- (16 = c s ) 2*(/>-0) (45 -a) 2m (jfj /?) p-a 6~&^ _fj a, 2?i (^j )S) jj a ( 6' = Cn 382 A MEMOIR ON CUBIC SURFACES. 44. We have ,Y = 0, Y=0, Z=Q, F=0 for the equations of the planes (12.34.56 = 4 (42' = ?,), (14' = z\ (12' w); and representing by f=^+- Y+-Z+ W=0 the equation of any other piano 7/t ?i v I the equation of the cubic surface may be presented in the several forms : 0= tf= -here .^ , ,, w)! , T= = Fhh + kzxj', TKniT, + ]ixj"/,, TflJ -f kyzx, TTnijin + kzxy, Fnjn + kxyz, = Fqq, = T7rr, +k out of tlle r.^ s r Ai if:^:r therefore ^(= V ) = 54. d uble n S ent of * spinodo ourvo, a,,,! . The ^ of the , may Me ^ A MJJSMOIB, ON OUEIC SUKPACES. 383 of a double-sixer is a double-sixer. Hence the 27 lilies of the reciprocal surface may bo (and that in 3G different ways) represented by 1, 2, 3, 4, 5, G 1', 2', 3', 4', 5', G' 12, 18,.... 56, whoro 12 is now the lino joining the points 12' and 1'2; and so for the other lines. The lines 12, 34, 56 moot in a point 12.34.66; the 30 points 12', l'2...5fl', 5'G, and the; fifteen points 12 . 34 . 56 make up the 45 points t'. The above equation, S a - 2'* = 0, shows bhat the cuspidal curve is a complete inter- section 0x4; o' = 24, Section II = 12 - G a . Article NOB. 47 to 50. Equation W (a, b, c, f, r/, h^X, Y, Z}*+2hXYZ = Q. 47. It may bo remarked that the system of lines and planes is at once deduced from that belonging to 1-12, by supposing that in the double-sixer the corresponding linen 1 and 1', &o. severally coincide; the lino 12, instead of being given as the inter- section of liho planoH 12', 1'2, is given as the third line in the plane 12, which in fact ropvoHontH tho coincident pianos 12' and 1'2. 384 A MEMOIR ON CUBIC SURFACES. 48. The diagram is [412 Lines. tt tt- >t Oi W W K> to bS M 12 13 14 1; l(i 24 25 ffi o <)(j 1 S 34 35 36 45 46 56 12 . 34 . 56 12 . 35 . 4G 12.36.45 13.24.56 13 . 25 , 46 13 . 26 . 45 14. 23.S6 14.25,36 14 . 26 . 35 16.23.46 15.24.36 15.26.34 16.23.45 16.24.35 16.25.34 15x2 = 30 6xl = Jiimilinl oiicli pair of inyo, onclt Uirco nioro lines. 412] A MEMOIR ON CUBIC SURFACES. 385 49. Putting the equation of the surface in the form T, Y, 2f , , , where for shortness a = mn I, finl m, <y = Im n, 5 = linn 1, p = linn, then taking X = as the equation of the piano [12], 7=0 as that of the plane [34], 2 = as -that of the plane [66], the equations of the 30 distinct planes are found to be Z-0, [12] 7=0, [34] Z = 0, [56] m Z + Z 7 + = 0, [23] m-'Z + Z m = 0, [24] l^Y+Z^Q, [18] 20, [45] ^-0, [46] ^0, [35] ^O, [36] Z=Q, [16] # = 0, [15] ^-O, [26] ^-O, [25] F = 0, [12.34.56] Tf = 0, [12.36.45] F-0, [12.35.46] TF = 0, [16.25.34] r_ ^877=0, [15.26.34] ^r +a/3 F = 0, [14.23.56] ^_ 7 S Tf=0, [13.24.56] mnZ + ttZ FH- Zm ^ + a/3 7 S Tf - 0, [16 . 23 . 45] pX + n F + m ^ + ^878 Tf=0, [13.26.45] wZ + jp Y+l J? + 75 ^=' [16.24.35] mZ + Z 7 + ^ + ^ F-0, [16.23.46] Z + ZmF+to Z-fat F=0, [15.24.36] ZmZ+ F+m5-7S ^= [13.25,46] ZZ + m F+ ^ - /9S TF = 0, [14 . 26 . 35] ZZ + m Y + n Z-afa F = 0, [14.25.36] 49 386 A MEMOIR ON OUBIO SURFACES. 50. And the coordinates of the 21 distinct lines are [41! (a) () () (/) (?) (A) whence equations may bo taken to bo I - 1 1 (!) X=Q, 7+ IZ = Q T71 1 - 1 (3) 7=0, #+Jr=0 n -1 1 (6) ^=0,Z+7=0 I- 1 -1 1 (2) Jf=0, 7+ ^-^=0 m" 1 1 -1 (i} 7-0 y i. )-' r n \tj t. u, ^ + ni A = u -' - 1 1 (6) ^ = 0, X+ jrU^O 1 n m Wy n () ^r + ^.^ +/9yI ^o n 1 I I ya n ya /-I /1\ T" / ^ 1^~ A Tr in * 1 I a/J m ()^ + W + ^o^.,.^iro \ 1 n * m 1 naS (46) z+ W-UN-W^-O, ,r-srr-o n 1 j 1 "^ (26) 7+ l-V + nX^Q, 7-/?Srr^O 1 I 1 i yS ??iyS (24) ^r + m -ijr+ /r,, 0, ^ - yftjr,, o 1 71 1 m ___ maS aS /*if)\ y v i ^ A 1 n 1 I JL 1 (15) r,. W+B-.JT.O, r-^Hr.o 1 1 m t yS (13) ^ -f- inX 4 ^ l J r ^ 0* % vS n^ 1 n * ' " *- J. * v/j ^/ *y(j j f ^j y " . -^ 1 1 1 1 1 /Ojn\ - i ir m W/?y nfiy V^OJ Ji -1* flJ ^I^Nfl>i"^'f^*_fl T^ i /I lir ft \ u, A -f TO 1 -I- m /J = U, -ft -|. /3y || r =: 1 1 1 n * fya (25) 7 + l-^ +n ^=Q, 7-,-yaI-r,,0 1 1 1 1 1 la.fi mya 1 -^^ ~ i 1 - _ " 1 .- (34) 7 = 0, WsO , 412] A MEMOIR ON CUBIC SURFACES. 387 51. The six nodal rays are not, the fifteen mere lines are, facultative. Hence 52. Resuming the equation W(a, b, o, f, y> K$X t Y, 2) z +ZkX7Z~Q> the equation of the Hessian surface is found to be 6, o t f,g t h1X, 7,%)* 2kW{(a, b, c,f> g, Ji$X, 7, Z)*(FX + QY+HZ)-3KX7Z] 7c a a*X* + whore (A B, 0, ^ (?, ff) = (6o-/ f (W-^, K = a&c - ct/ a - fy a - The Hessian and the cubic intersect in an indecomposable curve, which is the spinode curve 3 that is, spinode curve is a complete intersection 3x4; o-' = 12. The equations of the spinode curve may be written in the simplified form W(a, b, o,f, (/, liftX, 7, Z)* + 2kX7Z = Q, -8KXYZW + SkXYZ(afX + bg7 + ohZ) - 7c 2 (Z 4 + 6T 1 + &Z* - 26c7 3 ^ 3 - 2caM 3 - 2a&Z a F 3 } = ; and it appears hereby that the node O a is a sixfold point on the curve, the tangents of fcho curve in fact coinciding with the six rays, Eaoh of the 15 lines touches the spinode curve twice; in fact, for the line 12 we have X G t W-0' t and substituting in the equations of the spinode curve, we have (6I ra ~c^ a ) :==0 > tlmt is > wo have tlie two P oinfcs of contact ^-' ^=0) 7*Jb = Z^c. Hence ^'-30. Reciprocal Surface. 63. The equation is found by equating to zero the discriminant of the ternary cubic function (Xa+Yy + Zg)(a, 6, o, f, g, h^X, 7, Z)*-2kwX7Z, viz. the discriminant contains the factor w a which is to be thrown out, thus reducing the order to n' = 10. The ternary cubic, multiplying by 3 to avoid fractions, is X*, 7* , 3cJ, 3%, , 49 2 388 A MEMOIR ON CUBIC SURFACES. [412 Write as before (A, B, C, F, ff, H) for the inverse coefficients (A=bc-~f a , foe.), and A" = ale - fl/ 3 - Iff- - ck- + %fgh ; and moreover <P=(A_0, 0, ^ G, HJa, y, ,) P^Aa Q = Hx U = aft/2 + Igzeo + clmj, V - 2/%ir - flP?/2 - 6Q = ~a Efz ~ b Fz* ~ a - a Gye* - 1 HzeP ~ G + (~ abc - a/ 2 - bcf - W=(A t , 0, F, G, M=kwU+V, JT = 2i'fl&C K1JZW -)- W ; Then the invariants of the ternary cubic ar are is a - . 0) and the required equation of the reciprocal surface viz. this is which, arranged in powers of kw, is as follows; viz. Coeff. ( _ fl 412] A MEMOIR ON CUBIC SURFACES. 389 = 2a6o mjz (12i# - 8i 3 ) + W (- 3* + 1 + U s (- 2$ + 4i a ) + 2 DT (- 4) + F a - 86060 fljy F-18tfTP + 72a6o coyztU -16 a* 1 (Aw) 1 = -18QVW -1G7", bub I have nob carried the ultimata reduction further than in Schliifli, viz. I give only the terms in (IcwJ, (kwf, (Iswf, and (&w). 55. I present the result as follows; the coefficients doduciblo from those which precede, by mere cyclical permutations of the letters a, 6, o mid /, g, h, are indicated by (). = (Aw) 7 . -I- (Aw)" . wfss ct?bo -I- 1. /i/tfl-f 1 A, \.tf\f\ti J. J. " i) a a %-6 o%oA-6 06 V - 6 B io/ - y/i+ 2 o^ -I- 2. V -i- 42 abofgk+ 5cW + 2 a<j/"/i a - Ixsfgl - 24 flj/'V* + 390 A MEMOIR ON CUBIC SURFACES. [412 56, In explanation of the discussion of the reciprocal surface, it is convenient to remark that we have Node <7 3 , X = Q, 7=0, Z = Q, Tangent cone is (, 6, o, / ft AJJ, r,2)'0. Nodal rays are sections of cone by planes X=Q, 7=0, Z=0 respectively, viz. equa- tions of the rays are 2 +cZ* = 0, 7 = 0, cZ' + Z =0, .J* 6'r the lint Reciprocal plane is w = 0. Conic of contact is (A t B, G, F, 0, /fjj>- y, 2 y= o, w = 0, Lines are tangents of this conic from points (y=Q, = 0), (2 = 0, = 0), (,v=0, 7/ = 0) respectively, viz. equations arc w - 0, w - 0, y ^ plft f * he Six s to , tf-tyyz (reciFooal8 f tha TO i 12 and points 1 and we thus have meet: and the or, what is the same thing, (equivalent to two equations) t the second and third equations. 19 -- - hllPS m-r. '19 V-l rrt ^ni i* ' ltj "" fi(s - -Wio Unoa written Jlf aUrTe ' in the conic w = o i-0 ~ w =. i . " ", or say it is a ourve 4 x . 0l which _ 2 . O '_ 18 412] A MEMOIR ON CUBIC SURFACES. 391 Soction III = 12-jtf a . Arl,i(!lu NOH, 00 to 72. Equation. 21f(Z-|- Y+Z)(IX + mY + n%)-\- (i(). Tho Hyntom of linos mid pianos is at ouco deduced from that belonging to II -12 -(/a, by mipposing thu tangent cono to reduce itself to the pair of biplanes; It of thn pliimiH (n) of II = 12 O a thus coining to coincide with the one biplane, and tlmio of thorn with fcho othov biplaiio, (II, Thn diagram in Linos. Q) CM W IJ IO K> M M (-J OJ CJI rf*. O> 111 It- OV Ol It- O til it" 0) IsO f- 1 ill Mia 11 ^1 Ol l M Oi X 1W II CO wit 4M1 . . . Biplanes, l-l * Ifi ' ' 1(1 , ' y-i ui * a 1 yfi HO -' ' BiraiUal plnnoa each con- taining a ray 1, '2, or 8 of the one btplano, and a my 4, fi, or fl of tlio othor biplftiio. an M , an . tto 14 . H(l . Jlfi lfi.UH.H4 16 . U . ild 10 . aft . IM flxltstt i7 4fi t Plftnos oaoh oontainiiiR tlu'co moro linos. if? s ** I! 392 A MBaroiE, ON CUBIC SURFACES. [4J.2 62. Taking X+Y+Z=0 for the biplane that contains the rays 1, 2, 3, and lX + mY+nZ=Q for that which contains the rays 4, 5, 6, we may take X~Q, Y=0, Z=0 for the equations of the planes [14], [25], [36] respectively; and thon writing for shortness m n, n ~ I, l m \, ft, v, and assuming, as we may do, k**\pv, so that the equation of the surface is W(X + Y+Z) (IX + mY+ nZ) + (m - n) ( -1) (I- m) X YZ= 0, the equations of the 17 distinct planes are Z = 0, [14] J'-O, [25] ^=0, [36] A r + 7+ ^=0, [123] lX + mY+nZ=Q t [456] IX 12=0, [25] Q, [26] Q, [35] . IX + lY+nZ = Q, [36] F = ' [14.25.36] F-f (XZ^O, [14.26.35] F + ^y=0, [10.26. 34] TF+ n^^O, [15.24.36] ImX -i- m7 + n?^ + F - 0, [15 . 26 . 34] mnZ- W= 0, [16 . 24. 35] 412] A MEMOIR ON CUBIC SURFACES. 63, And the coordinates of the fifteen distinct lines are 393 () (*> () (/) (0} (A) whence equations may be written -1 1 (1) X = Q, Y + Z=Q 1 i (2) 7=0, Z + X=Q - 1 1 (3) # = o, js:+r = o ~n m (4) X=Q t ?7>?fcZ=0 n -I (5) r=o, # + ZJT=O m / (G) ^=o, ;x +r=o 1 (U) JST = O, ir=o 1 (25) r = o, ir-o 1 (36) 2T=0, ir=0 I n n n^v - nlv (16) lX+nY+n = Q, IP"+?iv^=0 I in HI l)i?p Imp (16) UT + wr+m2' = 0, IT-!-w/i7=0 I m I a -Imp (26) /JT+mr+/^ = 0, ir + tt,JT=0 n in n tnnv - v (24.) X+ mY + nZ^Q, IT+ nf ^= m m n mn/j. Wl 8 /* (34) wZ + wF-f- w^=0, ir+m/tr=0 I I n nl\ -2\ (35) ZX+ jr-i- #=<), IF + /XZ = Tho rays are The equation not, tbo mere lines are, facultative; hence b' = p'-Q; t' = (j. of the Hessian surface is k (l*X* 64. G5, + kXYZ {(I* + 81m + 32?z + -nm) JT + (m fl + 8mn + 3ii + nl) 7+ (?i a + 32 + 3?im + toi) Z\ = 0. The Hessian and cubic surfaces intersect in an indecomposable curve, which is the spinode curve ; that is, spinode curve is a complete intersection 3x4; a-' = 12. The equations may be written in the simplified form W(X + Y+ Z} (IX + mY + nZ) + IsXYZ = 0, - 4A r YZ [I (in + n) X + m (n + 1) Y + n (1 + m) Z\ = 0. We may also obtain the equation X a Y *& >\ Y = 0, C. VI. 50 394 A MEMOIR ON CUBIC SURFACES. |\| L. which shows that there is at B 3 an eightfold point, the tangents boing givoti by (JT+ Y+Z) (IX + m7 + nZ) 0, (V, & v\ -fw, ~v\ -X/$7#, ZX t AM r )' J - 0. Each of the facultative lines is a double tangent of tho spinodo ourvo; wlumcn ft* i* IN. Reciprocal Surface. 66. The equation may be deduced from that for 11= 12 - (?, vk writing (o. &> o, f, 9, APT, F, ^) a 2 (X + K+ Z) (IX + ?F+ ^f), that is (, &, c,/ g, h) = (2^, 2m, 2, m+n, w-M, we have (4,5,0,^, ff, ^) = -(XS ^ ^ ^ , ,x, Writing also X, /t, v = m~n, n-l, l-m as before, I (m + n) 2/2 + w (ft + ;) m + w (; + Mt ) ^y*+ mpz!s-\- nv (l + B)ffl+ (ft+i)y+ (i + W we have and then i = fc' so that the equation is L eualbn i7 *""' S W fc throw out ft the whole factor equabn i W trow out ft the whole equation tho 412] A MEMOIR ON CUBIC SURFACES. 395 or, what is the same thing, it is (7c 2 w a - Zkwt + <r) 9 (/cV0 + kw - 36 (Afttf 3 - 2&wi + <r a ) (kwv + o- - 32 (fewu + <rty ) 3 - 108&w (4rfcw0 - f )" = 0. 67. This is (fay) 8 . 5 + (/cw) 5 . - ^ ~ G<0 + u a + (kwy . <T H . 30 + <n/r . 2u + f a . Gi + (kw) s , - 2<ri/r B + o- a (2u a - . 37 288W - 96u a ) + f 2 (Si 3 - (Jew} . - o^p + o- 4 (- QtO + u a ) + o- 3 f (- 8tv - 1445) o-^ 3 . - 72 + f * , - 108 which, reducing the last term, is m) (lz - nes) (m> - ty) = 0. 08, I -verify the last town in the particular case = as follows: the coefficient of CT" is (0, 2tt(Z + m)ffiy, 2(m + w)w + 2(ft+Z) which is {(I + 7)i) (Jus + /*P) 9 + [(^' l + w) + (n {[(I + m) \ + (m -f ) c] Xa* 1 + [2 (Z + w) X/* + (m + ) /w + (n + i) i>X + 2?iy] which, substituting for X, & v their values m-n, n-l, l^-m, is = ZtfwY . - 2X/i (> - y) (m - ly) ; or for ^ the coefficient of o- 8 is agreeing with the general value 396 A MEMOIR ON CUBIC SURFACES. [412 69. In the discussion of the equation it is convenient to write down tho relations of the two surfaces, thus : Cubic surface. B a , X = 0, 7=0, Z=0 Biplanes X + 7+ Z=0 intersecting in edge. Rays in first biplane, X = Q, 7+ = 0; 7=0, Z+X-. rays in second biplane, X=0, mY+nZ=0- t 7=0, nZ Reciprocal surface, Plane w 0, Points in 10 0, viz. in line (m -n)a + (n - 1) y + (I - m) z~Q, that is, \o) + py + vz 0, or <r = 0, Lines in plane <u) = 0, and through first point, viz. i/-z = 0, z~x = Q, /y-7/ = 0; lines through second point, v\x. 70. The equation puts in evidence the section by the piano w = 0, vk thin in the hue <r = (reciprocal of the edge) three times, and the six linos (reciprocals of the rays) each once. Observe that the edge is not a line on the cubic; but its reciprocal is a line, and that an oscular line on the reciprocal surface; tho six linon reciprocals of the rays) are mere scrolar lines on the reciprocal sm-faco; they PBHB three of fa, through the point a = ^ and the other three through the point li/J v i : ; n; ff* 13 < th y ave six tangents of the point-pair (reciprocal of tho pan ot biplanes) formed by these two points. , - put in evidence the nodal CU1>VO on at P-edes ^t oo |t g g ^^ rf ^ mere ^ i, -, d and 4, 6, 6 the hnes which pass through the points 0, o: by g 0*0 anil 72, The cuspidal curve is given by the equations 9 , 24 (kwv + o-1/A _ 3G Writing down the two equations, = 0. = 16. _ J i: s a partial intersection 4.8-4 412] A MEMOIR ON CUBIC SURFACES. 397 Section IV =12-2(7,,. Article Nos. 73 to 84 Equation WXZ + P (yZ + S W) + (a, I, c, d$X t 7) 3 73. The diagram of the lines is l(i lO- CM rf^ W W Lines. (- M M U> tO OS to a* e rf2 i tf^WtOH^.COI.OI-' 01 IV = 1 2-2G' a . Ml Gi M M X X X bO o ^ [0] 1x2= 2 Plane touching along nxis, 11' * Planes through axis, 22' * * eitoh containing a 4x4=10 * * ray of the one node 1)8' * and a ray of tlie * other node. 41' 12 111 l'I: . 23 8 1 24 PH 84 Biradidl planes, each containing two rays t B two rays of the other 12 . node, I'd' * I'd' * 2'8' 2'4' 12.84 18.24 8x1= 3 i , . Planes each contnin- ing three mere lines. 14 , 23 20 45 g W s f" |p |p h-t ,*s (b p Sul^S-a IB (D " m lr ransversa tangent ; the axis. ' Qj 'I Ci S'e ^ 5 * B rt^ w E 1 & B * o P pUft ^, S B' B 1 *~** p *^* p 3 o fmt. ! .-(- . IH- H ^ H hj ^4 < <D T TO i fO 398 A MEMOIR ON CUBIC SURFACES. = 0) , And the 16 linos are [0] [irj [22'] [33'] [44'] [12] [13] [14] [23] [24] [34] [1'3'J [1'4'J [2'3'J [2'4'] [3'4/J [13 . 24] [14.23] [412 <*) (ft)- ' , () . _ (/) (?) W whence equations may be written 1 "(oT^oTr^ -y d (o) .A = 0, dY + yZ + 8W =:Q * o n rt n V 1 f i -8 (\\ Y ~ f v n &v fg -S (2} * V f s -8 ( ) V ft -S ( 3 ) t, 1* -^ V / M j, MI;M(IIII (N nnutf .SUHKAIIKS. 39!) (") iM ' () ! (/) t.v) (M ''. '. V n n v (i 1 ) -Arf,r..o, y r-i.i' l ir^o I'j I/ I" 11 n v (-0 I* f, IV II II n /'I' V y (' ) ' * ' M it f, I, " it v M'l r \< i * \ r t) I* A ., , 1 1 \ / 1 1 \ ,,.'<- I i - ( i \ V ' ( l ) ^ ' /' ^ 1,1; M. i..J ' Mr, r/ Wi (la.n'j* <S M' ' ) : i ' - /' l \ 1,1; > ! '> ' J Mi',' < t ) V " /I l\ 1 /I lv Ci'Af, ''rj 1 rjAr/ 1 '^ (!. fi ; M ' - ' 1 i ^ ' \ M t " ( i, ^' ' v(f/J V i /i i\ i /i i\ f^Ar, iVVAr/fJ (..i.^),, .i ' ; '/,-.!) ] *(!/;.) f J- 'l'( fifAr/'J t^'Ar, '''];) (a.r.r )n ' i M 1 - ' J i ( ' . l \ M. ; vt, ',' MI, ij r f.f.(f B '"0 f.i'Xi;' 1 '!;) < i ') n fi (r'r) ' >(! '!) v , iif/r/'i) f,Ui'"r) (3.1 . 1'2') uiu ttvr .!)'! *-i ith- liii.i 1^, :tT, ..hs.'i'vn Unit llu- twu i|iintiuim Eh.- P> A* lliul. n . t ss 3, es an- tliw mul thy wix iiiui'o liiii); A .';' i i /'|(.V - y,,') 1 \'t' **'.(('.' i yii'}.,- 4 loo A Ml:\!nll ,, N , | m , , , ''"" H'-mi, ,.!!,, , ,,... 7!l ' <! "'iiliiiiii.ir will, (I,, 1111 ..... ' ........... ^--n- ' J') 1 . whrjv ((,,. n^,,.,^ ir* 111 '.U II I',!' ,, 1 Ih ir.-.!j' ;!- I . y HM. Tl A* - X I A MM.MOIK OX i-HMIC N Sivtinn V \"}., /i', H.V Tin- iliiii;iain ..I I),,. MM, : , tlll | |,|, IMI ,., ;., '/. - U, V" J ^ II. % \ / /i 0, f iro. I'tnin 'i '-;!i'li lUi.'ii H)t> llllll'Al'liiJll, i ' 1 1 r *t^* | 1 I .MAT U^' 4 I ' H , t ON cmin< HUUKACKM. . i)v , A, U lt F, , _, liquation i -./.'-I /'". A(A/t - A l ;i r |- //,")- Ml.-./, u, whom UK, torm II nmy 1,,, oalrukto,! wi.J.nu,, -lillin.lrv: ,). ,,.,. (l . nil ., ,,,-, n tho (lovoldpii x,n, n H nf } (' Av ... A \r i //n ,. I r ' } "' 7 ^^^ ..... , ..... i.., ,,,,, i, ( ' ; ; ( " "i" l " 11 :*-'" ( '-,, (lll .i liM1J . Uio plan,, c^-0. ' V ' <! " 111 Nl "' ( ' A 'l '"""1'H-l.v H.r il,n ,. ,, v HH. Thu tioilul iitirvn w iimitt* un ,<(' n,,. i; i- , . I ..... " nn,l tin, ,,, u , N v al; ' ,''J' '"',', wl "" 1 ' ,'"'" ": ' ..... ' !. i ..... in,,, "" !'" '" .. """ T " t! l " fl " f "'" ..... ....... '". ........... ,,,,,,, ,,, ,;,, A" |.2.H/ii.|. |.fl/,tw- i), 412] A MEMOIR ON CUBIC SURFACES. 405 The equations of the spinode curve may be presented in the form XZ, flZ 8 + bZ* - Y 2 , aX 3 + IZ* = ; X + Z, W , Y* it is a curve 3.4 2, the partial intersection of a quartic and a cubic surface which touch along a line. The biuode is on the spinode curve a singular point; through it we have two branches represented in the vicinity thereof 'by the equations 8 x_ mv^A ' w~~WJ (w) ) respectively. 90. The edge counted once is regarded as a double tangent of the spinode curve (I do not understand this, there is apparently a higher tangency); each of the four mere lines is a double tangent; the transversal is a single tangent; hence '=2.2 + 2. 4 + 1, =13. Reciprocal Surface, 01. The equation is found by equating to zero the discriminant of the binary quartic X<s + Za)XZ(X + Z viz. multiplying by G to avoid fractions, and calling the function (*X, Z)\ the coeffi- cients are Qw (a; + 2aj), f } 'A (.7; + z] w 4- 4 (a + 6) and then writing L =if + 4 (a; + z) w + 4 (a + we find and then the equation is - 54wW) a } 0, viz. it is 9 = 0, 404 A MEMOIR ON CUBIC SURFACES. [412 87. And the lines are (i & c / * A equations may be written 1 (3) ^ = 0, ^=0 1 1 (4) X+^=0, TT=0 V& 1 (1) X=Q, 7-^V6 = -V5 1 ^ J / ' ~ > ~ 1 V5 (V) ^=0, -ZVa+r=0 1 -V (2') #=0, Z\/ft-i- 7=0 \ / ' 1 Vi 1 1 Va 2 2 ( Vrt - V&) -2 (11') but for bho otlior lines the coordinate expressions are 1 1 1 tlio more convenient. ~7^ 2 2 <- V - Vi) -3 (12') V o 2 2 ( V + Vi) 2 (21') Vfl& i i Vl 2 2 (- Vn + V&) - 2 (22') 88. The four mere lines and the transversal' are each facultative ; the edge is also facultative, counting twice; p' = &' = 7, i' = 3. That the edge is as stated a facultative line counting twice, I discovered, and accept, d postwiori, from the circumstance that on the reciprocal surface the reciprocal of the edge is (as will be shown) a tacnodal line, that is, a double line with coincident tangent planes, counting twice as a nodal line. Reverting to the cubic surface, I notice that the section "by .an arbitrary plane through the edge consists of the edge and of a conic touching the edge at the biplanar point; by what precedes it appears that the arbitrary plane is to be considered, and that twice, as a node- couple plane of the surface: I do not attempt to further explain this. 89. Hessian surface. The equation is 6; Combining with the equation. X2W+(X + Z)(7* - etX* -!>&)'= 0, and observing that from the two equations we deduce it appears that the complete intersection of the Hessian and the surface is made up of the line .X" = 0, Z~Q (the edge) twice (that is, the two surfaces touch along the edge), and of a curve of the tenth order, which is the spinode curve ; c' = 10. 412] and wo thus have or, what is the same thing, A MEMOIR ON CUBIC SURFACES, 407 LM+ wW =(), L, 12M, -fl/V = w 9 , L t M for tho equation of the cuspidal curve- Attending to tho second and third equations, those aro tjuarbics having in common w* = 0, L = 0, that is, tho line y = 0, w = four times; or tho cuspidal cnrvo is a partial intersection 1x4-4: e'~12, Section VI ~12-.B a -0,. Articlo NOH. !)5 to 102. Eqiuilion WXZ+ Y*Z + (a, 6, c, f)5. Tho diagram of tho linos and planes is W >( rf K*. N> rf* W l-S M O Visi3-7i 8 -a a . (-i ^ CO w K* M f-j X X w X X l ^l 1 II II CD II en 1x0= Biplane touching nloiig axin, nntl containing transversal ra.y. 00 Ixflss , . . Olhoi 1 biplnno. 22' , ' B 9 , PlnnoH oaoli tln-ouRh tho nxiB and jjlj' (J xfl = 18 . , containing a ray of tlio binodo 4 ' M ' ftiid a my of tho onionado, 1 * 12 ' * Bii-nclinl pianos of tho binoclo, 18 8x8= . ctioh containing my of axial biplnno tincl a ray of othor \)i- * t piano, 14 3'8' 2'4' 8x2= t ' ' Diraclial planca of tho ouicnodo. 8'd' n * * K g tr& W Et a"**! p. S g 'I'i' |>d g iL K Q a BE " S. P * G 3 'I " ^ oa *3j a. a i-s. & * <o S 1 B. 3" g t* P- o 406 A MEMOIR ON CUBIC SURFACES. [412 92. This, completely developed, is 64?<J 8 . ab (a + 6) a {(a + 6) f - (te - s)-} + 32W . 2t(6 f 3 (a + 6) [(a - 26) as + (- 2a + 6) z] f \+ (IB - zf [(- Sa + 56) x + (5ft - 36) z\ 3a6(a a - 7o6 + 6 s )2/ 1 [6 (9 a + 266 - 6 3 ) $ - 266 (ft 4- 6) aw + ft (- " Cffl - zf [6 f- 12a + b) a? + ZZdbKZ + a (ft - 126) z V / L \ * + 8?w 3 r 36 [(2ft - 6) w + (- a + 26) 0] # 4 26) a; 3 + Sabxz + a (- 2a + 96) + 6a% - (a + 6) V + cwa 3 2w + 4,'cV (w + z] f 96 s ) &} f - where we see that the section by the plane w = (reciprocal of #,) is imulo up of the line w 0, y = (reciprocal of the edge) four times, and of the lines w^O, ttya-aPeOj w~0, 6?/ a ~s a = (reciprocals of the rays) each once. 93. The surface contains the line ?/~0, w Q (reciprocal of the cclgo) ; and if wo attend only to the terms of the lowest order in y, w, viz. 2 a w 2 + 8 which terms equated to zero give we see that the line in question (y = 0, w = 0) is a tacnodal line on tlio auvfacu, fcho taenodal plane being w = 0, ft /a'etZ p/ans for all points of the line: it has already been seen that this plane meets the surface in the line taken 4 times; every other plane through the line meets the surface in the line taken twice. Wo havo in what precedes the d posteriori proof that in the cubic surface the edge is a facultative line to be counted twice. 94. Cuspidal curve. The equation of the surface may be written - (LM + Dsylff) 3 = 0, 412] A MEMOIR ON CUBIC SURFACES, 4C 99. The equations of the spinodc curve mi\y be written in the simplified form XZW+Y*Z + ( (t> b, c, cl^X, Y) s ^0, 4(a, 6, c, d$X, F)-H3(4flo-3& !l , d > H cd, d^X, Y)* = Q, the line X - 0, Y= here appearing as a triple lino ou the second surface ; tl curve is a partial intersection, 3x43. The node Q is a triple point on the curve, the tangents being tlio nodal raya The node ,83 is a quintuple point, one tiuigonb being AT = 0, RdY+^Z^Q, and tl: other tangents being given by Z = Q, (4ae - 36 3 , ad, bd, ad, d^X, Y')***Q. Each of the facultative lines is n double tangent to tho curve, or wo hivvo /3'~ G. Reciprocal Surface. 100. Comparing the equation of the cubic surface with thai; for IV =02- 2(7 it appears that tho equation of VI = 12- H s - O, in obtained by HubRtitiiling iu fch equation the valnsa S = 0,7=!. But instead of making this substitution in tho ruin formula, it is convenient to make it in tho binary qiwi-tic (#"5A r h K}', thus in fac working out tho reciprocal surface by moana of tho funotion a, >, o, f the coefficients whereof (multiplying by 6 to avoid fractionn) are Sony -{ IRbz-w, 6w a . We find where i 2/ a + 6 (a 1 M = Zdmj + 6 N = - 4d B ffl a - Sd (3w - 2tt?/ -h 2d^) w - 1 2 (36 1 - 4a o) w 9 . The equation is lOfte 5 ^ K /j2 - 122WJlf ) 3 - ( /jB ~ 18*w/Jf - 64s^JV)a| - 0, viz. it is i a (LN+M*) - ISzwLMN - IQtwM* - 27^ 3 w'W a = 0, where however ii\ 7 +Jlf u contains tho factor w, = wP suppose ; the L*P - l&zLMN- IG^Jlf 3 - W&vN* = 0. Write A G = 66tZw - 4rfy 4- 4^3 + 3 (36 s - 4ao) w, c. vi. 52 408 A MEMOIR ON CUBIC SURFACES. [412 96. Writing (a, I, c, dJ[X, 7) 3 = -d(0 2 Z- 7)(0 3 Z- 7)(^Y- 7), the planes are Z0, [0] Z = Q, [00] 2 Z-7=0, [22'] ^ 3 Z-7 = 0, [33'] ^-7=0, 7 (03 [12] [13]- [14] [2'3'J [2'4'] 97, And the lines are a b c / A equations may bo written . 1 (0) . JT0, 7=0 -1 rf (!) JtT = 0, dY *%~ 1 o. (2) o 2 A r - r = o, ^r o 1 8 3 (3) f? 3 Z-7=0, ^r==0 1 04 (4) ' 0,X~ 7=0, ^T0 0-2 ' -1 2 a (2') - 7=0, a "A'+ )r0 6 3 1 3 a (3') 8 A r - 7=0, O a *X+ ?r=.0 04 1 . 4 fl {4') ^- 7=0, ^A'H- ir-0 -de, -<ie t d d 1 1 -(*.+*) -<,+**) ~Q$4 -OA d(e s e,~e z e 3 -o,e,) d(OA-OA-OA) n 2 g' j/\ but for the remaining linos (13 . 2'4') the coordinate oxpvoasiona -de, d 1 -<*,+*.) -0A d(e,e a -6A-oA) nA o'o'\ aro mt >ro convoniont, ^1'i . A 1 The mere lines are each of them facultative ; 1)' = p' = 3 ; f = 0. 98. Hessian surface. The equation is ^+S(cZ + dP)}(Z^F+I^ + (a, 6, c, dJ2T, 7)'} -42 (a, 6, Cj dJZ, 7) -8(400-36", ad, H od, d'JJT, 7)^-0; and it is thence easy to see that the complete intersection is made tip of the line A-0 1=0 (the axis) three times, and of a curve of the ninth order, which ia the spmode curve; & Q. 412] A JMUMOIR ON CUBIC SURFACES. 411 l/ = 0, Next as regards the surface LM 4- 9zwN = ; the line ?/ = 0, w = is a simple line on the surface, the terms of the lowest order being Qzw (- 4>clW) = ; that is, we have ?y=0, and for a next approximation w = Aif, viz, L=y\ M^- :/V, and therefore - 2r%' + 9.s. A 2/ 3 (-4rfV)= 0, that is, A =---\- , or ., 18rfnw 3 ; there is thus a threefold intersection with one sheet and a simple intersection with the other sheet of the surface IS-IZzwM = 0. The surfaces infcorseot, as has been mentioned in the conic = 0, ?f + 4't,w = ; or we have the lino y = 0, w-0 four times, the conic once, and a residual cuspidal curve of the order 4.4-4-2, =10; that is, c'=10. Section Article Nos. 103 to 116. Equation WXZ+ YZ + YX* -%* = 0. 103, The diagram of lines and planes(') is Linos. j_i M O3 t^ 05 W l-l VII-] 2-B D . Oil IO X to X X X H 1 Ot On IOI M. o Ot K* O 01 1x15 = 16 4 Toi'flnl biplnno. 00 . . 1x20=20 . . Ordinary biplnuo. si a 03 * fe IB' 2x 5 = 10 Plfinos eaoh containing EI more line. T 45 I a ! f! td f en * 3 P S" B V s i The marginal symbols in the preceding diagram* constitute a roal notation of the linos and but hare, and still mow BO in somo of the following atagmme, thoy ai e anero marke of veforonoo, which are the lines and planes to which the several equations respectively belong. 522 Mum wti have A MKMiili! UN t II1H M Ki \. j ,. A : M ' -A/.- ..... 101. Coimiiln- HIM M'ttMni, l.il.M ,!, , tlur f,uv Ill ft 10m Ay, VIK. Wli f ., , ^ ; '' ' ( " 7'' ' ' ' " lM '' It ^ r , v .,, ,.,, ,., t . - . . liu, ,(' ||,, ,v,i,,n,.l ,,,,,-ih.v. lH ,,r rl,..,.....;.!.!! .-.,- (hat i, A - '' ,; 1 "' / '""'' *"""* ""'' M "" S "" *" - m ^/-*%'"0. 3SSS & 412 A MMMOUl ON (Until! HUHKAdKH. 4 \ ft 10!). Kach of tho linu.s (A' ~ IF'- I), Y-\-X.-0) , u .l (A' -|- W() t ,K --#-<)) is iloublo liiuigonli of fchn Hpiuniln ontio; in lar.fc I'm- l.lin Ih'Hl; nl' MmHn Ijm.'s \vn huv<' Unit i,s, HO I'Jmli bh lino UnudhuH uli (Jin l,wu poinU W iviii by ^^.|-hM); mu l ittlior lino toimhoH ub lilio hvo pniuU ^ivnn by U0 J I : : 0. in nbUin.'il l. vk Ottlliii K tins (*^A r , AfV, Mm nioniKiiMtLH (tnnllij.lyinit l'.V ti) nn- (({< - M//WJ, y I 4,,'m, ([/HI, 2 )'); . , aw! tlitiii wnl ,A , we hiivn \1 Tbo od^i A'-O, ^r-0 IIII vix. Mm U|im(iionM A%, 0, ^- , nniHt bu riiokiiiioil only UH n tlntibhi l.iui^riiL ll.'iir,- //',-,,;. o .,. >> p , , (j, mid fcluj i'i|iia(:i.ni IH, an in lii ,A' J ( f,N -|- ,U y ) I Hw^A J/ A r - 1 1 iv-0/ J ... ^TwW ^ ; but ^VH-jV- ,ut(l tlmrororn Uu- whnh- ..pmlim. ilivi.|i-s by . ami v ll.n .,, 07/M'A'-','-: o ov, (Htinplotoly (Iuvcluii!il, thin IH w' . (i-t + W.:)2 ( ,V-.kc) + ?"'. l(i.i'( fly- 1 4. Jis') + <" ( // 4 + at)// 1 ^ -( KiO//*.^ ^7^' >|< lil..*) 4- 7W' J . >kr ( 1 1, V ^ + 1 2yV ^. !l,y^ - .k a ,.^) -h w . i' 412 A MEMOIll ON CUBIC! SUUKACfW, 104. The planes arc The linen uro Z =0, [10] A r -0, A r = o, [oo] yo, 0, [12'] A'-O, y-i A r -lF^O Y-\- %::(} JL It ""* \fj ,1 [ fj --"' W> 105. The two mere lines are facultative, and tho od^i; in uho litmil|.al.ivn ; /i';i //-.'I 106. Hessian .surface. The equation is X*-&)~ 4A r 'r# + A 1 ' + \ft ; , (). The complete intersection with the tuirfiico i,s thus givoii by t;ho (>i|Miil.itiiiH WXZ + Y*Z + Kf ~ jgs. o, - &*Y% -I- P -I- -l^ 1 s 0, which i B made up of the lino X**Q, %-(} (tho I W ) loin- ti m ,. i UI ,| a rnrvu ,,f Mi,* eighth order. To see this, observe that tho luHtrimmUuiiuil mirliiiiCH li.uv in nu ......... i the line A r = 0, ^ = 0, which is on the first miife,, u t,,, wi i |j nt , ( l!l|Ul iti llM j,, vi,,;,,!^. being ^--A^.and on the second surfnco n triply linn (nqiiiLli.utN in vininit. lin,,- Z- l yX* and X*=] 7 Z*}, But * ^A'. tonoto, Jgr-.^V, un.l U... lu... (2 + 2 =) 4 times, 107. I say that the complete in Demotion is th,, linn ( A' *..-., ^^0) Mtivc linir, ogether with a spinode curve made up of this mm lit,,, ,. and uf t| lt , mnv ,,r the eighth order; and that thus o-' = 9, The discussion of the reciprocal surface in f ll(!t show, M,u|; ,;| in .v.m.rnral ,,f (h,. edge is a smgulav hne thoreef, counting onee us a nodal ,! ,,wi,, , H 1< 108. I find that the octic comnonont of Mi ^..;, ,,i the equations of which may be written ' " ' '" ' irv " ' H u IUlblWlU m "' V1 '' X ; F : - . ; the values of ^ at the binode B, arc tf tf- i ,1 , bourhood thereof the two branches ' WU th " H " htllin iu thu > 412 ] A MEMOIR ON CJUBIG SURFACES. 415 that is, the surface has the nodal lines (* + y=sO, jy-w = 0) and (z-y = Q, #-[- w = 0) which are the reciprocals of the lines 12' and 13' respectively. The nodal curve is made up of these two lines and of the line y=Q, w = Q (reciprocal of edge), as will presently appear; so that we have b' = 3. 112. The equations of the cuspidal curve are L*- LM+ 0. Attending to the two equations 48 w 1 = 0, (8 - 72 =) - Oiaiio 3 + 18si^ = 0, these surfaces are each of the order 4, and the order of their intersection is = 16. But the two surfaces contain in common tho line (y Q t w = Q) 7 times; in foot on the first surface this is a cusp-nodal lino 4auw + ?/ + Ay* = ; and on the second surface it is a nodal lino w (Amy + I8&o) ** Q \ the sheet w = is more accurately 4aMO + 2/ 9 H-J?^> 1 .. s 0; whence in the infcorsootioa wibh tho first sin-face tho lino counts 5 times in respect of the first shoot and 2 times iiV respect of tho second shcot together (5+2=) 7 times, and tho residual curve is of the order (16-7=) 9. 113. I say that the cuspidal curve is made up of this curve of tho 9th order, and of the line j/ = 0,^ = (reciprocal of tho edge) once; so that c' 10. In foot] considering the line in question ^ = 0, w = Q in relation to the surface, the equation of the surface (attending only to the lowest terms in y t w) may bo written giving in the vicinity of the lino and then _ that is, ^.= = -2- or 4>%w + f*= V-2. *-y* ; wherefore the line is a cusp-nodal lino, counting once as a nodal and once as a cuspidal line; and ao #ivin the forcgoine- results &'=3, c'=10. 114, I revert to the equation which exhibits the nodal lines (ai-w^O, , 10 = 0, y-z-fy for the purpose of showing that they have respectively no pinch-- points; that is, that in regard to each of them we have j' = 0. In fact for the first 414 A MEMOIR ON CUBIC SURFACES, J'41.2 111. To transform the equation so us to put in ovuluiicn tlio nodul cum 1 , I collect the terms according to their degrees in (>j, z) and (;, w); vix. tint u<|tmtlim thus becomes & a w fl 4- 4- 4- ^ 3 2/ a . w + f . - tu + z . w and if fa momont we write . ^a momon we wte . + 7 , y.- 7 mM , a)llBOtj u l (; . ilmit(1 , |wji , 7 by their values (* + ,,), ^-y), the equation can bo oxpivHso.1 in l;hn ii.nn 4- ( 4- y Y (tO + W ) a (* + y Y (z - y) ( W + W ) ( - W (* 4- 7/ ) ( Z - y)3 ( ftl 4- lw (2 -y^(fl;^ w . and observing that we have f the 8coond wte h ' + order m *- y and W + M conjointly; 4I2 .1 A MJOMOIU ON CUBIC SUWACKS. 417 115. The cuspidal uinthio curva is a utiicursal curve, tlio equations of which can be very readily obtained by considering it a* fcho reciprocal of tho spinorlo torso: wo in fact have (a : y : z : w = ZW+ 2 AT : VYZ + X* : WX K a -3^- or substitufcing for X, Y t Z, W their values (= 1(>^, 40 + 100', 1G#, - 5 - 80' and omitting a common factor 100 3 , wo find for the cuspidal curvo A- : y \ z : w80 + 240 s -160 : 240" + 820" : -4-480* ; 160" (values winch verify the equation A'+ >> + #* + Fw - 0); tho spinoclo ourvo boiiiff thus of tho ordor =!) as montionod. For 0-co we have the singular point (y 0, 0-0, w - 0) (rociproeal of torsul biplane), and in the vieimty thereof at : y i x : w=*l : - 20~ 3 : JJ0-" ; - 0-", tliereforo For = we have the singular point .v-0, ?/ = <), w = (reciprocal of tho other biplane), and m tlio vicinity thereof > : y : g . w=*-~ %0 : - fi0 a ; 1 : -40', therefore 110. Tho section of tho surface by fcho piano Q in an interesting ourvo. Writing 2 = in the equation of tho mirfuco, I find Unit the rumilfciiie ommtion inuv be written * -h ^"itf 9 + 27fl, J , T/ IJ - 82rt;?y) a 0, where observe tliat HO that tho curve has tho four cusps w a +27^-0, ^-32ro/0; the piano intcrsocts tho cuHpidal ninthic curvo in tho point (y~Q> js^Q, zy = 0) n,.Mni-.{ and in tho last-mentioned four points; in liict, writing in the equal;! curve 3 = 0, that is 1 + 120' = 0, \vo find w f ?/) w = 40, M0J, 1 + 120*) = 0, if - 82ww 0, 41G A MEMOIR ON CUBIC SURFACES. [412 of these lines, neglecting the terms which contain ce w, y-\-z conjointly in an order above the second, the equation may be written 82w" (z-y) ( + w) as (w - w) (z + y) -I- iw (2-y) 4 (ffi-w) 9 + f*(*-y) (<e - w) (s -|- ?/) ,.. . ~ ^' (*-#) ( 2 + y)^0, this is ^ (^1, 5, 0$fl;-w ( 2 + 7/) a =0, vo, collecting the terms and reducing the values by mean* of tho c<matioiiH = Q,z + y = Q, or ^y by writing w = W) -y = e, we have -, . Hence the condition 440-fla-n rt f n i points (if any) would be at the :^' U !'. 1 -.|7' 1S (f-8)' = 0, 80 that tho = 1, - 2 A/2, 2 V2 1 But t. , ' ^ + * = ' ^ - 8w ' J = ' "' 8 y >' ti S (y tho equals' I^s W- ilf/r,/' " f> JV = 12 ' ]2 ' - 1C i l "' ' J..WUVAVUI me obviously no f /in tl r " "" v> fu " 4 ' vo "J 1 - com l )0 '^ the cusp ^^^^ *% to on tho The line y=0 w = * pmoh-pointe. J>eing part of the cuspidal curvT no ^ t^ ^ * vefy poinfi a pinch-point, but M, in regard to this line also we' have i'-o A i ^ rGgarclGcl ^ ft pinch-point; that -? ==0 - ' ? Aud thorefom for tho ontivo nodal curve A iMUMOW ON (HrillO t ,H. Tukn iiin -in,, IIH Um 1'ootn o|' Mm equation (i i I, Mien Mic jiluncN are A'-O, wi - 4fH, HO tlmt % K o, t J) ] }' % ! A r 0. 1. ll ';' A'.f ir r= 0, [ .'!.] r! (JH, - 1 -v, : S r " (/ a i A; ' i/'i r O^-i) \ , H'J [ 25) r (.-!) ^ . 4(i| 1 7y(t 1 LIU. It And tlic ines un , h I/"".! (tquutiniiH miiy lu 1 \vi'i(lcii 1 (7) A' =0, l r (I i t) () ^ 0, I' 1 I) n (!') II' O t J': 11 1 I 1 (7) r-1-^.i.A o, ir n 1 1 1 t) () r^A'-i- ir ii, ^ r tt 1 1 I <!>) )'.i./r.i. ir o, A' 11 7/1, 1 I 1 (i) r <., D-r, C , (j i)^ 1 . o I//., 1 1 1 till -" I (B) r-:( M , a l).V (, r .|)^f 1 I HI, - 1 1 !,.- 1 () y (wj-.i) ir .( W| IJA* I 1 w., - 1 t) 1 (i) 1' OH, 1) ir, (m.-l)A' 1 1 1 (fl) r (m, i)jsr *( Wl i) r __l_j 1 1 <rt, * v |IJX ^ r 418 A MEMOIR ON CUBIC SURFACES. 412 Section VIII = 12- 3G' a , Article Nos. 117 to 125. Equation P+ 7(A r + # + W)-\- 117. The diagram of the lines and pianos is Lines. |O H td| 031 -)| I=! 2-3<7 2 . M. X X X to M iK tOl M II II -ql bj w LO 7 \" __... I'luiU'H cuuli LoimliiiiM 3x2= C alotift an iixia nntl (, ' (ion tiilnin^ Uio oiii'i'ti- " HlHUUllllH LriUlHVIH'Mlll, 12 34 8x2= 6 1 ' ' Bli'ftdml plnmiH onoh t (iontiuiiiiiif two nivH G6 " ' . of tlio Hiimu innlit. 18 10 26 6x4=24 ( 1'lfinoH oiiuli {ie)iitniiiiii (in axiH.nnd two myH ItiroiiKh tho Utnnliial 4(i nlii mniiooUvcly. 36 ' 80 1x8= 8 , . Piano Uii'oi^li Uio tlirco ItXliK. 89 1x1= 1 H 46 . I'lnno tliroiieli tho three trnnHvoi'Balfi, _ f 3 H W B* H m fi Cul-i m CA J" - - J~^ (3 S =71 (6 S> w 5" fr si 05 tfl B E. tf *o S S ff 0!) a Q. S * i O W- C ' 9 1 o E, 09 - ~~ ~ <1 tD CO ~ -_ ~ , O A MUAIOTU ON <!IMW! HIMIL-'ACJUH. ... UllH 1H -i- <r.' J (('I 2 7 -I- 2. f((i 7 ' J - (tv - i)' J (; ~ w) g (0 ~ VW)' J ^ 0. W. Tilt) nodiil Miii'vo i^ mini.) up uf |,ht! liur (t/ -.-.-. t ,t }.* ), (y ,>=,,; ,- 3 , w ), KwilH of Mm llnvn H o Hhow thin I mi nark Lliul., writing l-lio (!i|niLtion uf Mm Hiirfiuui may bo wriUun 4y (// - j) (/y - *) ( (V - w ) + . (,'/ J (12/8'fi' 7 ') -|. rt , , iH 7 'fi' -|. 278'*| I- 2 |; V (- <i#'*S' ). 2/j VK H. !) 7 '') - h V + ^7^' J - W) (.'.' - ?)' J (rtJ - WJ)' J (j H/)' ra 0, whoiHfo ttlwui'vhitf Muit 7' JH oi 1 t,lin unl.'i- I inul ^' !' |.)iu onlor 2 in (. i/), (;..- y ) ronjtiinlily, i<m;h U'l'i.i tif Uio (]tliUinii in ul, Inmt uf the wrrun.l unlnr in (;-'/). (j '//) itnijointly; or w hnvti y-,!^;, a iiu.lal linn; an.) Nintilnrly Mm nth,-, Mvn ' MIM-H urn uoilal liin-H, 121. Tho fim^oiii^ LrniiHluniidt i|iiu|,i.in is must iviwlily oliUiitotl by vnvnrtinjj in tho cubio in 2 1 , W, vi. writiitK ;i ".*!/. ;*-//, w iV( un.l LhcrHnro * + ), -J^'/H-r, wy*M, Mnj ttnlm; f'uncdi.ii duaLi>f tin-mill J'm V-\-ylf) I { K+ ////) 1^ -|- ( K- / V writing /3', 7 ', 6' m Jt + ,- -|- , /-!; -h r. ;/, Uiu fut.'l1i<nciiiH unt (,'( (,t -i- 1), - // f ' HIM! tho uquation of tho Hiii'laou in thun ohtainml in tin; I'unu 27 <. + ])"' which, arranging in powera of . ad ravoraing thi mgu. in tho fnroguinu on result. 420 A MEMOIR ON CUBIC SURFACES. [4.12 120. The three transversals are each facultative; p'~b'-3\ t' Q. 121. Hessian surface. The equation is The complete intersection with the cubic surface is made up of tho lintw (F-0, X = Q), (F=0, Z=0), (7 = 0, W-0) (the axes) each twice, and of a aoxbic oiirvo which is the spinode curve; o-'=6. The spinode curve is a complete intersection 2x3; tho equations may in iuoli bo written P + P (X + Z + IF) + 4&XZW = 0, the nodes D, G, A are nodes (doable points) of the curve, tho tangnnts at cnoh nod,, being the nodal rays, Each of the transversals is a single tangent of the spinodo ourvo; in mt f..r e transversal 7 + J + Z-O. F=0, these equations of couL satisfy thi equation of fcllt ^ ' the equation of tho Ho.snian, wo havo = > ^-0, F = is a point on the axi.s IF=() r . i i . , ' J v " '" MU"ib uji me axiH r = () ) = n not belongmg to the .prnode onrve ; wo have only tho point of contact .F + ,T-U = o' M' = 0, JC -^=0. Hence ^ = 3, ' Reciprocal Surface. 122, The equation is found by means of the binary cubic, viz. writing for shortness this is a binary cubic (jy f ff)", the coefficients whereof are 3(0+1), -%-j8, o^ + % .3^ and the equation is hence found to be = 878-} ^ A M15MOIU ON (MJIJIC) HUIIKAOEH, 127. Writing ( ( /,, b t o t tl$X, Kys-^/.A' - Y)(f.X- V) (./',, AT - V ), Mm [ 7] [ H] , , ,. and tho ILIUM art! /, A' !'(), [2fi] ,/U'- y0. |;i(i]; A' - 0, y o, (()) /,A r - K(l, % -0, (I) / a A r -y0, # 0, (2) /aA'-y-o, ^ -o, (:i) /,A r - T.--0, irt(>, 4.) m-u I2H. Thovo in no lli<!nltulivn linn; p' liif), H'dHMitui mirliini!. Tlio c(|tiii|,inu X %]\r(oX -h (/l r - HA" (j H. bluit Uid JluHHinu liivukH up int.ii ill.- into a unMo nf Mm Tin! i:iniil(i| ( n Mm lino A r (), F() (Mm iixiw) lour t-iim-H; Uiu HphmiUi c.iirvn; o'^.S. In that c and tlm oi|iiuliiHiH (, 6, (t. (/'ox, rr and ulnu 9' = . 0. 0, (axinl .i- <-.iinmnu l.m>luno) witli thn o.Hblu sui'lmi.s is i.uulc np .if J nl' u Hyntom nf tnur wmit-.n, wliinli in f" l,li , ry o, order wliiuli lnvakn into four <smicHj fur wo <tuu fnnn thn two i!i|imti<inH (a, 6, 0| dJtX, y)*(oA r -Kn'}'hHA r (o-6. that IH (4ao-afc, ad, bd, cd, tF&X t y )'(), a system of four plnncn each inUswisntiupf fcliu cubic A / ^lK + ((/ J ft, a, tl$X t K)>0 in tho axia and a conic; whence, a* above, Bpiimdti curvo is four conies. It ia oaay to HOC that tho tangent planes along tiny conio on tho tmrfoco JIUHH through a point, and form therefore a yuadrie cana; henco in particular tho wpmocto torso is made np of tho quwlric conca which touch tho Hurfucu along tho four rGBpootively, 422 A MEMOIR ON CUBIC SCTHVACI'H 125. The cuspidal curve is given by thn equations S(ft + l), -2ay-& ?/- + '? or sa}' by the equations that is a ( - 3) f + 4y - 3 (a. -|- 1 ) 7 0, and ' 3 (a + 1} S H- (2y H- $) ((/* -|- iy) =3 Q I consequently o' = 6. It is to bo addod tluit the section, 2x3. Section JX = 12 Article Nos. 12G to 136. Equation 126. The diagram of the lines and plimoH in 41 a oumi ix a ,n,mIulo inln- ' Linos, IX 1 2-2J? 3 , -^ en X M X CD || II "^1 CO II (O 1x0= ' 1 Ooinmou hljilnuv, OH. oitlfiriiluiiKtluinxk 7 , 8 2x6=12 I Otlior bii>luium of tint tWO blllOtluH I't'lJllllot. ri ivol.y. <8 ( 25 3x9 = 27 * I I'liuiCH onoli Hiionuh llioaxlmuuloonlnlii. 30 ' ' jii({ nyn of tho two 45 11? !r P si Cl| o f K-&^ 3t w, a (0 H o ?& ^ F rt (B 7 ~-~ _L " "-""I-'.I.I.M 412] A MEMOIR ON CUBIC SURFACES. 425 133. The cuspidal curve is a system of four comes; in fact from tho preceding equations written in the forms (ad -bo, eliminating zw, we obtain t l) a =0, 2 (- ad* - Sicrf + 4c a ), \CtfGCb ~T" u (L -"" jD(s ff 6 (6c arf) 6, = 0, which shows that the cuspidal curve lies in four planes, and it honoo consists of four conies; these are of course the reciprocals of tho quadric cones which touch the cubic surface along the four conies which make up the spinode curve. 134. The equation of the surface, attending only to tho terms of the second order in y, g, w, is 27dV,sw = 0; it thus appears that the point y = 0, &**Q, w0 (reciprocal of the plane X = 0) (which is oscular along tho axis joining the two biuodcs, or .SB-axis) is a binode on tho reciprocal surface, the biplanes being z = Q, w = 0, vis:, these are the planes reciprocal to the binodes (A' = 0, Y=0, 1^=0) and (X = 0, I 7 = 0, #=0) of the cubic surface ; we have thus B' = i , It is proper to remark that the binodo ?/ = 0, #=0, w = is not on the cuspidal curve, as its being so would probably imply a higher singularity. ISO. A simple case, presenting the same singularities aa the general one, is when a = d, 6 = G = : to diminish the numerical coefficients assume a = d = -fa, the cubic surface is thus 12^^^-!- A' B + P= 0, mid tho equation of tho reciprocal surface, multiplying it by 4, becomes viz. this is the surface (a; 3 - f) = 0, (o (a? + Szw) (3ffi s + 0w) a = considered in the Memoir " On the Theory of Reciprocal Surfaces." is, as there shown, eomposed of the four conies y = 0, 3a; a + sw - a 3 - ssw ; and it is there shown that the tw o points (to = 0, y = 0, z 0) each reckoned eight times, are to be considered as off-points of the re C. VI. 424 A MEMOIR ON CUBIC SUIIFACEH. 412 Reciprocal Surface. 130. The equation is obtained by means of the biimry wibio XfaX + yVy + faufa, b, o> d$X, )') ;1 , viz. calling this (*jA r , F) 3 the coefficients are The equation is found to be 432 (fftp - Galcd + 4c 3 -1- 216 [(ad* - 36cd 4- 2c 3 ) ^ + 9 [3tfV - 12<K&fy + (lOh? + Sc a ) ic-f - (lad -I- Me) w?/ 3 + (4(w - ft") i/'J j - I/ 3 (c/ft? 3 - Scflft/ + 36ffiJ/ 3 - fl.7/ 3 ) =s 0. The section by the plane w = (reciprocal of # s = .0) is tho of edge) three times, and the lines w = 0, eta 8 - Se<ify + B/M^" biplanar rays). And similarly for the section by the planu c The section by the plane ?/aO is made up of the linos each once, and of two conies, ?/ = 0, (), // (ruri i|miiial j . 0), i|>mi'iil of thi 16 (a 86V) 4- rfW * 0, 181. There is not any nodal curve; &' = 0, 1S2. Cuspidal curve. The equations may he written ? Forming the equations (od - lo) . 144*V + (Sate" - 2 Cffl y - 6^) . ;i 2 2W a ^a , these are two quartic mrfiu* having in common tho HnoH (. 0| W 0) 0/-0 -0). attending to the he (w = s^tn tin's k n M, \ H/ - U J' ^^". * M ( U : : 5 u ' * u * tms 1S on "w second Hiii-fuci) mi oHoular line, ' on fche fil ' st "&ce it is a nodal line, tho ono tangmifc t ,lano boi, W .^ 0| the other t,ngent plane being ,,0, but tho Huo being in regard to this sheet an oscular lino, ^ 42G A MEMOIR ON CUBIC SURFACES. [412 136. The like investigation applies to the general surface, and wo have fclnw & = 16 ; the points in question are still the points (a = 0, y = 0, e = 0), (ss = 0, y = 0, w = 0) ; viz, these are the points of intersection of the surface by the line (us = 0, y = 0), which points are also the common points of intersection of the four conies which compose) the cuspidal curve, that is, they are quadruple points on the cuspidal curvo ; it does not appear that the points are on this account, viz. qucl quadruple points of the cuspidal curve, off-points of the surface, nor does this even show that the points ahotild be reckoned each eight times. As already remarked, the singularity i-oquiros a uioro complete investigation. Section X = 12-^-(7 a . Article Nos. 137 to 143. Equation WXZ + (X + Z)(Y*~ X*) =0. 137. The diagram of the lines and planes is M FO M Lines. w o X=12- <]l M X 1-J II __ X if X *>. II co X en II H" X CO II CO 1x12=1 * Biplane touohtiiK along fixin, and containing odgo, 3 1x12=12 1 Other biplano. g 22' 1 1 2x 8=16 Planes oaoh through iho axia and oontainiiig a hi- planar ray and a onieno- dal ray, 8' lx 3= 3 * ' Plane touching along tho edge and oontnining tho more lino. 6 45 - _ Bimdial plane through tho two cnionodal rays. & ID CmcnotJal rays. Biplanar rays in the non- axial biplane. Edge of binode, being a transversal. O $> l> O a < I A MEMOIR ON CUBIC SURFACES. 427 138. The planes are and the lines arc X = 0, [0] A r = 0, F=0, (0) Z = 0, [3] A r = 0, #=0, (3) x - r- o, [irj x - F= o, z = o, (i > A r + r - o, [22'j x + F= o, z = o, (2 > Tf = 0, [3'] A r -F = 0, W = Q, (!') X+^-0, [1'2'J A r + F = G, Tf = 0, (2') A r + # = 0, W**Q, (12). 139. The facultative lines are the edge counting twice, nnd the mere lino ; p = Z>' = 3; i' = l. 140. Hessian surface. The equation is X (X + Z) (ZW + 3 X* ~ XZ} + F* (A' - ^) = 0. The complete intersection with the surface consists of the line (A r = 0, F = 0), bhe axis, four times; the Hue (A r = 0, #=<)), the edge, twice; and a sostic curve, which is the spinode cui've ; c' = 6, Writing the equations of the surface and the Hessian in the form X (ZW + F fl ) - Z* -h # ( F u - A' a ) = 0, we see that the equations of the spinode curve may bo written viz. the curve is a complete intersection, 2x3, Y f Z\1 y f 7\n There is at B t a triple point ^^-f^-J . w == ~(~W) ' Mld afc a n double P" lb * the tangents coinciding with the nodal rays W = Q t F" A' a = 0. The edge and the mere line are each of them single tangents of the spinode curve. But the edge counting twice in. the nodal curve, its contact with tho spinode- curve will also count twice, that is, we have $'=2.1 + 1, =3. Reciprocal Surface. 141. The equation is obtained by means of the binary cubic 4wZ (X + ZJ + fa)Z (X + Z} (&>X + zZ) + tfXZ* ', or calling this (*JX, ^) 3 , the coefficients are 542 4-8 A MEMOIR ON CUBIC ;uid thence the- equation is found to be ft' - 5*) f--2 (as ~ 2*) (a; - where the section by the plane = (reciprocal of bimxlo) in the line ^0^ = (reciprocal of the edge) four timon, and the (reciprocals of the biplanar rays). _ The section by the plane ,= is found to bo * - ti <. >>- v\ r rhi, we have thatis, n .,/, ,, two values, . 143. Cuspidal curve. The equations are pwon " complete = onco, A MKAKHlt ON <U!|II(! HMItl-'AOI-X Hwl.ion XI,:, |2- ff,. Arl.idn NUM. ill m 111). Kpml.i.m \\' X Z *\- K 3 #H- A'" Tim itinerant i' rim liaoH run! plnui'H is !-! H r-', M ii || II !'i' P ,p p 1 .1 a t' !' 11 F t rf Nj Nj li ii "3 II IF O 'S> . K. O XI li. ./V 0=1 U, ~< >: . t.' II Ol 1'lmii'n nic li h-1 -. i II i~ ?ji .V' (1 i] ] ;-: Ifi 1 T. Oxiiiilni' l>i|ilniii'. .V II It !:.. m Ol'diliinj,' lii|i|illl( l . a -ifi ,,.{? N b w r-' |;;; i ^ n ' flj IIM oj 1 Ihn li ni'N utiit | llUH'S lill! .nllUWJI j i tlin margins < f l.lui Uiu^v gn in a lilt nlliiUvii 1 till' (''Mill liitf Ilin "' L.iiitc.4; ih in vilt npjinitr 145, Tim l.lw ilimtiiHKion ni' lint )v(.!i|ivoi'iil nm'liin', Tlit^ri-Hini (>' ft' -. M ; i'^. ]. 14(1. Hosriiuii mivtiuio. Tliin in ii]i iiitti #(), Mil! iiHtiular biplniii', ami intu u cnhio Hiu'l'uco (itsnir n, Hiirlhai XIil2 /i,,). Tlio mtiiijiltitii infcijiww.tion with tho itl)in HiirDw^ IN nuulti up |' (Jin lino A'"0, ^r<() (tlio wl^o) nix linion, uiul of a iVHidiml HxU (^^ M (tonioH), wliiiih in tfiu s[iiit(nlo (inrvij; o' (I. 'Hiu ctipiatiniiM uf Liu.) Hoxbiu uro in I'mit A r ^H-r j :^0, A'-' 1 -f- J?* se 0, HII that Lhin !ti- uf tlirtsu ituiiifH, uiuili in a jilaiiu paHsing tln'ciu^li UK; (.td^u. Thu otlgo timohoH enah of bliu fchrou nonicw tit thu puini A'^0, J?f ii nuiHt bu ixickuijud jw a ningla tiuiguiit of tho Kpinodu uurvu, and ihmo 430 A MEMOIR ON CUBIC SURFACES, [412 Reciprocal Surface. 147. The equation is obtained by means of the binary cubic . . L , VIZ. It IS 482w" or, completely developed, it is + w" ,r 148. The nodal curve is the Hue = ft w - th ^- & In to the fo - ^' Vlz - ther e are two aheets oaoulatinir alono- in thi S H Qe taUen three tin,, " 3- For the cuspidal curve we have giving = 0, the = 0, . ' """"" - ' - * - r ,, ,, 6*" s= o 412] A MEMOIR ON CUBIC SURFACES. 431 Section XII = 12- U s . Article Nos. 150 to 156. Equation W(X + 150. The diagram of the lines and planes is XYZ= 0, Linos, XII.12.-tV W to M W 10 M OTl W w X X M co 1 to * 1x32 = 32 . . . Uniplano. a o) 3x 4=12 Planes uaah touoliing along a ray, and con- taining a mei'oliiiQ. 3 ; Ix 1= 1 5 45 Piano through the three more linos. i I s a> ta t ! p EJ' tf (D 161. The planes are X +Y+Z=Q, [0] A'=0, [1] ^=0, [2] Z = 0, [3] W=0, [1'2'3'J The lines are o, (i) = 0, (2) -)- F0- ^ = q, 0, (2') z = a, w = o t (3'). 162. The three mere lines are each facultative: ^ / = &' = g; t' = l, 153. Hessian surface. The equation is - Jl "- J via the surface consists of the uniplane X+F+# = twice, and of a quadric cone having its vertex at U 6l and touching eocli of the planes X=0, F= 0, #=0. 132 A MEMOIR ON CUBIC SURFACES. The complete intersection with the cubic surface ia imulo up of (I|H; my* omsh l,wi< find of u residual sextic, which is the spiuode curve; o-'=(>. The equations of the spinode curve are X* + F + #- 27^- 2&L' - 2XY = 0, vix. the curve is a complete intersection, 2x3. Each of the mere lines is a single tangent (as nt imoo appoint* by writing fur instauce W=0, Z = 0, which gives (7-^=0); that is, #' = & Reciprocal Surface. 154. The equation is found by means of the binary cubits - writu* for shortness 4 ( r - Z 0(^0)(^'D) /3 = rc + 2/ + 3j 7 = y + zts + <sy t _ then the cubic function is ~* (12, W _4A Ay, -12SJ2', aud the equation of the reciprocal surface is found to bo 432 S 2 -I- 64 y (w - 4/3) 3 S + 72 (w- 4/3) 7 $ expanding, this is " ( w -*/?)V0; w'.-S - above. 12 J A MEMOIR ON CUBIC SURFACES. 156. For the cuspidal curve we have 12 , w-4/3, w - 4/3, 4y , 48y- ( w 433 or say whence the cuspidal curve is a complete intersection 2x3; c' = 6. Section XIII = 12 - B., - 20,. Article Nos. 157 to 164. Equation WXZ+Y*(Y + X 157. The diagram of the lines and planes is o H> to ( W Linos W M o O5 CII Col M M M to X to II fii II X II to X IT M 10 1 2 2x = 1 ' Biplnnca. OCR 1-1-12 = 1 Piano through the three axes. C s c 1 2x 0=12 * Pianos each through an axis joining the biuoclo with n omonode. M Ix 4= 4 Piano through the ftxis joining the two onia- nodos. 12 Ix 8= 8 ' ' Planes through the hi- plnnnr rays. x 2=i S 8 46 ?lano TransversaL Cnicnodal rays, one through each cnionode. Biplauarrajs, one in each biplane, and being each a transversal Axis joining the two cnie- nodes. 1 I J Axes, each joining the bi- ! node mth a, enicnode. | O. VI. A MEMOIR ON CUBIC SURFACES. [412 158. The planes are The lines are *-0, [1] Z = 0,7=0, (5) z => [2] ^ = 0, [056] Z = 0, 7 = 0, (S) 7 = 0, TF = 0, (0) X = Q, 7+^ = 0, (1) = o, r+^-ao, (2) > [6] 7-F=0, [34] W=Y=-Z, (S) X + Y+Z-Q, [12] F=7=-l r , ( 4 ) lf = - [] W = ; X + 7 -H Z = 0, (012) 159. The transversal is facultative; />' = &'=!, (' = 0, 160. The Hessian surface is The complete intersection with the surface is made up of the line 7=0 X (fl*r three tnnes; the line 7=0, ^=0 (^-axis) tlL times; li.o F=0, (CC-axis) twice, and of a residual quartic, which is the spinode curve ; ff ' = 4, 161. Representing the two equations by U=Q, 7/=0, wo have and (37 + Ar ^)^-^^ a (^ a ^7Z+47^ + 4^X -JfT- suppose, ^^ whence - + 28^^+80^) 27P (,Y + ^ 27 (X + ^) fr+ 9 ff + (_ 9X7 or, as this may also be written, 277' _ ^ )}.0 J and we thus obtain thp pmioi-iA ~e >.i. , , equat.cn of the res.dual qu ftrt io, ov spinodo curve, in the form 4Z + ) = '* l ^J A MEMOIlt, ON OUJUO HU.HFAOIW. 435 Thu Hpinodu uurvo is bhiiit a uomplotn intm'soolion, 2x2; HIM! ninao bho lirab surface a comi having UH vni'tox on bho Hucond Hiirfiujo, wo HUD moreover that tlio npinodo J'vo iH a undiil (|iuitl)'i([(iiidna IiiHtoml of bho Itml; (iquabion \vu may wrlto inoro om-vti 0. 'lilt! iH[ual,ii)iiH nf him iionH iif l,lu> HpiiHiilu uro IK^ 0, A' -I- Y -\- %=* 0, ami HiibHti luting in vti \vn obtiiiii from ouoli ixjualion (A F -#) 3 0, thai i, l; of tho 1(12. Tim (tiiuulion of Uni oiibic m dorlvod IVoiu l;hab boloii^ing to Vlralg-J^-f/., hy writing thin'itn u**b - 0, o^, (Z 1. Milking bhiM cluuigo in the Ibi-inulio for b1u> iHKiiiM'iJt'iil HurOitio if tin.- cumi jimt roHiirtuI lo, u-u huvn JV ..a - 4;H 1J , V . l(U- 1J ami HiihsliUitih^' in llir i<i|iiaLion =- 0, (<i|iiiiLion divuioH hy c^; in- Lliruwinjf IhiH oiib, thw n thw i !l . Hi ; + w { ?* -H 45 s ) 552 436 A MEMOIR ON CUBIC SURFACES. The section by the plane w = (reciprocal of B 3 ) is w = 0, y=0 (the edge) three times; and w = Q, y~x = Q] w-Q, y~s = Q (reciprocals of the CB-axes), 103. Nodal curve. This is the line y = x ~z\ wherefore &'=!, To put the line in evidence, write for a moment x = y + ct, z = y + y, then the equation is readily con- verted into = 0, which, each term being at least of the second order in , 7 (aj_y ( z -y\ exhibits the nodal line in question. 164. Cuspidal curve, Multiplying by 27, the equation may be writtou (1y-3x-3z~5w, -y + Qw, - w% 2 + 162/zy-12aw-12swj + Uw\ - 20/ + 2%aj + 24?/2 - 27^ - 8mo + IQw^ = 0, where and we have thus in evidence the cuspidal curve, f + 1 Qyw - 12 (a; + ) w + 16^ = 0, - 20/ -I- 24y (* + *)- 27a - 8j/w + IQ W * o, which is a complete intersection, 2x2, or qadriquadrie curve; o' = 4. 412 A. J1KMOIU ON 437 Suution XI V~ 12 - Ai'ticlu NOH. 1(!fi to 171. ISipiatimi Tho diagram of tlm HIU>H and plains in ^ N i II ii p II p II p <n *i M N '-I In (fl II 11 o o P ro w .0 M o il-l H M M M \ I V ;.: 1'' . - // ( ' X x >: >; H M 1-1 II O o ii II l-i M II -1 1-1 Cl O O I'lnntiH nro /:..-() I lxlfiM ... Tin-Hill hi])liini!. A':.:0 01 1K*., Onliimvy l)l]ilfuu', r. :0 onit 1x10:- 10 ' I'liinii lluoiiuli uxin niul Din Lwo riiyH. 'ii -i/i O ft) M 1" (3. *^l' P.. H j p? 9.1 !"' 1 (T '?' "pi.U 'i'o l.lii 1 i'i|naM(iim of tlm jtlitiiMH niul MUCH n! Hhitwn in l;hn l(i(i. Tho <tlg{s IH a tiic.nltal.ivii lints HH will appear I'nua tlm diwiussiitn of t.lm 107. IlcHHiaii Hurfiuus. The equation is m'# u + )'"^ - 8 A' 9 )'# -I- A' (I. Tho (intnploin inturHtiOlion wifih the Hiirftum IH intultt up of Uio lino A r 0, K (tlut nxw) live liuiow, the lino AT0, ^?^0 (tho eilgo) four tinuiH, and a nkaw nuliic, tho orpmtionH nf wliinh may liu wriLlon A', I', If 0. , A", -5F Tri fact; frarn th oipuitiorm CT-0, jff0 we doihico H-ZUm X* (X* + *Y%) Q \ and if in Z/0 we write Jf41'J8f ( it bocomcs ^ (AT IV + :)') 0; and thuu in nU~i) t writing 5F a ^ A'TK, wo have 438 A MEMOIR ON CUBIC SURFACES. [412 1G8. I say that the spinode curve is made up of the edge X = 0, Z= once, and of the cubic curve; and therefore a-' = 4. In fact in the reciprocal surface the cuspidal curve is made up of the skew cubic, and of a line the reciprocal of the axis, being a cusp-nodal line, and so counting once as part of the cuspidal curve : the pencil of planes through the line is thus part of the cuspidal torse; and reverting to the original cubic surface, wo have the axis as part of the spinode curve : I assume that it counts once, The edge is a single tangent of the spinode curve; /3'=1. Reciprocal Surface. 169. The equation is obtained by means of the binary cubic tw& (Xas + Zz} + X ( YZ - wX}\ or, as this may be written, (3w>, -2yw, f + 4*ow, 12mJ_X, ZJ, The equation is in the first instance obtained in the form tys (f + 4& * (f + 4,-CTy) but the last two terms being together - 4w s a (y 3 + 4flw)", the whole divides by and it then becomes -f 4 + ffi(2/ 2 or, expanding, it is w" . The section by the plane w = (reciprocal of 5.) is w = 0, y0 (reciprocal of edge) four times, together with w = 0, ^ = (reciprocal of biplanor ray). The section by the plane = (reciprocal of ff.) is (f + 4flw) = 0, vi this is *-0, ^ + 4w = (reciprocal of nodal cone) twice, together with *0, w0 (reciprocal of nodal ray). l 170. Nodal curve. This is the Jinej y = 0, y-0, reciprocal of edge. The equation in the vicinity is ^~^^/~~w^ showing that the line is a cusp-nodal line counting once in the nodal and once in the cuspidal curve; wherefore &'!, 412 ] A MEMOIR ON CUBIC SURFACES. 439 171. Cuspidal curve. The 'equation of the surface may be written where 4aj . 3w - y* = 1 2a?w - if, This exhibits the cuspidal curve I%aw ~f = 0, $zw + 8a# = 0, breaking up into the line ?o = 0, y = (reciprocal of edge) and a skew cubic; the line is really part of the cuspidal curve, or c' = 4. The equations of the cuspidal cubic may be written in the more complete form 8a; Section Article Nos. 172 to 176. Equation 172. The diagram of the lines and planes is = 0. tsj H o b B 1 o o o (D w ta ^ XV = 12- r/ 7 . Wl M M M X X X |_A o ftnos aro 10 -ll M o M Ci XssQ 21 . . 1x40 = 40 * Uuiplano. #=0 28 Ix 6= 5 * ' Plnno touching along the single ray. 2 IF to- K H S? B' i p V e. P 1 4 where the equations of the lines and planes are shown in the margins, 173. The mere lino is facultative: p' = &' = !; f' = 0. 174. The Hessian surface is viz, this is the uniplane Jt = twice, and a quadric cone having its vertex at U~ yt 440 A MEMOIR ON CUBIC SURFACES. [412 The complete intersection with the surface is made up of X = Q, 7=0 (torwil ray) six times; A' = 0, = (single ray) twice; and of a residual qu'artic, which is the spinode curve ; a-' = 4. The equations of the spinode curve are XZ-Y* = Q, X W + 2# a = ; the firat surface is a cone having its vertex on the second surface; and the curve 'is ttuw a nodal quadriquadrie. The mere line is a single tangent of the spinode curve; /3'=1, Reciprocal Surface. 175. The equation is obtained by means of the binary cubic viz. throwing out the factor y, the equation is w 8 (- G4s; 3 ) + w (- 16aV + 72<e^r -h 27?/ 1 ) + %V = 0. The section by the plane w = Q (reciprocal of ff 7 ) is W = 0, ^0 (reciprocal of tori ray) three trmes, and w = Q, y = (reciprocal of single ray) forioe. llMlplOClU Nodal curve. This is the line .^ = 0, ^0, reciprocal of the mere lino: 6' = 1. Cuspidal curve. The equation of the surface may be written whera ( 64a > " 162 > -Swja' + aflw, OT,* 4 . 64a;(- 3w) - 25G* 3 = - 26fi t* 176. {The equation resembles that of a quintic torse, vi z , the equation of a quintie torse i s y j zztsf = which equation, writing 9y for y> -8* fo r , and ^ y fop W) or, what is the same thing, and developing, this is only ia tl ,c A MNMOIR ON (JUHIC HURFMMS, 441 Huotion XVI-lii-'l.a,, Arli<iUi No. 177 to 1HO. Munition ll r (A r r-|- XZ+ Y 177, Tho iliiigniiM ol' Uio UIIHH und plains in ! !' -I-' ||' ||' |f if if || "-I N N , P P P P o P P P ,' M K N o N -1 ^ II II II O O o =1 O O O O if- lU ,u w ii to XVI.. M'J.. I'huitm nut lr: 8 . l W X II ^1 W X tl- li '/,-\" IIV.'II IU r-l- M r ;i|) III . r-i-tf no n A-'l-ll'^O Mil llx'J.-U'J I'lfuii.'H cnoli tmiiiliiiiK nloi itn nxiH, niul conlninii n tvuuHvui'mil, A*.I-XMO ui ' A'-M'-O ill .v.-.o i * > \ :-..0 U .U.H,,,, ! ' . I'lftllCH (Jllflll Ull'Ollull till' ftXOH. A'.|. r-i-x-i ir-iit iviin 'ili . . - I'luiin tlirou^h llio llu p i i jr. .' .' I Li'niinv(ti'Hnln, H ||| o B Q * L iB-'el K M^ ffi is rt s !- a E, o f I whoro the oi|uatioiiH of the linc nnd planes arc shown in the margins, 0. VI, 442 A MKMUlll ON ('inili: m 178. Tho tmiiHVOiwlH nro wmli tiuiiiltutivn: /' // ;i ; /' , |, 179, Hosaian miri'mso, Tim njimtioii I'M *XYZW-(X+y+%\. II')(,II'AT I \\'XfS i mV? , or, what is the siuno tiling, ,Y(1^ >l I'll' i ^11') -i-^.v .(. ir.vi The complete intonation with Mm oiihi,, H.trliin, i, .MU,],. (t|( , twice, and thoro i. no H pino ( lo um, | ,'.-,. ^, w ..... ,, i i 180, Tho equation w inu.u.tmtuly uhiniii,.,! in ,| l( , ,,!.. . . /';! /v 4'^- 1 -\Vt,(t, oi 1 ratioiwhfflng, it i.s (^+^ + ^-^. ,,,,.. .,,, ...... ,,.,,. ^, so that this is ;,, | llct , S t oilU)1 .. H , 1111H . tin m|| . |i|( , t , Nodal ourvo. Thin c IIK iH|, a (lf u,,, n,,,.,, -,. = (); |i ""' l '''-". 'To pit any ono of tl mi ,, f m . ,', ,,,,, wnto tho 0(1 ,,ati, m ,,f t l u , Hrtitt, in lln, f,,,',,, * '" """""' '" " vi>l ........ ' w " There is no cugpidftl curve; '-0. 412] A, M18MOIU ON CUBIC! SUBFAOR8. 443 Hoobion XVn12-27f fl -6 Y fl . Avtiolo NOH. 181 to lfi. K<iuiiti<m WX%+ A 181. Tim diagram of tho limw and plauon i,s '" = 0. n n ll' ll' ll' r O O p p P B -t * ID H H 53 *N IV "> -1- ! if II n' a n it o ro 11 li O rt. W N l- o xvu-.ia y lly-'Cy. 1 X X X OS II CT1 II II I'lnnoH (mi -^ll O> IP M to A'nO ix Oe 11 . . . (Jin ni mm liijdiuio, tln'ongli tliii 'iixiH joining tho two hinodoB, tfi'iO lit ir-0 ill ax iiwia ' Itonuiinlng ItipliiniiH, ono for cnoli liinodu. r n o <ua ' 1 1'lnno Ihvoutth tlio tlivoo * HXOK. A'MV;0 Oil I lx .,-: I) I'lllllO tllVOIIRll lllO ftXJH fi 'ifi joining tho two liinodon. ffi "a* -6: WK " 3'^, O' _ p. So' ft H I* 6 -!. i* H' |'5! h ^ o (D P O E d rj- P CR A CO p*tr O whovo fiho e([imliuiiM tif tho HnoH and pianos arc Hhowu in tho nmrgiiiH, 182, Thoro is no fncultfttivo Urns; b'**p' = Q t i'0. 183. Tho Hossinn Burfiwo is X ( WXZ + 8 7ZW + X r j ) 0, vis*, this breaks up into JT0 {tho common biplnno), and into ft cubit -14 A MEMOIR ON CUBIC SURFACES. [412 The complete intersection with the cubic surface is made up of Z=0 t Y~0 axiH) four times, of 7=0, Z=Q and 7=0, F=0 (OS-axes) each three times; and of a residual conic, which is the spinode curve; a-' =2. The equations of the apinorlo curve t are F'-;JW = 0, 4^+37=0; viz. it lies in a plane passing through the /M-axis; since there is no facultative line, /3' = 0. Reciprocal Surface. 184. The equation is found to be or say this is 16V + (%" - 3<ky + 27tf) ^ + y 3 (y - a;) - 0. The section by plane W =0 (reciprocal of B^D) is w==0) y(w-)0 vi this is he^hne w . 0l -0 (reciprocal of edge) three tinL, and ihe line Li) y- v - o (reciprocal of ray) once ; and the like as to section by plane * = 0. ~ The action by plane .. (reciprocal of C^) is itf== o, (^ + 4^)^0 vi, this is the come (reciprocal of nodal cone) twice. ' ' There is no nodal curve; i' = 0. 185. Cuspidal curve. The equation of the surface may bo written i 1 ^ 5 aUd thei ' 6 is thus a G 8 P^ oonio p--ia,o Attending only to the terms of the second m ^ ftl , fe"*0j that is, the point V = 2 -o n y> *' W> tho e( l llfltio11 ^ocoinoa Knodo of the surface; or there is'the"^ 1 ^ 1 f the C mm " bi l 31 ) ^ <l 12 A aiKiMOIH ON (JUIUU 445 Snc.lion XVIII- I :!-/* AHiuio Nus, LSI! t,,> IN!). KqimUmi W X % -\- P* (X -\- Z) Tim din^nun of Mm limm iunl pitman urn 01 =8 o O 3 1* II I? P ii" Muno tltn (lin><! o mul unn- titinhiK I'ltuiu tflueliliiK ftloiiK uxiH tlm l\vu cnii'iHiili.'H uitil coutiiiiiiiiK line, whuro tho tiijuatiioiitt ol' tho lino and plunou 111*0 uhown in this iiiurghtu. 187, Tho inoro Him in fiuuilUitivo; tlu; erlge in ttluo fiwultafcivo counting twice (this will appear froim the dwcuwion of the reciprocnl aurfiice) : &'/>' 8, i'l. 446 A MHMOM ON CI'llli' .'iriM'.Vi f 188. Tlio UOHHWII Hiirfacii) in (.V'|./o ir.v/i i.v t\ \ .1 Tho flomploto inl(>rai'i!|.i<m with lJn< ntltir Miilti.-,- j, J n (tlio 6^-iwoH) ouch lour liinoi; I'.-n, II' <i ,/W/,ui,i ?v,i,^' odgo) twice. Tlusn; !H nu H]iin,>i!<> ntrv. ,' it; Hl h ,|.,|,,i.. ,,!-.-. ; ^ ..... r , ( ,. , . . f nodal cmio) bwi(H , . mitt Hil||i|(|r|y ( . (| , ^^.^ ; ' - -'., * -I n ' t * 1 J|'i*n-.!l3| t,J Nocl.il cum,. Writing U,, t , ( and writing tho cinmtiuu j|, t j,,. j;,,. or Kiiy ' J ' * " * M * s ^ :J|( * f/'.V/i'. Tho o(|iiutiou I'M thus (//*'|-4H'.r | IfCi-j-.. li.|[,.>- ... u oi 1 in an mutioiml liinii , ,. (li , () lour fcinioH. ' ','i> -I* ;/ ...n n,,. ,1,1,..., | ,,, 12] A, MEMOIR ON OUimJ 447 Suction XlX:i2-yy -6V Article No. !!>() to 1IJM. Kquatiuu WXZ+ UK), Thu diagram of l,lio linen and pliimiH in ll' .P ii 1 e N N P II o II ID O XIX1-. -,-, lil M M X Ul t| 1'llUlOH lll'O II -^ll Ol K - OHcnliu 1 biplfinc. fc o IK HO ^110 Oi'tltimry l)i|)lniio, 'H -ir' W fc, ;e of tbe biao c. 1 ?' S B 01' K. F JT> O" g' ft whcns fcho otjiuitiouH of Iho IhioH and pltuiun avo nliiiwii in tho I!) I, Tin; uxm to a ftwjulttvtivo lino cmuUin^ Lhrots tiuicH (aH will appouv from tho reciprocal Hnrfticn); f)'-~~b'~~- 1\, i'l. 1!)2, Tin; Histwiim Hurfticu IH vi/,. lihiH in tho iimulftv biplano ^>0 and a cubic Hiirfacu. Tho complete intovHocitum with tho cubic mirfaco is nuulo uji of A'^0, ^0 (thu odgo) nix timuH, and A r 0, K0 (tho axin) six tunoH. Thoro is no Hpinodo curve, cr'^0; whence nlno /3'wO. Surface, 103. Tho uquntioH is at unco found to be (f + 4ww) 0, 44B A Mi;,MnMt (IN Nnilnl curve. Tlit- i><|iiati>'H j;i\." ( Tlii'iv is iin rii',|iii!ul \\ llUlt' in X H ii UHI. TJ 1111(1 tlu ' i lll JH i in, 412] A MEMOIR ON CUBIC SURFACES. 449 Reciprocal Surface. 197. The equation is at once found to be 27 s 4- 4icw) a - G4fy = 0. The section by the plane w = Q (reciprocal of the Unodo) is w = t * = (reciprocal of ray) four times, There is no nodal curve ; 6' = 0. But there is a cuspidal conic, y = 0, a 3 + 4wtv = 0. The point 2/ = 0, s = Q, w = (reciprocal of the imiplaue A" = 0) is a point which must be considered as uniting the singularities '=1, #' = 2. I give in an Annex a further investigation in reference to this case of the cubic surface, Section Article Nos, 198 to 201. Equation WX 198. The diagram of the lines and planes is N *< *!} 1-1 o g xxi=i2-a 3 . II II II & o o o cwj w X CD Planes are r=o o #=o s ir=o s Oomi tin 1x27 = 27 t . 3x6 =18 4 46 . . . Honii oat o rt a' where the equations of the lines and planes are shown in tt 199. There is no facultative line; p'=ti / = 0, f' = 0. 200. The Hessian surface is XYZW=Q, the oomir biplanes each once. The complete intersection with the s each four times ; there is no spinode curve, o-' ; whence C. VI. 450 A MEJTOIH ON CUBIC SURFACES. 412 Reciprocal Surface. 201. This is 27^0-^ = 0, viz, it is a cubic surface of the form XXI = 1.2 ~ &#,. There is no nodal curve, &' = 0, and no cuspidal curve, c' = 0. Moreover .#'= 8. Article No. 202. Synopsis' for the foregoing sections. 202. I annex the following synopsis, for the several cases, of the facultative lines (or node-couple curve) and of the spinode curve of the cubic surface: aim, of tho nodal _ curve _ and the cuspidal curve of the reciprocal surface, It is to bo observed curve ffj* 8 6 ? 1 * a curve ' for instance > 18 = 4x6-2, this moans that it is a JZAnf 6 ', ' 1S) the I 31lrtial "^-section of a quartic surface and a quinliic fliuiaLt, out without any explanation of the nature of tho (Mint 01*1 causes the reduction, viz. without explaining whether this is a eonicor a^mir of Tinm SLl 1 " e1 ' """' "" ^ be ^ b >' 1 '*" CO to P^-loL of 2 Facultative lines. Nodal curve. Spinodo curve, CuHpldal onrvu. 1=12 27 27 12 = IJ x 4 ~" 2-i~(\ i "" 111=12 -IL 15 15 12 = 0x4 184xfila IV-12 W 9 9 12 = 3x4 104xfi--l V-12~ a 7 7 10 = 8x4-2 4 7 = 6 + edge twice 7 = 6 + I'ec.of edge twic I'co. of edge taonodnl 10 = 8x4-2 12=4x4-4 ~ 3 ~ C 2 3 n y TT 10 -n " s: 8 x 4 H lOsa-t x4~d3 s 3 = 2-f.edge 3=2H-i-eo. of edgo, = (3iko-M ' roc. ofedgoisouspnodii 841)10 Uffft 10=reo. of cdpfl H- uniQiirxal (t-Uifc, 111=12-3(7,, a rec, of o(l(|o ja oimpnodnl IX = 12-2B u 8 = 2x8 <!2x B x=i2^;, a none 3=l + edge twice none 3 = 1 + reo, of edgo twice veo. of edgo is tnonoda 8 = 4 oonlcs = 2x8 8 = 4 coniofl XII = 12- tf fl 3=edge 3 times 8= roo. of edge 3 times 'eo. of edge is osonodal 8 oonios O.^l.. 3 8 = 2x8 0-2 XIV-12 1 1 4 = 2x2, noddl qua- 1=2 xS qnndiiqundrio l=edge l = roo. of edge, ce. of edge is ciispnodnl ^Z 4=84-1-00. of odgo, xvi=i 2 -. 40a 1 1 = 2x2, nodal qua- drlqundvlo 4=2 x 2 ouBpulftl qnn- diiqiifuliio XVII=l2-2B s - 0a o none 3 nono nono SI 3=l+edge twice none 1 + i'ee. of edge twice, wo. of edge tnonodal 2=conio none 2 = oonlo nono XX=i2-t; fi 3= axis 3 times =i i eo. of axis 8 times, i'eo. of axis osonodal none noo XXI=l2- 3j5s - ,, none none nono 2=conio 2 ss. oonlo I pass now to ~ . the two fflooo n c none " _ 310110 none A MEMOIR ON CUBIC SUMTAdm 451 Article No. 203. Section XXI]: = (1, 1). Equation X a W+Y*JS = & 208. AH this is a Hcroll there is here no question of the 27 lines and 45 pianos; thoro IH a nodal lino A r 0, Y = Q t (&=,!) and a single directrix: lino, # = 0, TK=0. The Hessian Hiu-faco is A r *7 a = 0; the comploto in tors oc bio n with the cubic surface is mado up of X=0, r=() (the nodal liuo) eight times, and of the lines A r = 0, #=>0 } and y=0, W = t ouch twice. Tlio reciprocal siufuco is ;%-^y = 0; via. this is a like scroll, XXII = 8(1, 1) ; c'-O, &'!. V " Artiolu No. 204. Section XXIII = S (17 1). Equation A r (A r !K+ YZ}+ Y* - 0. 204. ThiH ia alKO a scroll; there is a nodal lino X = Q t y = 0, and a single directrix lino united therewith. Tlio licaniau Hiu-fiioe i A' 4 -0; the comploto intersection with the cubic surface is A'0, Y~Q (the nodal line) twelve times. The reciprocal nurfaco iw ^(aw + ^)-^ = 0; via. this i,s a liko scroll, XXIII = (171) ; o'0, i'l. .-w Gontuininff Additional lleiieawhes in rcyu.rd to the case XX = 12-Z7 8 ; equation Lull the Hiirfaco bo touched by tlio lino (a, b, o, f, g t A), that ia, the line the equationH whereof arc -ft, 0, /, b ff> ~f> 0, G ~ a, - b, - G, Writing the equation in the form oW . cX + X (oZ? + &Y> = G t and substituting for oW, c% bhoir valucn in termn of X, Y, wo Iwvo that is or say vix, the condition of contact ia obtained by equating to zero cubic function. Wo have thus 270 4 (* - cffY -f 4o 9 (2a6 -H o/) 3 452 A MEMOIR ON CUBIC SURFACES. [412 viz. this is + 27 o j o a 36 ab a cc/ + 18 b + 27 c which is the condition in order that the line (a, b, c, / g, h) may touch tho aiufoco X*W + XZ Z + 7 3 =0; and if we unite thereto the conditions that tho lino almll pass through a given point (a, ft 7, S), we have in effect the equation of tho circumscribed cone, vertex (a, /3, 7, S), Writing (/ ffl h, a> b, c) in place of (a, b t c, / g, h\ wo obtain + 80/yAa + 18 <7 B /i + 27 A as the condition that the line (a, 6, o, /, </, A) shall touch tho reciprocal surface and if we consider a, 6, G, f, g, h & standing for W-fa az-yx, fa-ay, Sa-mo t $y-/3w> Sz- 412] A MEMOIR ON CUBIC SURFACES. 453 values which satisfy the relation ( 0, h, - ffi (i J, & 7, S) = 0, -A, 0, /, b /* /\ a, &, - c, then the equation in (a, b, c, / y, A) is that of the circumscribed cone, vertex (, /3, % ^); the order being (as it should be) ft' =6. The cuspidal conic is y = 0, 4aw + 2 a 0, and we at once obtain a a 4cgr = as the condition that the line (,, &, c, / </, A) shall pass through the cuspidal cone. Hence attributing to (, 6, c, / </, A) the foregoing values, we have a* - 4c0 = for the equation of the cone, vertex (, (3, 7, S), which passes through the cuspidal conic; this is of course a quaclric cone, o = 2. I proceed to determine the intersections of the two cones. Representing by <B)==0 the foregoing equation of the circumscribed cone, and putting for shortness X = 27/i a (/ 3 - bk) - % a (2/0 + A), = : I find that we have identically (/a _ M) X + ((/ - 4a/i a - BfgJt?) ( whence in virtue of the relation af + Iff + oh = 0, we see that the equations = 0, (t a --4c</ = 0, arc equivalent to or the twelve lines of intersection break up into the two systems 1 and a/0-0, To determine the lines in question, observe that we ( 0, A, -g, a$>, /? -A, 0, /, 6 454 A MEMOIR ON CUBIC SURFACES. [412 and we can by the first three of these express a, 1), c linearly in terms of /, g, h ; the equations / 3 -&A0, a 2 -% = 0, 27A 9 (/>- Wt)- 2&*(%fff + ah) = become thus homogeneous equations in (/, g, A); the equations may in fact be written = 0, ^ 0, = 27A 8 (S/ a - A S -I- 7 A/) + 2# 9 (/3/t 2 - 7(7/1 - 28/0) - 0, viz. interpreting (/, (/, A) as coordinates in piano, the first equation represents a conic, the second a pair of lines, find the third a quartic. We have identically 4/3 a S (S/ 3 - /t u + <yA/) and it thus appears that the two conies touch at the points given by the equations we have moreover = 4/3S (S/ a 2S/- 7 A) [2/3S/- (f 4- hence at the last-mentiowed two points - J SA 3 + f y<//i + 2S/r/ is =0; aiid the quavtic Jt thus passes through these two points. The conic (a a -c#) = and the quartic A' = intersect besides (as is evident) in the point t/ = 0, A=0 reckoning as two points, since it is a node of the quartic; ami they must consequently intersect in four more points: to obtain these in the most simple manner, write for a moment then we have identically (jh + 2S/) 3 , + 4aS) (? 3 AJ.CUUB wiiBii a' ^cg=i(), we have 412] A MEMOIR ON CUBIC SURFACES. 455 and substituting these values in the equation X = 0, it becomes viz, multiplying by l(i/3 3 8, and omitting the factor O, this is 27/t a n + 16.<% 4 = 0, or finally 16'/% 4 - 27 (7 3 + 4aS) (ftf + 27W = 0, a pencil of four lines, each passing through the point y = 0, h = 0, and therefore inter- secting the conic (7" + 4S) (f + /3 W ~ 2/%/t - 4S/i/ = . at that point and at one other point ; and we have thus four points of intersection, which are the required four points, Recapitulating, the conic a s -4c</ = meets the sextic (f 9 -bh)X~0 in the two points 2/9S/- (7* + 4S) g + jftyA = each three times, in the point #=0, h~Q twice, and in the four points 27 (7* + 4aS) ^ + 27^ = 0, = each once. Or reverting to the proper significations of (a, &, c^/, (/, A), instead of points wo have 2 lines each three times, a line twice, and 4 lines each once; the line 00, A = 0, that is, f/ = 0, A0, a = 0, being, it will be observed, the line #=- drawn from (a, A 7, S) to the point y = 0, = 0, w = 0, which is the reciprocal of the uniplane ,Y = 0: the twelve lines are the aV lines of intersection of the circumscribed cone a! with the cuspidal cone c', viz. aV = [aV] + So-' + %' ; [a'c']-4 referring to the last-mentioned four lines; </ == 2 to the two lines; and %' = 2 to the line </ = 0, A = 0, a = 0, which it thus appears must in the present case be rentf^ twice. [413 413. A MEMOIR ON ABSTBACT GEOMETBY. [From the Philosophical Transactions of the Royal Society of London, vol. CLX. (for the year 1870), pp. 51-63. Received October 14, Eead December 10, I860.] I SUBMIT to the Society the present exposition of some of tho elementary principles of an Abstract m-dimensional Geometry. The science presents itself in two ways, us a legitimate extension of the ordinary two- and three-dimensional geometries ; and as a need in these geometries and in analysis generally. In fact whenever wo ore concerned with quantities connected together in any manner, and which are, or arc considered as variable or determinate, then the nature of the relation between tho quantities m frequently rendered more intelligible by regarding them (if only two or throe in number) as the coordmates of a point in a plane or in space : for more than throe quantities there M, from the greater complexity of the case, the greater need of such a repre- sentation; but this can only be obtained by means of the notion of a space of tho proper dimensionality; and to use such representation, wo require the geometry of such space. An important instance in plane geometry has actually presented itsolf in the question of the determination of the number of the curves which satisfy given con- ditions: the conditions imply relations between the coefficients in tho equation of tho curve; and or the better understanding of these relations it was expedient to consider the coefficients as the coordinates of a point in a space of the proper dimensionality. A fundamental notion in the general theory presents itself, slightly in plane geometry, t ! !^ ~: V S0lid ^ - - >-e here the 'difficulty as I " CU1 ' Ve n SpaCG ' r < to 8 P eak mora wemrately) to the U f the tW fold relation bot * coord nates of a that of a *-feU relation.-as dis- elation8 > se S ^ ^e expression a stern o f e ni "^ BU h ^ tio " * "* of a system of equations (or onefold relations). Applying to the ease of solid geometry f i d ^ , 7 atl nS ( r nefold relation8 > se S e expresson * "^ BU h 413] A atJOMOIK ON AUSTItACT GEOMMT11Y. 457 my (joiiohiHion in tho gonoral thuory, it may bo mentioned that I regard tho twofold rolabiou of a onrvo in H|KJU iw being uompletnly and precisely oxprefwod by moans of a HyHtom of (M|iiatioiiH (7 J ^0, Q=*() t ... T-Q), when no ono of tlio functions P, Q....T in a linoiir f'umstion, witli oon.stant or variable integral oooffioioiits, of tlio obhora of them, uiiil whim wwi/ mtrfitce whti tenor whioh passon through the curve hn its oquation oxproHsiblo in bho fni-in If ^l./ 1 -I- .A 1 ^ . . . + AT, with coiwbanb or variable) integral (souttiuiimtH, A, Jt, ... 7i r , It in Imrdly nouonHJiry to rumark tliat all the fnnobionH and urn ljukttii to bo rational fim<stionn of the coortlinataw, and that kho ' word linn I'idttniimo 1m lihu Artiiili- N(m, '!. to .M(J, Gwioml .f&nplnnulions ; .Huhtiion, Loons, it-c. \, Any in (|uantUirs may bo rqirowMitod by nwaim of m + 1 c[imntitioH nn the m of m of thcno to tlin niinainin^ (in. -\~ \ )th cinantity, and thus in places of the al)Holul;o vjdnoH of linn in ojmntitioH wo may (ntiiHidor tlio ration of m -I- 1 quantities. 2, III in to lit! notkiod that wo urn Ihrowjhmil <!onenriu:d with tlio ratio.4 of bho m -I- 'I. cinnntitii'M, not, with tint abnolnto valuta; thin biimtf undorHtenKl, any montion of tho ration in in ^cncnil nniuntcHnnvy ; tlnin 1 whall Hpnak of a relation botwoon the tn *\< I (|iianlitiun t nuiuiiug fchrnsliy a rulatiiui as rogardn the 1 ratioH of tho iiuatititioH ; at id HO in utlirr cascH. It may almt bo unbind [iliat in many itmtanctja a limiting or (KXtviMiiu caHo in HoinoliiiK'H inolndml, .soinotiini'H not included, nndor a gonural uxproHHion ; tho goiuTnl i!X|ii'('HMiini in iiitiindrtl t,n inohidi? whattwor, having ro^ard to tlus md)jcot- imitttM' and oontoxt, can bo inoliuhul mulor it, It. roMtulati 1 , W(s may <!nm>uivn butwiMin bhu m -{-I iniantition a relation^). 4. A rolatinn w oithoi' rttyitlur, that w, it him a dolinito manilbldiHws, or, Hay, it IH n /.:-lultl relation ; nr C()HD it w irt'ai/itlar, that in, onmpoaod of volatioim not all of th<! Hiuno inanif'oldnt'KH. AH to lint word " (iomnoHod/ 1 HOO jmt, No. 14. I'iH A MI;M"IU UN , (Illlt lilt- HI-:.- /.-N.-f. .,1 Wliri |..., ,., tltt |,, ..... , tl limy ^I..IM| IM i,, t <ln,|.. ,h., j,,,,^^^,^ , 41|1 , H Ml.' ruMhlinHlv,. , ,, M , ,,,.,. I l( il( J! . . ifjft< ,, is i ,. ! ..^ ^.j , ( A ..... ( ^, (j , ; ; ^^^ (!j f fa' H i I . i, .-toli.lN IM till' J hl; v IH ^,,^4 Hi th, ,,,.,,, ta 1(t , t. fl in,'h rvintion, * 413] A MEMOIB, ON ABSTRACT GEOMETKY. 459 14. Any two or more relations may be composed together, and they are then factors of a single composite relation ; viz. the composite relation is a relation satisfied if, and not satisfied unless, some one of the component relations be satisfied. 15. The foregoing notion of composition is, it will he noticed, altogether different from that which would at first suggest itself. The definition is defective as not explaining the composition of a relation any number of times with itself, or elevation thereof to power; which however must be admitted as part of the notion of composition. 16. A /(j-fold relation which is not satisfied by any other /ofold relation, and which is not a power, is a prime relation. A relation which is not prime is composite, viz. it is a relation composed of prime factors each taken once or any other number of times; in particular, it may bo the power of a single prime factor. Any prime factor is single or multiple according as it occurs once or a greater number of times. 17. A relation which is either prime, or else composed of prime factors each of the same manifoldness, is a regular relation ; a &-fold relation is e& vi- termini regular. An irregular relation is a composite relation the prime factors whereof are not all of the same manifolduess. 18. A prime /d-fold relation cannot be implied in any prime fe-fold relation different from itself. But a prime fc-fold relation may be implied in a prime more-than-fe-fold relation, or in a composite relation, regular or irregular, each factor whereof is more than /(.--fold ; and BO also a composite relation, regular or irregular, each factor whereof is at most /c-fold, may be implied iu a composite relation, regular or irregular, each factor whereof is more than ft- fold, In a somewhat different sense, each factor of a composite relation implies the composite relation, 19. A composite relation is satisfied if any particular one of the component relations is satisfied ; but in order to exclude this case we may speak of a composite relation as being satisfied distvibutiwly ; viz. this will be the case if, in order to the satisfaction of the composite relation, it is necessary to consider all the factors thereof, or, what -is the same thing, when the- reduced relation obtained by the omission of any one factor what- ever is not always satisfied. And when the composite relation is satisfied distributi the several factors thereof are satisfied alternatively ; viz. there is no o throughout unsatisfied. 460 A MEMOIR ON ABSTRACT GEOMETRY. 41.;} 22. There is no meaning in aggregating a relation with itself; miuh only occurs accidentally when two relations aggregated together become 0110 ami tho same relation; and the aggregate of n relation with itself w nothing O!HO than Un- original relation. 23. A onefold relation is not an aggregate, but itj its own .solo ooiiHtitnnnt ; IL more than onefold relation may always be considered as an aggregate) of two or mom constituent relations, The constituent relations determine, they in font coiwtitutu, tin- aggregate relation ; but the aggregate relation does not in any wwo determine lilu/ 11011- stituent relations. Any relation implied in a given relation may bo (;o]ini<loii!<l iw a constituent of such given relation. 24. The aggregate of a Mold and a /-fold relation is in general ami at mo*!, a (fc + J)fold relation; when it is a (& + J)foIcl relation, the mnntiDuonl; rolatioi.n an- independent, but otherwise, viz. if the aggregate relation ia, or has for factor, a I,*H than (A+J)fold equation, the constituent relations are dependent or mtaominotnd. 25. Passing from relations to loci, we may say that the compomtion of rnlul-innn corresponds to the congregation of loci, and the aggregation of relation, to (;hn ,',,,.- section of loci. 26. For first the locus (if any) corresponding to a given compoHitn rnlati.m is the congregate of the loci corresponding to the sovoral prime faotom of t li,, K iv,,, relation, the locus co 1T esponding to a single faetor being takou ,mo, an.l thn [ feetor bciug 28. It may be remarked that a Wold locus and a Mold loons here the aggregate re.ation h ,no re than w -fo,d) l.avo n ot givon not in involution is said to be "' mluiwn ' ll from the system any relatou until we arrive at an implied in the gil . -ens, and it 1 th, to * JJ W m V oralt '"" 8 ldatlora| ' and 80 "aoeaHlTOly * onofcW rolato 413] A MEMOIR ON ABSTRACT GEOMETRY, 461 be added on to and made part of the system. It may happen that, in the system thus obtained, some one relation of the original system is implied in the remaining relations of the new system; but if this is so the implied relation is to be rejected; the new system will in this case contain only as many relations as the original system, arid in any case the new system will be asyzygetic. Treating in the same manner every other onefold relation implied in the given Mold relation, we ultimately arrive at an asyzygetic system of onefold relations, such that every onefold relation implied in the given Mold relation is implied in the asyzygetic system. The number of onefold relations will be at least equal to k (for if this were not so we should have the given fc-fold relation as an aggregate of less than k onefold relations) ; but it may be greater than fc, and it does not appear that there is any [assignable] superior limit to the number of onefold relations of the asyzygetic system. 31. The system of onefold relations is a precise equivalent of the given Mold relation. Every set of values of the coordinates which satisfies the given Mold relation satisfies the system of onefold relations; and reciprocally every set of values which satisfies the system of onefold relations satisfies the given Mold relation. But if we omit any one or more of the onefold relations, then the reduced system so obtained is not a precise equivalent of the given Mold relation; viz. there exist sets of values satisfying the reduced system, but not satisfying the given /c-fold relation. 32. In fact consider a Mold relation the aggregate of less than all of the onefold relations of the asyzygetic system, and in connexion therewith an omitted onefold relation j this omitted relation is not implied in the aggregate, and it constitutes with the aggregate not a (k + l)fold, but only a Mold relation. This happens as follows, viz. the omitted relation is a factor of a composite onefold relation distributively , implied in the aggregate; hence the aggregate is composite, and it implies distributively a composite onefold relation composed of the omitted relation and of an associated onefold relation ; that is, the aggregate will be satisfied by values which satisfy the ^omitted relation, and also by values which (not satisfying the omitted relation) satisfy the associated relation just referred to. 33. Selecting at pleasure any k of the onefold relations of the asyzygetic system, being such that the aggregate of the k relations is a Mold relation, we have a com- posite ife-fold relation wherein each of the remaining onefold relations is alternatively implied; viz. each remaining onefold relation is a factor of a composite onefold relation implied distributively in the composite Mold relation. Hence considering the fc+1 onefold relations, viz. any fc+ 1 relations of the asyzygetic system, each one of these is implied alternatively in the aggregate of the_ remaining k relations ; and we may say that the jfc + l onefold relations are in convolution. 34. More generally any A + l or more, or all the relations of the asyzygetic system are in oonwteto'on, that is, any relation of the system is alternatively implied in _the aggregate of the remaining relations, or indeed in the aggregate of any k relations (not being themselves in convolution) of the remaining relations of the asyzygetic ** G2 A MEMOIR ON ABSTRACT GEOMETRY. U I 3 system. It may bo added that, besides the relations of the system, there is not any onefold relation alternatively implied iii the asyzygetic system. 35. The foregoing theory has been stated without any limitation as to tho valuo ot t.aud it has I think a meaning even when k is >m; bnfc the ordinary case is km Considering the theory as applying to this case, I remark that the last proposition, viz nut no reduced system is a precise equivalent of the given /,-fold relation, is aeuerallv true only on the assumption of the existence or quasi-existcnce of sets of values fold relation - For iet * be m * more tha11 "-*>* relation th r "-> reaton the coordinates ; the number of relations in the system may be > w +l- and if s - "">= -- '= :rz (In illustration of the foree-oine- Nm 9Q t q- T " But when the fac VB C g ftnch ni1 r ec l uafcions i f /I of them constat ' . 8i = is implied ; T'" I " e remam 'g equations Q = , .B = 0,..., = ffi-0 equations, and the relations (Q = 0, 7eLo ) ad V U, -ft = U, , . . I H,rP Pnillirolanl- 4-^ 1 i ^ " J U| . i i I Uilill -0 equivalent to each other. But in the ease of a convolution then the relation the equations Q = _R = im I -P = 0; or, what is the same thing, the relation ' '(0 = V T, >_!" \ ^ ^ A ~ r els of the two relations (.4 = 0=0 ^ = i -"-U,..,) is a relation composod any" 'of tf eXPreSS d ^ m re tha *"^Sl ( fp"- i 0-0 p'S'^ {" ^ any t ot these equations which are not in c 1 t * "'). selecting the remaining equations, we have a convolut^ fT' ^ Unifcin8> tllGrefco any one of relation is precisely expressed by means of ^Vf e e . quafcions ' and lvh0 n a A-fo]d Q=0,...) ( then ev ery equation O = imnliprl -T r more ec l uatio ^ (P=0, -same thing, the equation of any onefold 1 ^^ relati ni 01 *' what is th(S the A-fold relation is in involution with the ^T^ D gh tho locus & b 7 have identically O = ^ + ^ + ^+ f ^ P-0, Q- 0f ... ( that is, we coordinates.} ">-*. A 0,... being integral f unctions of fche 413] A MEMOIR ON ABSTRACT GEOMETRY. 463 Article Nos. 37 to 42, Omul Relation; Order. 37. A Af-fold relation may be linear or omal. If k = m, the corresponding locus is a point; if k<in the locus is a fc-fold, or (m ~ ^dimension al omaloid; the expression omaloid used absolutely denotes the onefold or (m l)dimensional omaloid; the point may be considered as a ?ft-fold omaloid. 88. A m-fold relation which is not linear or omal is of necessity composite, com- posed of a certain number M of m-fold linear or omal relations; viz. the m-fold locus corresponding to the m-fold relation is a point-system of M points, each of which may be considered as given by a separate m-fold linear or omal relation ; each which relation is a factor of the original ?n-fokl relation. The given ?n,-fold relation, and the point- system corresponding thereto, are respectively said to be of the order M, 39. The order of a point-system of M points is thus =M, but it is of course to bo borne in mind that the points may be single or multiple points; and that if the system consists of a point taken a times, another point taken /9 times, c., then the number of points and therefore the order M of the system is considered to be <10. If to a given /c-fold relation (k < m) we unite an absolutely arbitrary (m~/e)fold linear relation, so as to obtain for the aggregate <a m-fold relation, then the order M of this m-fold relation (or, what is the same thing, the number M of points in the corresponding point-system) is said to be the order of the given fc-fold relation, The notion of order does not apply to a more than m-fold relation, 41. Tho foregoing definition of order may be more compendiously expressed as follows : viz, Given between the m + 1 coordinates a relation which is at most m-fold; then if it is not m-fold, join to it an arbitrary linear relation so as to render it m-fold; wo have a m-fold relation giving a point-system ; and the order of the given relation is equal to the number of points of the point-system. 42. Tho relation aggregated of two or more given relations, when the notion of order applies to the aggregate relation, that is, when it is not more than m-fold, is of an order equal to the product of the orders of the constituent relations; or, say, the orders of the given relations being >, /*',..., the order of the aggregate relation 464 A MEMOIR ON ABSTRACT GEOMETRY. [413 really as well as formally Mold; if they do satisfy certain relations in virtue whereof the formally fc-fold relation is really less than Wold, say, it is (k l)fo\d, then the relation is in fact to be considered ab initio as a (k )fold relation : there is no question of a relation being in general fc-fold and becoming less than /u-fold, or suffering any other modification in its form ; and the notion of a more than m-fold relation is in the preceding theory meaningless. 44. But a relation between the coordinates (x, y, ...) may involve parameters, and so long as these remain arbitrary it may be really as well as formally /u-fold ; but when the parameters satisfy certain conditions, it may become (k-l)tol<\, or may suffer some other modification in its form. And we have to consider the theory of a relation between the coordinates (a, y ,...}, involving besides parameters which may satisfy certain conditions, or, say simply, a relation involving variable parameters. If the number of the parameters be m', then these parameters may be regarded as the ratios of m' quantities to a remaining m' + 1 th quantity, and the relation may be considered as involving homogeneously the m' + l parameters (to', ?/,...). And these may, if wo please, be regarded as coordinates of a point in their own m'-dhnensional space, or we have to consider relations between the m+1 coordinates (a;, ?/,...) and the m' + l (parameters or) coordinates i/', ..,). It is to be added that a relation may involve distinct sots of parameters, say, we ^ have besides the original set of parameters, a set of m"-\-I parameters /,...) involved homogeneously. But this is a. generalization tho necessity for which has hardly arisen, Article Nos. 45 to 55. Qualities, Notation, &c. 45. A homogeneous function of the coordinates (0, </,..,) i a represented by a notation such as (*$>, #,...)<> (where (*) indicates the coefficients and ( . ) the degree), and it is said to be a quantic; and in reference to the quantic the quantities or coordinates <, ...) are also termed faw.nti. More generally a quantio involving two or more sets of coordi- nates, or iacients, is represented by the similar notation t ' ite ' &c " accMdi ^ the number of 18 one two, three &c,; and with respect to any set of coordinates, it is binary o es - ^ bee, foui, or + ; and it is linear, quadric, cubic, quartic ..... according as the degree m regard to the coordinates in question is 1, 2, 3, 4, ... . 47 A quantic involving two or more sets of coordinates, and linear in regard to 1 if i taVLt it V antiparfcit6; - - l-^r t when there a. oTy t sbe mear; may 413] A MEMOIR ON ABSTRACT GEOMETRY. 465 48. Instead of the general notation (*<, </,... )<>(< 2/',...)' !) .. "we may write (a, ...)& */,... /(.< y', >, where the coefficients are now indicated by (a,...), and the degrees are p, /i', ... 49. In the cases where the particular values of the coefficients have to be attended to, we. write down the entire series of coefficients, or at least refer thereto by the notation (a,...); and it is to be understood that the coefficients expressed or referred to are each to be multiplied by the appropriate numerical coefficient, viz. for the term ffl"P ... ftj^'... this numerical coefficient is 50. It is sometimes convenient not to introduce these numerical multipliers, and we then use the notation (a,. ..$0, y t ...Y(af, jA ...)"., (a j ...i 1 y,...> 2/,..- >'.-.. In particular (a, 6, c]j>, y}\ (a, b, c, $%, yj &c. denote respectively off" + 2% + cf, &c. ; but (a, &, ojffl, y) 3 , (a, &, c, djffi, i/) 3 , &c. denote aa; 2 + 6ffy -f cf, a B + btfy + cwf + dy 3 , fee., and so (a, &. o, /, ffl ^ y. *> and (, &, c, /, ^ A]fe y. S ) fl denote respectively oo 9 + aa; 3 + 61. To show which are the coefficients that belong to the several terms respectively, it is obviously proper that the quantio should be once written out at full length ; thus, in speaking of a ternary cubic function, we say let U=(a,...$a> y, *) -(a, &, o,'f, g, A, , 3> k > 1 1> y> ^ = cw? + bf + 3 (ffz and the like in other cases. 0. VI. 460 52, A nuufnlil n'tal-inii oquabinn of (,ho form A MKMUIH (IN' AIM'llAiT i.J"MMUY, i in '> m. nii'f .-I it)l nil. Tins oxprtiHHiim "mi an equation ,,f ihn f,, nil j n ,,,,,(!,.. vi. it,., ,.,,,.,* aom; th ly Hft y proHsiblu by an rn! itor, iviul to UK,, a that Lho ti. K ln I.. h, // t mrly ti. wril, Wo may aim npnak uf M, ( , tl ff=0, orfcho rm H/i. A /,-ful.l ivInMnn b,.i,w.-..|| il,,, r-Mhln,.,* a HyHloin nf k ur in..r,, ,,| M .|;,],| n-l..!).,,,,, ,,,}, 0, and UKI Mi.|,l ,-nla. ' >'Jtf-h tf-allv *!nf iv- .,,, ', ;;, , r - 11 'b" * |M*i<.ii ' , ';" , tho m>fta , ,, l '"' v - h ' Uli ON AUSTHACT GKOMKTUY. 4G7 will mproHont u ] ](l)illt in tho ^-npaeo, nay, tho oo^mon, point, fl. A m-fol.1 relation, or tho loeus, w pnint-sy.stom tluu-oby reprinted, Z t 'TiV" 1 '. r ; ! ht \ lliliutM , r thu iwn! may b, gonouilly ii Mold rolutiou (fc ;f> ,), or tho locus thoroby ropro.soutod may l.uvo n or / pun,,; f r lob tliu ,-oladon If U,s than JfohM .^oTfoW by *> it n mmr iH-h l ' l"ii.W *> it n mmr (iH-^hl lvUt iou ^infi,,! by the, coordinate h ; "tho ; ; ; Hin v7 iMri ^ ^ if t!i imiiit ^^ bi ^ ^w I , r th V'fHyH ta ^^ -P-untod, tho point i 11 ' Il< ] 10 ol * ilml "" Ii)ltl ! ' olrtbl<mi or of 00. A givn-f|,| rrlati.m (^ ^ w ) bo.Am.-n thn W1 +l owmliiurtoH. oi< the IOOIIH Ui nby , l .., l .a w IUIH nut m ^.orul a ,ulnl point. But if thu Ration involvo thu i lmmt. !W ( y ,,..}, thui,. If a uortuin nnofold rotniinn lo Ha^fiud btwooii tho 1 " "' ft ' ll , I t, rchttunt of 1,1ns ^i V( .|i /:-lbM rolation. t. Tu l)h WIHO m .jnnstion, /,-:(>, thu diwi-imhuint relation i tho ronnltunt o Uon nl a (*+ )lol,l y.hitmn whioli i.s tho )W -O K ^O of bho givou ^fokl relation with a iHu-fian. rclntiun c-ulh.l tlu, .Aa^w/t .rf(/ (H , t or w |,on tho di^iuotion IH romm-wd fine t/acobuni rcltttwn in i-ogiml tu tho (a; t ?/,...). (12. On,,Hi(l,r n /.-(old nHatu.n (k > m, > m') hot^oon tho wn- 1 eoonliiuittw (; v ) ami ho f /i +1 nwmljimh* ( ; ,; ,/,...). i, ; ^ b(!(m H , " t i , s " ' -nji.ii UIMIU uw u, ^ivuii ti\ii ui vcunoH oi _ (<, y,...) or, Hiiy. (a a given point in the >aeo, thoro corn-spun.]* a Jb-fold ' IOOUH in tho_ wwinncu. and that to a tfvon wot of valuon of tho (a,, y,...). or to a R ivcm point m tlm ^-Hpmto, thoro wirix-HpraidH a M.ld lociu in tho m'-Hpace, Tho /-fold Ueus m tlm w'-Hpaco may haw. a nculal point; thiH will bo the OHM if thero in witiflfind boh thu (/,', ?1 ...) a curtain one-fold relation, the discriminant rotation of Urn givon /.-fold rohUmn in regard to tho (a-', y' t ,}. This onofold relation repre^-^ in tho wt-Hpaee a onefold looiw. tho envelope of the /.--fold looi in tho m - Hponding to tho Heveral points of tho w'-Hpaoo. Tho property of the euv to uiufh point thereof thoro coiTOHpondn in the wi'-spnco a A:- fold locus ho point. Article Noa. 0309. Consecutive Points; Tangent Omuls, 08. As tho notiona of proximity and remoteness have been thus far altogether ignored, it seems necessary to make tho following- _ Postulate. Wo may conceive a point consecutive (or indefinitely near) to a given 592 4(18 A MEMOIR ON ABSTRACT GEOMETRY, [413 04. If the coordinates of the given point are (at, y, ...), those of bho consecutive point may be assumed to be (x+Sae, # + &/,...)> where So;, By,... aro indefinitely small in regard to (,, y, ...). 65. It may be remarked that, taking the coordinates to bo (,* + X, ?/4- Y, ..A thoro is no obligation to have (X, Y,..,) indefinitely small; in fact whatever tho UKignitudoH of these ([imntities are, if only X ; Y:...=x : #:..., then tho point (,v + X t y -\- Y, ...) will be the very same with the original point, and it is therefore clear that it con- secutive point may be represented in the same manner with iimgniUuloH, howovcir large, of X, Y,... But we may assume them indefinitely small, bhnfc in, tho mtiiiH .v+$,v : y +%,..., where Sx, %,.., are indefinitely small in regard to (as, ?/,.,,), will represent any set of ratios indefinitely near to the ratios (m : y t .,.), The foregoing quantities (&p, %,...) are termed tho increments. _ 06. Consider a fc-fold relation between the m + I coordinates (to, ,...) k > m tho increments (&, fy, ...) are connected by a linear A- fold relation. ' The linear A-fold relation is satisfied if we assume tho increments proportional to tho coordmates-this is, in fact, assummg that the point remains unaltered. Wo may writ, y " S1UC " in8UC an e the tios ''0 a to. r ,. os ''0 aomor to. ut F0portlmml to tho " as by writing ,... t j ; c ^'-^ r io f r' s r ses through th Wold locus at the point fe ,, n f th<3 to .'/ (! *-< of tho original !<"' (, y, ...), which pomt 1S said to bo the po{ , a of ^^ ^^S^Thir 06 ^/')"y( X ,y,...),and combine elation (containing as paramete ' ^ ^ f"* J ate s <, 7,-.) a M .fbld the original A-fold relaHm. ,^i -, ^2/.--0); these pammeberB satisfy . but the point in qu^n 7sTn Z T Z? V- ^ - i d not also follows that the tangent-omal locus con d P m a 2 *" fold looiw ] cs. ng e ori ^ inal A-fold relation, has for its envelope the 413] A MEMOIR ON ABSTRACT GEOMETRY. 469 69. We thus arrive at the notion of the double generation of a A-fold locus, viz. such locus is the locus of the points, or, say, of the ineunt-points thereof; and it is also the envelope of the tangent-omals thereof. "We have thus a theory of duality; I do not at present attempt to develope the theory, but it is necessary to refer to it, in order to remark that this theory is essential to the systematic development of a m-dimensional geometry; the original classification of loci as onefold, twofold,... (m-l)fold is incomplete, and must be supplemented with the loci reciprocally connected with these loci respectively, And moreover the theory of the singularities of a locus can only be systematically established by means of the same theory of duality ; the singularities in regard to the mount-point must be treated of in connexion with the singularities in regard to the tangent-omal. These theories (that is, the classification of loci, and the establishment and discussion of the singularities of each kind of locus), vast as their extent is, should in the logical order precede that for which other reasons it may be expedient next to consider, the theory of Transformation, as depending on relations involving simultaneously the w + 1 coordinates (to, y, ...) and the m' + I coordinates (of y 1 , ...). 41 !, ON I'OLYXOMAI, fUMlVKS. uTIIKiaVls s'/'W i )*,. ... "S "< .*./ .V,..-,rtV ../ A HO. |(,,,,,l |,i,|, I.,,,,,,!,,, , " 'K' -. ...... . ,.n. v ,, ., ... .. * r, , ,.,,,! i,, u, ......... ..... ,., ..... - . .'/' "' ""' curve W/W|'.|. (jr.. ,,ii ' ," ' ' ' ' "' " I-I.V/..IIH I;,, hav,, U !lm , ,,!. ,, ' *a If Un, n,,,,,!,,, ,- - ....... ' > . mu! o u , " ' . mv , ,, . o, w.irthy ,,r (.<)i.i l l l , m iin it j, ; , , ....... '" '- " "'" '"" ' Hiir on,,, t in,c, Bmmiiullt ,, w ,, , , '" " * ........ ,"""" ...... ' ">" .. i...l , ,, , ,, &- are o,,,, Ntall( , Tl)0 Ml^ 1 ' ^ ' ?' M "< "'" ' ....... ',*,.., ,. <mrvl! i *u bm ..... 1M Uwr.n.r. U, H , ..... ,, ,,r I, '"" "'. """' '" in i'og,ml to thu ^.omul mtw WlMt/.mj e ' '"" ""W" 1 " 1 '" iu p"! "wwii a ...... ,, tH ;, r i,i,^;i, ll ;,;" 11 u -* lii - hiiiiD *'''' . onion , ftllll r _, ,,, loct| . V( . | ' r *" '' l., ,,>, M,,,, .J..,, ,,r * " ' ''' - of ( . to of th, ourvu bo , . . p. curto 41 4 ] ON rOLYHOMAfc OUBVEH. two trramml oui-voa, each oxpvo.sniblu by ntoaiiH of any thrno of the four fmiotmiiH y, V, W, .'/'; for example, in the form */l'U+ Vw'F> Vyrr= 0. If, in |,hi theorem, wo tolco _p = 0, then tho original curve M the tmomul V^-|- VwiK-H -vVl'K 0, .'/' m any function -(aZ/.|-bV>eF), whore, couriering J, m, us givon, a, li, o ,im quantities Hiibjoot only to tho eomlition { + + .'- 0, un.l wo havo ll.o theorem ,,f 11 I) " tho variable HOI mil of a tribunal oui'vo," vix. tho (Hpuifciou of thn tmumal yW+* / V'+<SnW**Q l may bo oxproHWid by nioaim ( ,f miy two of |,lu. throo functioiiN (/, K, JK, and of a Aniotion 2' dotonninoil an ubovo, for t-xnmplo in I In- tonn Vm+vWl>>vV.7'0! whonco alHo it may bo OXIH-OHHIM! in torniH of tl.m, now functioiiH .7', clotonnincd an abovo. Thin thnownn, wlu.ih oeimpion a ]i-oiniuonl, |.u,iii im in the wliolo tlioory, wna mi^.mtod to mo by Mr (Jiiwy'a tlicnruin, pnMuiitly ivfml to, for tho csoimlii'iiotion of a bicirouliir quaHio as tho onvolopo of 11 vtirinblo injlc. In tho i^onml curve Vi ( + //li) + & c . 0, if M bo a wmio, 0> a Itn-. tho annuls H + 7/1)^0, foe. arc TOIIKIH punsing thn.ugh tho HIUIIO l,wu pnintn M - 0, <P-0, and fchoi-u in nu real I,,HH of - om!m lity in baking lihesu to bo Llut (liroular pointn ut iiihmty that in, iu taking tho OUIIJUH tu bo cimloH. .]),.h. K thin, and HHI]I K a H )Miiiil notation *?["() for tho oration of a oirolo having i(, H cnntro ut a K ivcn ju.inl, J. mid winilarly Sf0 for (ibo equation of an nvimow-imt oirdu, nr nay .f Hi,, p,,i h f, ^1, wo have Mm iMsomul curvo VAT^&e.-O, and iho nioro Hpuciul form VU &.-<).' AH rogardn the last-mentioned unrvo, V^l + & (! . 0, the iioint yl to wlueb Mu- Pf0 boloi^H, in a oc.iw of thu curve, vi/. iu tho OHNO ^-,'1, in | H an ..vdinary and in tho UAHO ,$, it in a special land of f, muH , whioh, If tho tonn w.uv roi miffhb be called a fotso-fooim; (} M Memoir <!0)itaiiiH an i^planntion of tin- theory of tho foui of piano (jurvoH. Fur i.-tt, tho equul^.n V/vf+ Vwi (- V H .<J roully e(|iiivalent to tlio apparently inoro general form ^I"H- At^' 1 -I- "/(.*;' !). fit liwifc thiH hiRt in in general a bioircmlar quartio, and, in regard to it, tho bi>lnni.iuniiiinii( H i thoorom of thu variable mmml btiw.inoK fttr OVsey'H theorem, that " thn bi.^iroular quartie (nnd, OH a particular 0,1*0 ehcniof. tho oimdar cubie) i H tho uivolopo nf a variable en-do ^ having j tH ounlim ()ll a ^ iv(!11 0()nic mul (m(iti , IK (lt ^^ tU} ^ H . Clrclo< " Z lllH tjie(ll ' (!111 iH Jl wHUciont luiHi.H lor the cjoniiiloto thoory of tho ' 472 ON 1'OIVYXOiMAI, OUIIVMH. i ,. , Investigations; Part III., On tho Thtsory of J'Vi ; MIX! IW IV., (),, (J,,, TnVnnm! Mlll | Tetrazoraal Curves whore tho /omuls urn mivlr-s. Tlinv JM, |,,,uvvrr VMM,,. IM-.V. ,, intermixture of tho theories troutcd of, and |,h,. i,m.., K n,.n.l. will ',,,,. ..... n . :!. detail from the heading* of thu Novoral m-tii'lm Tit.. purm-mi.!.* an- ,n,,, ..... IVI | nm tinuously through the Momoir. Thorn aro (! Ann-s.-s. H-luring ,,, ,,,,;, .,,:,.,. . seemed to mo moro convenient to (,ivat oi' UIIIH wpimih'ly, It is right that I Hhould explain U, (! vrry K n,,i ,. X |,,,|. ( w position of the present Memoir, I an, in,!,!,!:,,! , A | r , Wv - ;'0n the Elation, and l^ wrtiw (,) ,- tllc . s^,,,,, ,, ri in a plane; (2) of tho SyHtom of Sph.n, ron.-hin, li.nr nph-n-H n System of Circle touching tln,o ,i,,| ( , on , H,,^.; { ,,, ' ,,," ^nbed in a conic and touching tlnvo innn-iLnl ,(,, in a. 2 , s?=jssi; ::,K:";;;!,,;:: TV-- -""'v;;",;;::: in hia lotto of Marcl, a IHOi ,, ' """""""'W ' Inn, U,,. ,,,,,l,l,,, ...... | v ,,,| with ca^ Eqnati ,;; ^:: , ; /:: ll ," )if : liiv ..... '" ...... ........... - a , , / , ,, also a lw Annex No. IV o f ^ Z M,t ',! '''' ' :MI< |:I " S| ....... In connexion with this ilu-nr,,,,, i ' s u-nr,,,,, i ,, , t ,, ';: ....... i"":; 1 " 1 ." ? ir ...... > ...... ' - **> ., ,, ( - ohl! U1 :,: V ':;,S"; ;';;: ';'*, "- .-- ^, denote ^ mutlml , |iHto|K ! ()S ,. ,; ;' 'V o ' ""' '"' ''' Mr Casey, in a lottor U, mo d.ilod MOII A ,", u '^ ""'"" '" " (1 ""'i'. of viewing th,, , iuo8[irai ^ ; 1(1 ' A I'"L IW-T. ,...,,, ........ hi ................ fc'.'./.ff.AjfeAtf-o.wi ' 77 1 ,/""'; '"""" ........ "" - ......... ' "i-- the equation (, 6, e , ^, *V' i-/ n ? "' '" " l ' i " i '"" l " r ...... f ""' "' "'- " '' ..... " ..... " ' ..... o, , . ,., conic; denoting thin conic by /,' ftll , '''"" '.' , a ' ^ 1 "" "} I ..... i. i, ,!,, '.I Foved that, if tt variabl c , , " """ '""" "' "' "''' ........ v '-V orthogonally, it. onvcl wm ,,; ( '"' -I ''i,.r ...... ,, ir ..... , down above;" aild J Mm , '" >l" l.' ..... |li..,, i. ,|,,,, wrf,.,,.,, double foci of the quartic and II, ' ' " ..... """" """ ll "' '"'! "I' F HIV lh, ^ *, the ttcd'irv 1 ":: i: 1 "" 11 / r " ^ M >* : ^ Memou. on Bieiroular Q lmrt T, tl " '"' '''"'"'"' '""' '"' * I- Memon; as road before the ttoya ,,' A ""'" J ' , ' i " i " 1 "" 1 ' A " Al-nlm,,. ,,r ,|,,, sit, r -f""'^' "" H S h ;;;";;, """ " <iim ""^ ^ ..... '> ..... .1 30th Apnl, and 8omo othor thooron T i ....... '''""' " ll '" li "'""' i" Hi,. ,,. ,,r d u t r wl i * - r^n ;'c;r w "^ u r ........ ^< Memo;r U ,n m h , ^^ ^J^^y r ,,, |; POLYZOMAL CURVES. 473 PART I. (Nos. 1 to 65).-0 POLYZOMAL CURVES IN GENERAL. Article Nos. 1 to 4. De/Vm fl rf Preliminary Remarks. 1 As already mentioned, ZT, 7, &c. denote rational and integral functions (*$, f/ all of the same degree r in the coordinates (, y, ,), and the equation then belongs to a polyzomal curve, viz., if the number of the zomes *ffi t VF &c is -*, then we have a ,-zomal curve. The radicals, or any of them, may contain 'rational factors, or be of the form f*/Q. but in speaking of the curve as a ,-zomd it is assumed that any two terms, such as PVQ + P'Vg involving the same radical V0 are united into a single term, so that the number of distinct radicals is always * m particular (r being even), it i s assumed that there is only one rational term P But the ordinary case, and that which is almost exclusively attended to, is that in winch the radicals </U, W t &c. are distinct irreducible radicals without rational factors. 2. The curves V=V=Q, &c. are_ said to he the zomal curves, or simply the zomals of the poly.omnl curve ^+Vr+fra-0j more strictly, the term zomal would be_ applied to the functions U, V, fa, It is to be noticed, that although the form Vtf+VK+&c.0 is equally general with the form Vjff + VmT+ &c. = (in fact, in the former case the functions U t V, &c. are considered as implicitly containing the constant factors l t m, &c., which are expressed in the latter case), yet it is frequently convenient to express these factors, and thus write the equation in the form V7?+ v ' c 1'or instance, in speaking of any given curves [7=0, 7=0, &c., we are apt, disregarding the cons ant factors which they may involve, to consider U, Y, &c, as given functions; but m this case the general equation of the polyzomal with the zomals. 17-0, 7 &c., is of course V#? + VmF+ &c. = 0. 3. Anticipating in regard to the cases = 1,^=2, the remark which will' be presently made in regard to the p-zomal, that V!? + VF+&c. = is the curve represented by the rationalised form of this equation, the monozomal curve VF= is merely the curve Z7=0, viz.,Jhis is any curve whatever U= of the order r; and similarly, tho bizomal curve Vi7+VF = is merely the curve Z7-7-0, viz. this : ~ whatever Ii = 0, of the order r; the zomal curves U=b, V A - 1 - not curves standing in any special relation to the curve in may bo any curve whatever of the order r, and then 7=0 is a curve of the same order r, m involution with the two ourves_fl =- Q, 17=0; we may, in fact, write the equation fl = under the bizomal form VZ7_+ VflTD 1 = 0. In the case r even, we may, however, notice the bizomal curve P + Vtf=0 (P a rational function of the degree r); the rational equation is here a=^-P^=0, that is U=fl + P* t viz P is any curve whatever of the order r, and 7 = is a curve of the order r, touching the given curve il = at each of its r a intersections with the curve P = I further 0. VI. 474 ON POLYZOMAL COBVES. remark that the order of the i/-zomal curve ^F+&o. = is =2-r; thin in right in the case of the bizomal curve Vt/"+\/K=o, the order being =r, but it fails fur tho monozoraal curve >JU=Q, the order being in this case r, instead of ), us given by the formula. The two unimportant and somewhat exceptional cases v = I, i> =. 2, aro thus disposed of, and in all that follows (except in so far as this is in fnet applicable to the cases just referred to), v may be taken to be =3 at least. 4. It is to be throughout understood that by the curve </U+ VK+foc. = | H meant the curve represented by the rationalised equation Norm(VZ7+ VF+ &c.) = 0, viz the Norm is obtained by attributing to all but one of tho KOHIOH V / -J V &,. each of the two signs +, -, and multiplying together the several rolling values ,','[' the polyzome; m the case of a ,-zomal curve, tho number of factors is thus =2"-.,. (whence, as each factor is of the degree |r, the order of the ourvo is 2 ..... Li,. = 2 r. as mentioned above). I expressly mention that, as regards the poly,mal cnnV we are not m any wise concerned with the signs of the radicals, which sifins , m , aad nnuun essentially indeterminate; the equation VF+ VF-h &e. = 0, is a mon) ' Hyml,l Ir the rationalised equation, Norm (V(7 + VF+ C .) = 0. Article Nos. 5 to 12. The Branches of a Polyzonal Ouna. 5. But we may in a difeent point of view attend to tho signs of ,;!, radicals: -f for all values of the coordinates we take the symbol ^ m . c(mHillm , ^ jy &c. as signifying detennmately, say the P o^e valuos of W, ^V. fa,, then ,H ,h ol'' the several equations J U_ JV +ta ^ , fixing ftt ' . ' ' ofthe s x = ft , t Cations of Aginary, va, u , If for a? - ot , ^ we ON POLYZOMAL CURVES. 475 all one of the two Apposite values of V?+#, ca llmg it the positive value, and representing it by ,/+#, then, for any particular values__of the coordinates, U being = +#, thejalu^of Vff may be taken to be W^; and the like as regards JV, &c. ,/ff, ^F &c. have thus each of them a determinate signification for any values whatever, real _or imaginary, of the coordinates. The coordinates of a given point 01^ the curve Vff+VK+&c. = 0. will in general satisfy only one of the equations N/^s/F&o =0; that is, the point will belong to one (but in general only one) of the 2" branches of the curve; the entire series of points the coordinates of which satisfy any one of the 2-' equations, will constitute the branch corresponding to that equation. . 8. The signification to be attached to the expression J^Ji should agree with that previously attached to the like symbol in the case of a positive or negative real quantity; and it should, as far as possible, be subject to the condition of continuity, ,jjH-/9,' passes continuously to ' + /3'i, so V*T^ should pass con- tinuously to W+/3'*j but (as is known) it is not possible to satisfy universally this condition of continuity; viz., if for facility of explanation we consider (a ft) as the coordinates of a point in a plane, and imagine this point to describe a closed curve surrounding the origin or point (0, 0), then it is not possible so to define that this quantity, varying continuously as the point moves along the curve, shall, when the point 1ms made a complete circuit, resume its original value. The' signi- fication to be attached to V + # is thus in some measure arbitrary, and it would appear that the division of the curve into branches is affected by a corresponding arbitrariness, but this arbitrariness relates only to the imaginary branches of tlie curve* tbe notion of a real branch is perfectly definite. 9. It would seem that a branch may bo impossible for any series whatever of points real or imaginary. Thus, in the bizomal curve Vtf+VT-O, the branch Vtf+Vp-:=0 is impossible. In fact, forjmy _point whatever, real or imaginary, of the curve, wejiave^= V, and therefore Vff- V7; the point thus belongs to the otf branch VET-Vr-0, not to the branch VI7+V7=0; the only points belonging the^ last-mentioned branch are the isolated points for wlr ] - -' " 47 G ON POLYZOMAL CURVES. [414 would have opposite signs), but there is no apparent reason, or at least no obviously apparent reason, why this should be so for imaginary values of the coordinates, and if the sign be in fact -, then the point will belong to the branch VI? + V7> VF= 0. 11. But the branch in question is clearly impossible for any series of real points; HO that, leaving it an open question whether the epithet "impossible" is to be under- stood to mean impossible for any series of real points (that is, as a mere synonym of imaginary), or whether it is to mean impossible for any series of points, real or imaginary, ^whatever, I say that in a y-zomal curve some of the branches are or may impossible, and that there is at least one impossible branch, viz., the branch C. =0. 12. For the purpose of referring to any branch of a polyzomal curve it will bo convenient to consider VF as signifying detenninately + VF, or else -VFj and the like as regards^ F^&c., but without any identity or relation between the signs pre- fixed to the VtT, V7, c., respectively; the equation VF+VF+ & . - 0, so understood, wiU denote determmately some one (that is, any one at pleasure) of the equations ^tfVF&c. = 0, and it will thus be the equation of some one (that is, any one at pleasure) of the branches of the polyzomal curve - all risk of ambiguity which might otherwise exist will be removed if we speak either of the citrw VF+V7, &c. = or else of the branch Vtf + VF-f&c. = 0. Observe that by the foregoing convention, whou only one branch is considered, we avoid the necessity of any employment of the sign + , or ot the sign -; but when two or more branches are considered in connection ' with each other, it u necessary to employ the ^ign - with one or more of tho radicals u \ &G ' ; ,- US ^ the ,^ omal T ve ^+VF+V=0, we may have to consider vt I ^ t, "t Tf=0 ' ^+^-^-0; vis., either of toe equations the L,r ."I I 6 ? ^ ne bl ' anch at P leaSU1 ' e of thQ <. but when h oZ T 88 , \ , ne eqUati U IS flX6d ' then th bl ' a ch Putod by tne other equation is also fixed, J Article No, 13 to 17. 2fc Points cownon to Two inches of a Polyzonal Curve. 13. I consider the points which are situate simultaneously on two branches of thermal curve V^VF + & , = 0. The equations of the two branches may be taken viz., fixing the significations of Vff VF VF frn i. m i ,1 , e i , , ' Y> Y w > fflc - in such wise that in the- 414] ON POLYZOMAL CURVES. 477 V W t &c, to be those radicals which have the sign. -, The foregoing equations break up into the more simple equations VF + c. = 0, VF+&c. = 0, which are the equations of certain branches of the curves VtT-f&o. =0, and VT7 + &C. = 0, respectively, and conversely each of the intersections of these two curves is a point situate simultaneously on some two branches of the original y-zomal curve vU+ Vy+&c. = 0. Hence, partitioning in any manner the y-zome vl/+ VF+ &c. into an a-zome, Vt7=&c. and a /3-zome vW + &c, (a4-/3 = v), and writing down the equations of an -zomal curve and a /3-zomal curve respectively, each of the intersections of these two curves is a point situate simultaneously on two branches of the y-zomal curve ; and the points situate simultaneously on two branches of the y-zomal carve are the points of intersection of the several pairs of an -zomal curve and a /3-zomal curve, which can be formed by any bipartition of the y-zome. 14<. There are two cases to be considered ; First, when the parts are 1, v 1 (v ~ I is > 1, except in the case v = 2, which may be excluded from consideration), or say when the y-zome is partitioned into a zome and antizome. Secondly, when the parts a, /9, are each >1 (this implies y=4 at least), or say when the i/-zome is partitioned into a pair of complementary parasomes. IB. To fix the ideas, take the tetrazomal curve \/T7> VF+ Vlf+ V^^O, and consider first a point for which V77=0, VFf VF + V2 T = 0. The Norm is the product of (2 B ) 8 factors ; selecting hereout the factors let the product of these = U be called F, and the product of the remaining six factors be called (?; the rationalised equation of the curve is therefore FG = Q. The derived equation is GdF+FdG = Q; at the point in question VT7= 0, VF+ VIF + VF=0; G and dG are each of them finite (that is, they neither vanish nor become infinite), but we have and the derived equation is thus QdU = Q, or simply dU=Q. It thus appears that the point in question is au ordinary point on the tetrazomal curve; and, further, that the tetrazomal curve is at this point touched by the zonial curve^Z7 = 0^ And similarly, each of the points of intersection of the two curves V~Z7=0, VF+ VF + V^O, is an ordinary point on the tetrazomal curve; and the tetrazomal curve is at each of these points touched by the zomal curve U=0. 478 ON" POLTZOJTAL CURVES. [414 . y=0; to form the m this case the two factors let their product " f ^ remainin six factors be called <? ' &* rationalised atn * n ' uc^ r ^ th f deriV6(1 eqUati n 1S >+<W* I -0. At the point in hi n 1, , ^ ^ eaCh f theM flnita < tlmt is ' &** neither vanish nor become minute), but we have becomes iclenticall)r 0=0; the :B on ' 96e that ifc is in fect a node - OT *"7 doable prat, on the tetrazoml ourw And similarly, e ach of the points of intention of the two curves Vf7+^=0, VF + VF=0 i s a node on the tetrasonml curve. v I! 1 ' P1 'f ^ " le foreg , hlg * W examplcs ras P^ly are quite general, and T " "" Z0 ' ml UrVe ' emm te tlle res lte as f"o, viz., in a T' , F mtS SltUatG sim "l to ^ o two laches are either the inter- , tt / 0mal , OU1 ' Ve aml it3 mtizomal curre - * "y are the intersection, C .? m P lemental 7 P<1 curves. In the former case, the potato in ^ H T P , mt : n th0 "- Z01m1 ' bU * t]le * are P inte f ntoot f >B 1. T J . ' my ^ addOCl ' tlmt the "te^^ons of the zomal and In Z ' T tW10e ' ate "" the inta "ti f the omal and zomal. L 'r,' P mtS '" ql ' eSti n am nodes of the "- zoma li ;t >"V ^ ^ded, Lhe ! has no, m ffeiKm l, any nodes other than the points which are thu s " C mplementar y P""^ nd that it has not *, general any Article Nos. 18 to 21. Singularities of a v-zomal dune. 18. ^ It has been already shown that the order of the ,-zomal curve is =2"-,.. Considermg the ease where is =3 at least, the curve, as we have just seen, has oontaot. with each of the ,o ma l curves, and it has also nodes. I proceed to determine the number of these contacts and nodes respectively. 18. Consider first the zomal curve 17=0, and its antizomal VF+VF+&o=0 the lers n- w >- m S , points Hence the ,-zomal touches the zomal in 2-r' points, and reckoning each of 'these twice, the number of intersections is =2r, viz., these are all th? mtcrseotions of the ,-zomal with the somal 7=0. The number of contacts of the ,-zomal th the several zonmls [7 = 0, V= 0, &c., is of course =2-"rt, 414] ON POLYZOMAL CURVES. 479 20. Considering next a pair of complementary parazomal curves, an a-zomal and a /3-zomal- respectively (a + /3 = y), these are of the orders 2"~ s r and 2* 1 " 2 ?' respectively, and they intersect therefore in 2" + ^~ 4 T* 2 k ~ 4 r a points, nodes of the y-zomal. This number is independent of the particular partition (a, /9), and the v-zomal has thus this same number, 2"~ 4 ? ta , of nodes in respect of each pair of complementary parazomals ; hence the total number of nodes is = 2 1 '"- 1 r' 2 into the number of pairs of complementary parazomals. For the partition (a, /3) the number of pairs is [v] v -=- [] a [/3] p , or when a.~/3 t which of course implies v even, it is one-half of this; extending the summation from = 2 to a = v 2, each pair is obtained twice, and the number of pairs is thus = ^2{[y]"-^[<t] a [/9] p } ; the sum extended from a = to a = v is (1 + 1)", =2", but we thus include the terms 1, v, v t 1, which are together = 2^+2, hence the correct value of the sum is = 2" - 2i> 2, and the number of pairs is the half of this =2 ( "" 1 v 1. Hence the number of nodes of the i<-zomal curve is (2"" 1 v~l) 2 11 " 4 r fl . 21. The v-zomal is thus a curve of the order 2"- 9 r, with (2"" 1 - v - l)2"- 4 r 2 nodes, but without cusps ; the class is therefore and the deficiency is = 2"-' r [(v + 1) r - 6] + 1. These are the general expressions, but even when the zomal curves E/=0, F=0, &c., are given, then writing the equation of the y-zomal under the form */lU + \finV +&c. = Q, the constants { : m : &o., may bo so determined as to give rise to nodes or cusps which do not occur in the general case j the formulae will also undergo modification in the particular cases next referred to, Article Nos. 22 to 27. Special Gase where all the Zomals have a, Common Point or Points, 22. Consider the case where the zomals U = 0, F=0 have all of them any number, say k, of common intersectionsthese may be referred to simply as the common points. Each common point is a S^-tuple point on the y-zomal curve; it is on each zomal an ordinary point, and on each aiitizomal a 2"- 3 -tuple point, and on any a-zomal parazomal a 2- a -tuple point. Hence, considering first the intersections of any zomal with its antizomal; the common point reckons aa 2 1 " 3 intersections, and the k common points reckon as Z^Ic intersections; the number of the remaining intersections is therefore = 2"~ 3 (r 8 &), and the zomal touches the v-zomal in each of these points. The intersections of the zomal with the y-zomal are the /c-common points, each of them a 2"~ a -tuple point on the y-zomal, and therefore reckoning together as %"-*k intersections; and the 2"-" (?' a - lc) points of contact, each reckoning twice, and therefore together as 2"- a (r a -&) intersections (2"- 2 ft + 2*- a (r a -^) = 2 l - 2 ?' 9 J =r.2"-v); the total number of contacts with the zomals C7"=0, F=0, &c., is' thus 2*-" (r a - Is) v. 480 ON POLYZOMAL CURVES. [414 23. Secondly, considering any pair of complementary parazomals, an a-zomal and a /3-zomal, each of the common points, being a 2 a ~ a -tuple point and a 2 /J - a -tuplo point on the two curves respectively, counts as 2 a+ ^~ J , = 2"~ J intersections, and the k common points count as 2"~ 4 & intersections ; the number of the remaining intersections is there- fore = 2 1 '~ J (?- k), each of which is a node on the c-zomal curve; and we have thus in all 2"- 4 (2"- ) -i>-l)(?' 2 -&) nodes. 24. There are, besides, the k common points, each of them a 2"~ B -buplo point on the v-zomal, and therefore each reckoning as ^"""(S"" 3 1), == 2 ai '~" 2''~ a double pointy or together as (2-"~ 5 - 2"- 3 ) k double points. Reserving the term nodo for the above-mentioned nodes or proper double points, and considering, therefore, the double points (dps.) as made up of the nodes and of the 2^- tuple points, the total number of dps. is thus 2"- 4 (2'~ 1 -i/-l)(r 2 -&) + (2 a "- fl - 2"- 3 ) k, = 2 1 - 4 (2"- 1 - v ~ 1) r"- + {(v + 1) 2"-' - 2"-} k ; or finally this is = 2-* {(2-' - v -l)r+ (y-l)l; so that there is a gain =3r*(v-l)k in the number of dps. arising from the k common points. There is, of course, in the class a diminution equal to twice thia number, or 2"- 3 (i/- 1)&; and in the deficiency a diminution equal to this number, or 2"->-l), 25. The zomal curves ^7=0, 7 = 0, &c, may all of them pass through the same v 1 points; we have then k = i* t and the expression for the number of dps is s =(2'--2")r, viz., this is =2"-' (2--l)r. But in this case the dps, are nothing else than the ? common points, each of them a 2^-tupIe point, the ,,-zomal curve m fact breaking up into a system of 2"- curves of the order 7-, each passing bhrouffh the ,- common points. This is easily verified, for if = 0, * =, fl are somo two curvos of the order r, then, m the present case, the zomal curves are curves in involution with these curves; that is, they are curves of the form 1+ M> = 0, m8 + m'<I> &c and the equation of the c-zomal curve is ' '* + Vm -f m'$ + & c . = 0. The rationalised equation is obviously an equation of the degree 2- in , * therefore a constant value for the ratio 6 : ft; calling this q> or writing 0= V^ + I' + Vmq + m' + &c. == 0, ta 1- '? ^"fT,;: " eqU f n f the d8 ree 2M in ?. d g-es there- loie 2 values of j. And the ,-zomal clu , ve thn8 brealts into 8 6 9 - = e-O!. O X , curves U-0, $ = 0. The equation m g may have a multiple root or roots and the ystem of eurves contain repetitions of the same curve or ourree- an imtace of tn (m rfat.on te the trizomal curve) will present itself in the sequel but Ido not at present atop to consider the question. 414] ON POLYZOMAL CURVES. ' 481 26. A more important case is when the zoinal curves are each of them in involution with the same two given curves, one of them of the order, r, the other of an inferior order. Let- = be a curve of the order r, # = a curve of an inferior order r a; L = 0, Jf=0, &c., curves of the order s; then the case in question is when the zomal curves are of the form -|-.ZXl> = 0, @ + M*-0, fee., the equation of the v-zom&l is &o. = 0, where l t in, &c, are constants. This is the most convenient form for the equation, and by considering tho functions L, M, &o. as containing implicitly the factors l~ l , m~\ &o. respectively, wo may take it to include the form */l + Z<T> + Vm + Jtf* + &o. = 0, which last has the advantage of being immediately applicable to the case where any one or more of the constants l t m, &c. may bo = 0, 27. In the case now under consideration wo have the r(r s) points of inter- section of tho curves = 0, <I> = as common points of . all tho zomals. Hence, putting in tho foregoing formula }c-r(r-s), we have a y-zomal curve of the order 2 11 - 3 ?', having with each zomal % v ~*rs contacts, or with all the zomals % v ~ a rsi> contacts, having a node at each of the 2'"~ 4 ?'S intersections (not being common points = 0, <D=0) of ouch pair of complementary parazomals ; that ia, together 2''"-' (2'"" 1 v 1) rs nodes, and having, besidoH, at each of tho r(r s) common points, a 2"~ a -tuple point, counting aa 2 a "-o_2r-s djw., togoliher as (2 a "~ B 2"-) r (r s) dps. ; whence, taking account of the nodes, tho total number of dps, is = 2''~ 4 r [(2'- 1 2) r-(v-l)s], Article Noa. 28 to 87, Depression of Order of the vtovnal Curve from the Ideal Factor of a Branch or Branches, 28. In the caso of tho r(r s) common points as thus far considered, the order of the y-zonml curve has remained throughout = 2''~ a r, but the order admits of depression, viz,, the constants I, m t &c., and those of tho functions L } M, &c,, may be such that the Norm contains the factor <L> U ; tho p-zomal curve then contains as part of itself (<fc w = 0) the curve O ~ taken w times, and this being so, if we discard the factor in question, and consider the residual curve as being tho y-zomal, the order of the p-zomal will be = 2"~V- w (r -s). 482 (IN roLY/o.MA], ri'ltVKH. j il or, wlwli is l;ho Htiinn Uiinjj, it w which uxpiumion may duiihiiii lln' liu-tur <l>, or ti lii>{lhT |-ur, ,,\ -I- I>',.| iM-un.-.-, h wn Imvii V/- 1- tot^O, tlm pxpiiiiMinii wilt rtn- ..... mlitiu llu> l',t.-i. (1 l- , mnl ii' u. j||.,,, liavo .A V/- [&<!, :: () (uliNorvi' iJiin i]ii|i!lr-, t m :tl , u ,y r.,naii..n-. ,n 1)1,1.. ,ti.' ir.\.-u-.-fin ln'niw in |,h,> wlioln Hnrirn ,i|' 1'iiiu'lii.iiM / .!/, \v. ; ih.n., i! I,, M, A.-. ,.- ..-,.-1, .ViL,,, '| tl' l'"i'iii / J -f./^ M./I!, willi (,hr> mini' vnl.h.-t ,,r /'. I,J, /,', Ion v,,f!, .till. t,ui -,!-.,, < ( f tlui rnolliiiiKhlH , /, c, (Jinn it itti|ili<>tt tin- lliiv,. i- ( |imiiiit> K \ / ) ,v, , u, /. s ^ , ,v,. ..u, V/ -i- &(..:-.(); and KU in ,,|,|| t .r ras,,.), if I HIV /,\7i,\,- 1,.. lt |,,, u ,)...., , hl ! nxiBtHHinii will nmtmn Mm liichir K ,! ,. .. , ll,,- ,.. ,.,, ,,,| ^u.,,^^,,,, |.,. l(1 ,, tluil) Uiu uximiiHinn c.mliuns ,u. IUi-|..r n .vriuin jmw,,r !- i ! |, ltll ..,, 1( . ,,), 1|( - () ,M l j,, |J,i H .,. IUH | | (1 , ri , fiUll M , ,,, .; ...,, lli . , . il 'l-^-hWti.ft (l tlin NDHII will rniihiiu l.lir liiri.n- !". 30. U IHIH lh!un MM.ntin,,,,,! |,hnl, || l( . fi irill V /(H ,. /,,|,, ( ,,.,,,,,1,.,,.., is , ! H H .M., U,nt i, ],.,,/... ,|, ( . , iprlll v'M- If i,, ,!. ,. |iml|ll|l .., ^ t . i "y Hiiiilt torni- .li.r i.iHiuu,',, if i| |t , ,,,(,, ),, ^ l,\> , V )I:I H f ,, , ,. + fe . u,,. Nimi| wi|1 ,,, .- o Umt now .. A &,. ,,,, ...... - ,, , ..... v ,,',' ', - ...... ^ *,... vvii, " wi 414] ON POLYZOMAL CURVES. 483 being so, the general theorem is, that if we have branches ideally containing the curves 0" = 0, <[:>0=o, &c. respectively, then the vzomal curve contains not ideally but actually the factor <!>"> = (w ~ a + /3 + &c,), the oi'der of the i/-zomal being thus reduced from S"" 3 ? 1 to 2"~ 2 r <a (r s) ; and conversely, that any such reduction in the order of the u-Koinal arises from the factors <D = 0, <E>0 = 0, &c,, ideally contained in the several branches of the y-nomul. 32. It is worth while to explain the notion of tin ideal factor somewhat more generally; an irrational function, taking the irrationalities thereof in a determinate manner, may bo such that, as well the function itself as all its differential coefficients up to the order -l, vanish when a certain parameter <D contained in the function is put =0; this is only -saying, in other words, that the function expanded in ascending powers of <D contains no power lower than <I> ; and, in this case, we say that the irrational function contains ideally the factor <!>". The rationalised expression, or Norm, in virtue of tho irrational function (taken determinate!/ as above) thus ideally con- taining <I> a , will actually contain the factor <D K ; and if any other values of the irrational function contain respectively <l) 3 , &c., then tho Norm will contain the factor <l)+fl+&0. > 88. A branch ideally containing <D = may for shortness be called integral or fractional, according as tho index a is an integer or a fraction; by what precedes the fractional branches present themselves in pairs, If for a moment we consider integral branches* only, then if the jj-fjomal contain <I> = 0, this can happen in one way only, thore must bo some one branch ideally containing <E> = 0; but if tho u-gomal contain <I> Q =--0, thon this may happen in two ways, either there is a single branch ideally containing <D B - 0, or else there arc two branches, each of them ideally con- taining <T?eO. And generally, if tho y-swmal contain * M =0, thon forming any partition to = a. -|- /9 -|- &o. (tho parts being integral), this may arise from thore being branches ideally containing <l>" = 0, <1>^ = 0, &o, respectively. Tho like remarks apply to the case whero we attend also to fractional branches, thus, if the j/-zomal contain <D = Q, this may ariao (not only, as above mentioned, from a branch ideally containing <3> = 0, but also) from a pair of branches, each ideally containing <t>* = 0. And so in general, if tho //-xonml contain <D0, the partition w = 4- ft + &o. is to bo made with the parts integral or fractional (= J or integer H-^ as above), but with the fractional terms in pairs ; and then tho factor *" = may avise from branches ideally containing < n = 0, 484 ON POLYZOMAL CURVES. is, through a single branch ideally containing = 0), than the i'-zonml will havo U-n branches, each ideally containing = 0, and it will thus contain <I> 8 = 0, In fact, ii in the zomal and antizomal, or in the complementary parozomala, tho branches whiiOi ideally contain $ = are + &c. = 0, V^(0+ 17*) + &c, = respectively (for a zomal, the -f&c, should be omitted, and the first equation bo wril,l,i>n = 0), then in the v-zomal there will be tho two branches rn or ' &c.) = 0, each ideally containing O = 0. Conversely, if a ,-zomal contain $ by reason that it ha. two bmnolios m. ideally contannng $ = 0, then either a zomal and its antizomal will each of thorn else a pan- of complementary parazomals will each of them, inseparably contain * <).' 35, Reverting to the case of the y-zomal curve V((0 + i$~) + Vm(04TM) + &ci. = 0, which does not contain 4> = 0, each of the r(r-.) common points = 0, d> - u, n of i e JZ n , tle t r ZOmal; h of * oounto thorefL for 2- i - /r o 7 ia tr\ TT 0== f1SeCtl nS f ourvo pas.es 2- time t i ea h ^^^ ^^^ ' i h ; but c the ,-zo^al-passes therefc e only 2- ' T ^ Th r 8iflllttl each of the ,(,^) comnon , l ts if a f I^f *? 00min011 ^ J thftt !H ve *. meets the ^ i "" ' r . r the r(,- 5 ) common points, each of them a ?2- u "i } P - ' Vif! ;' thfl8 inolml11 therefore counting together as tor*- \, ~ ft))tu P 1 P oint n the wsomal, mirl (r-.) other intersections o/the ^^ StT fche fact01 ' intersections of any zomal with it an^Vn T" 1 ^ 0== ' *"- th0 the ,-zomal, and the ^jj h ei Z cT ^ ' "^ f the are nodes of the , 20 mal, it bif^Zd laTi ? ^ ' r*" 811 ^ contain a power of cD = such DOWP f <J 7 zoma1 ' ail(iiz oraal, or paraxomul residual cm-ves attended io. The n,l! V ^ fc b ^^^ed, and only th Articular case be investigated without diffl u 'f "^ f n dea V ^ ^'/ - - - -,-, L '1:14] ON rOLYZOMAL CURVES, 485 <i> w limy prismmt ibwslf, idoully in a mnglo branch, or in novoml branches, and lilio <!01H(!(|lUmli OniUlTOlKW 111 I,ll0 llllilX)! 1 (!IIH(! of jHnVOl'HOf t.D0 j a cOl'lilUll of tllO HOmills, imhixonmlH, or jmruxmnuH Uio MWOH to bn uoiiHulomd would bo voiy numoimw, and thorn IH mi I'oimim to luiliovn lihiif) fcho nwultH could bo prammljoil iu any inodomtoly Cdiiijitio Corn i ; .1 lilnmifin'o iibsLain IVtini on boring on Uiu ([ Ucl(! NOH. JtH and Hi). Ow tho Trisomtd Ouruo and the Tetmsomtd Curve. !tH. Tliu tri/i until wirvn for iliH I'lvUoiiidiHi'il foriii nf tf<+ K" + IP - 2 K1K- 2 IK/7 - 2Z7K ; in 1 u.s thin may idnu bo wrilton, (1, I, I, -1, -I, -l,}(f/, V, 117 = 0; nut! wo imiy 1'nnn tltin mtioinil I'ljualJon vonfy tho xontM'til roHidtH ii])plioablo to tlio OIVHH in hiind, vi/.., t.hai. l>ho U'ixonuil in u (survo of Llio ovdor 2j', and fchiit // :-(}, ill Uiush uf Us fl l<i itltoi'MIJtStioilH Wllill Y IT = 0, nm|H!Olivtily 1-nunh ilio ii'i/.nninl, 'I'lmro urn not, in gonoral, any notion or (umpH, and bho owloi 1 boiii|( -- 2/', dltti cluHH in 2r (2?- 1). !H), 'L'lui tnlnvxninul i 48G ON POLYZOMAL CURVES. [4 I 'I Article Nos. 40 aud 41. On the Intersection of two v-Zomah having the same Zomal Curves, 40. Without going into any detail, I may notice the question of tho intorauctiim oK two y-zomals which have the same zomal curves say the two tmomals V(7+ VF-h Vir () f M+t/mV+t/nW^Q, or two similarly related tetrazomals. For the trissomals, writing the equations under the form VF + VK+ V IF = o, VJ VF + \lm \/ F-f Vw V"F = 0, then, when these equations are considered as existing simultaneously, wo may, without loss of generality, attribute to the radicals \/F, V?, \/W, tho sumo values in bho two equations respectively; but doing so, we musf^ in_ tlio .second aquation suoeoHnivnlv attribite to all but one of the radicals V Vm, V, each of its two opposite, ior the intersections of the two curves we have thus viz., this is one of a system of four equations, obtained from it by changes of Hi K n -say m the radicals V^ and V. Each of the four equations gives a not of 7" poinln ' we have thus the comnlete m w, .4,.. of the -^ flf intoraootion of fc , () ; curves. the form 41^ But take, in like manner, two tetrazomal curves;, writing their oquatio.iH h, VF+ V^ VF + V;j Vf = . , m ra " fl " - ,. ,r u 7 , eCOnd equatlon attribute successively, suv hat' ea h f Ulei " ^ 01 " 3 Site U - ^ * "^-oction, of tho ^ tnzoraal curve, of the order 2r- rt. 7' . ^ ualloM raprosont each of thorn ft if each of these , a point O ' f ,1 -""''T f 4 180 ' thBteforo in """ P oi ts . " in all 8xV = 32,< intersections at, f "" tW teto20malfl . should havo the order 4r, and they intersect tl" Lf tetra mala ara each of them a ourvo of that not all tho 4^ P Lt s ta onf 2 ^f ^ n ' y 16 '" l )oi " te ' T1 ' explanation i.,, In fact, to find M the interactions of the * t i , n ' eraection6 f the tahwnmnh equations to attribute opposite sims tn 7 , tazomals - ll JL.neo6ssary in theii- two intersections from the equationa as tly 8 Z, 1 tM ^ S ^ W ' ^'' we obtai " 2 '" two equations after we have in , , remalnm S 2r- intersections from the the second equation reversed the sign, sav of VP 4H] ON POLYJJOMAL CURVES. 487 Now, from tho two liquations as they utand wo can pass back bo the two tetraaomal oqnationH, and tho Ih-Ht-moutiunod 2r a points arc thus points of intersection of the two totimomiil ourvon li'oni tho two equations after" such reversal of the sign of v'J.\ wo ouniHit PIIHH buck to the two totrazomal equations, and the last-mentioned 2r a pointH aro thiiH not points of iutomHition of tho two totrnxomal curves. The number of intm'HOutionn of tho two curvuH IH thiiH 8x 2r'\ = lo'r a , as it should bo, Artido NOH, 42 to 45. Tho Theorem, of the Decomposition of a Tetmzonud Curve. 42, I (soiiHidur ttio liotroxnmal ouvva- V/ if -|- Jin Y + */n'W -t- x/pT - 0, wlun'o tho HI mwl <mrvoH arc in involution, that in, whoro wo have an identical relation, atf-l- l)K-f isF-l- d2' = 0; and 1 jiroctMid to nhow that if t, m, n, p satisfy tho 1'olation 1 i m _i n a. ? J - n --I- .--(. --(- -r U, a It o d tho (jurvo hi-ciakn up into two ti-iximialH, In fact, writing tho equation under the form (V* It -I- Vwt V + Vn IF) 3 - ^S 1 = 0, and HiilmtiUitin^ for .7' HH viilun, in toniiH of //, 'I 7 , TK, tliiH is (id -|-?m) ?/ + (MM! -I- ;b) K + (?id +^o) If .|- 2 Vwrnl V K1K -I- a V/iiii V !Kf/ -|- 2 Vfowd V UV ; or, (ioimHUtvin^ tho loft-hand nidn as a qnadrie function of (Vtf, V K, Vlr), the condition for itH breaking up into Ihutora in j l(\~\-pn,, d d A/Hii , d ViHi, that in ,11% y/ J ((bed -I- 7/icdn. -I- dab +juabo) ~ 0, or finally, tho condition JH i + ?? !.,- L l + o. a + b ' c + -' 43. Multiplying by M+;MI, ftnd observing lihat in virtue of biio romtion wo , , , , i i (id -h pa) (wwl +pb) fond" - (hi + pa) (nd -I- yo ) ind" - 488 ON POLYZOMAL CUBVES. the equation becomes or as this is more .conveniently written an equation breaking up into two equations, which may be represented by where where, in the expressions for Vi, &c., the signs of the radicals may he taken determinate!)' in any way whatever at pleasure; tho only oHVct, >f alteration of sign would income oases be to interchange the valuos of (VV^Vw,. \ with those of (V/ 3 , V^ 3l Vtt a ). The tetrazomal curve thus breaks up into two tm.mm 44. It is to be noticed that we have , a m c + bd thati s and that similarly we have 414] ON POLYZOMAL OUEVES. 489 The meaning is, that, taking the tmomal curve */lJJ + VwijF"+ Vn 1 TT = 0, this regarded as a totrazomoJ curve, V^ U+ Vm, 7 + VnTl? + VO!T= 0, satisfies the condition - + ^ + + -,==0; t G Cl and the like as to the trizomal curve \ 45. The equation by which the decomposition was effected is, it is clear, one of twelve equivalent equations; four of these are ,o dV/' bed V ? 7i-i-Ji V?l VI }( a VHI ' ' '\ cda ?)i , u , "vn + ,-j ^ > ) 1 " b v / \ c ??i dab 11 n I i\ ./ ' i ^^ \ / Vi , VTO , , \'p + V- cVp \ abc p and tho others may bo deduced from these by a cyclical permutation of (U, V, W), (a, b, o), (I, in, ri), leaving T, d, 2? unaltered. 490 ON POLYZOMAL CUllVJflS. | -H -I viz., the trizomal curve JlU+ Vw7 + Vi? = 0, if a, b, c bo any (jiianhitiioH wmmutrd by the equation a b c ' (the ratios a, b, c thus involving a single arbitrary parameter); and if we tal<n 7' a function such that aZ/>bF+cTK+d2 ? =0; that is, '/'=(), any ono of l.lus wunvs nf curves af/"+bF+cF= 0, in involution with the given ourvon ?/=(), K0, IK" --(),-- has its equation expressible in the form that is, we have the curve 2'=0 (the equation whereof contains a variable as a zoraal of the given trizomal curve \/lU-\- \^iiiV~\~^n \Y(); and \V(i liuvn lliu-* from the theorem of the decomposition of a tetrazomal dodmiud (.lio Uujurtiiii of t!i^ variable zomal of a trizomal. The analytical invc.sfiigation in .snmowhat, HJitipliiii-il \ty assuming p ~ alt initio, and it may be as well to rupuut if, in UUH iorin. 47, Starting, then, with the trizomal curve nW = 0, mid writing as the definition of T, the coefficients being connected by a b c~ * the equation gives .' substituting in this equation for W its value in terms of U, V, T, wo Imvu (aw + cl)U+ (bn + cm) F + 2c ^llmUV -|- dnT = 0, hich by the given relation between a, b, c, is converted into ac , r be -f-ttiu (F+ 2c vlftiU'V'-t- dnT = ' Da * lat is n j i z,, this is G finally 414] ON POLYZOMM, OUBVES. 48, Tho result just obtained of course implies that when as above 491 the trizomal curve *JlU+ VmF-t- vnW Q can be expressed by means of any three of the four siomals U, V, W, r l\ and we may at once write down the four forms * _ A / -A/ Wvif VT vw \ c" ' V b a ' VabcJr^ VK) VIT) ^ abo* abc nd abc Wl abc' abc' the last of which is the original equation that if the first equation be represented by have and therefore = 0. It may be added w,F+ V^T = 0,that is, if we n _. LJ. -0- i r r " i r i "" h ~ v > bed bc\a b c or if the second equation bo represented by have lf + Vp a 5P - 0,-that is, if w& V4 = and therefore -r c d or if the third equation be represented by have and therefore nd T~ "T T b d 622 492 ON POLYZOMAL CURVES. then the equation of the trizomal may also be expressed in the forms ad ' ac Ajjbe 1 ~V acf' ab ac - ab ' Mac V bd- ab be ' ab ' and / "V jJ 3 ab ed ' 49, These equations may, however, bo expressed in a much more elegant form. Write a '=- a b ' = '- c ' ~ d ' 037*)' where, for shortness, (^S) = (/9 - 7) ( 7 - S) (S - /3), &c. ; (a, ^, 7 ) being arbitrai-y quantitioH or, what is the same thing, Assume : TO : re = pa' : erb' (7 - a) a : re' (a - /3) s ; then the equation - + ^ + - = takes the form a b c 414] <>N I'OLY/OMAL <: anil tho lour forum <if |,ho o(|uabiou ani found tu IMS vix., bin wo urn l>ho tiquivalnnti forum of bho original tt<piubion awnumod to bo (# - 7) V,m'tf -I- (7 - ) ^ II ' I' -I' ( a - #) ^ Tl! ' 1|r "= * 50. I rniiuii'k tiliuti tlin Mittoii'in nl' tlm variable minml may bn itbfciunwl us a (jiuiwrnniHil.iini Llu-nrcin- -vix,, mMipimiitf tlu' t|iuilii(m V///-I- Vml^-l- V/i.lF- with tho (!iiuiili()it \^/i"-l- ^mi/'\- V~ ! ; 0; l.hw liiHli boloiiKH t H tmi<i lumditHl by liht^ blmm linos , T/r:i(), IT "'' | Hill C(|Ulll.inil 1)1' |.}|0 HlllllH *ini)i(l IUHKt, ili W lcMlV, lit) OXpl'(!HU)l ill a Hiinilur I'nnn by ini'iiiw nl' any utlmr tlinm ttin^i'.nlH bliorctjl', Tmt tho oqimtiou of any laugonl. "I' Mm c'unir IH iui -I li/y -I- rc= '(), wluini a, b, art; any (|niLul.itioH HiitwlyinK bho romliUmi l |--'" !' ^O; whnniHi, wi-il.iiiK miH- ly/-H^-Hlw -^ 0, u ' (! llia }' infii-oiltuio ?w = (i h c- alontf with any Uvo iti 1 Mtit <r'^uml winiiiU /"(), ,v^0, s(>, or, instead of thorn, any tihmi AuHiliiniiH 1 nl' Mm lunn w\ anil Mum Mm nmvn (thivn^i "'' "', ?A ^i ""' int_ 'A K ^. -^ givon bho Llii'iii'i'iii. Hub it, w IIM oany Mi iMiiiiliinb l;lm analyHW with (U, V, W, T) m with (f, .i/, ~, w), unil. mi I'nniliic.l.nil, it in roaliy Lint Htinm iHialyHiH IIH bhab whoroby the (ilworiMii in 1'ni.abliMhnil KH(, Nn. '17. fil. Ib in worbli whiln in uxlubib bho iMiuutiinii of bhu *!iirvn m n , , . t . in a liirm luMilainiiiK Mnvn now wniialH. OliMorvo ft hub th( oquaUon ( - -I- - }) + - U is by aty;x. b-m^, ,-,% if mily -\~ $ -I- ^ - ; or .say, if '-(*", (( ( ^ it- rft". Thn ( M[iuibuni X V(. f (it - '") / ^H- ("' "") (' - ) " l ^H' ( ft " " (t ) < lt " - rt/) B 1V + ^V(/)-70(ft-/O^+X6'"'O(^-'0t^ + CA''"'0^^^^ -. p V ( ; - V o - ") it/+ (o 1 - ") (o' - o)*'K+ (?"- V) (0" V) If IM ciniHwiuontty IIH omiabion involving thvoo /.onialw of tho pi.-i ........... , - inino \,n. v in Hiu-liwim- UK In identify thin with tho original wpuitiiMi VH7-I- vi/., writing Hiiert-ssivuly //(), K0, ll r -0, wo find (it 1 - (O X -I- (V - b") p -I- (o' - c") i; = 0. 494 ON POLYZOMAL CURVES. [414 equations which are, as they should be, equivalent to two equations only, and wlriuh give X : ft : v= I, I, 1 ; 1, 1, 1 ; 1 , 1 , I b, b', b" C, C , G a , a , a ' c, c', c" a, a', a" b, b', b" ami the equation, with these values of X, p, v substituted therein, is in fticb tho equation of the trizomal curve >JlU + *JinV+ */nW = in terms of tliroo now xomuk It is easy to return to the forms involving one new zomal and any two of tho original three zoraals. Article No, 52, Remark as to the Tetrazomal Curve, 52. I return for a moment to the case of the tetrazomal curve, in order to H!IDW that there is not, in regard to it in general, any theorem such ay that of the variable zomal. Considering the form vV+ Vmy -|- V* + */~pw = (the coordinates #, y, z, w nni of course connected by a linear equation, but nothing turns upon this), tho curvo M here a quartic touched twice by each of the lines to=>0 t 7/ = ( z = 0, w (via., oiicjh of these is a double tangent of the curve), and having besides tho throe notion (K=IJ z = w)> (x = 2, y^w], (01 = 10, y = z), But a quartic curve with throe nodos, m- trmodal quartic, has only four double tangents that is, hesidea the linoH a=0, ^=0, s-0, w=0, there is no line <w + fy + yz + &w = Q which ia a double tunwmt f bho curve; and writing U, 7, W, T in place of ,, y> z } W> then if U, V, W, T um connected hy a linear equation (and, d fortiori, if they are nob HO connected), bhoru \* not any curve Z7 + /3F+ 7^+8^0 which i s related to tho curvo in the sanio way with the hues ff-0, 7=0, W = Q, 2'=0; or say there i s not (besides tho oum* 1 , J 31=0)l any ther zomal a^+jSV+V^ + Sa'^O, of bho liobrtusnnml curve The proof does not show that for special forms of U t 7, W, T thoro nmy not be Z omals, not of the above form ff+/97+ 7 F+ SZ'-O, but bolonging to iv separate system. An instance of this will be mentioned in the sequel, Article No, 53 to 56. The Theorem of &, Variable Zonal of a Tribal (hn 53 I resume the foregoing theorem of the variable sonml of bho tfuomal ourvo F.a The vanable zomal 2'=0 is the curve aZ7 + b7+oTr- 0| wh.ro a, b, c are connected by the equation i + * + ?_ ; that is, it belongs to a Bi , the of any one O f the om . vos 414" ON POLYZOMAL CURVES. 495 and of two of tho thrno givon em-vow, in identical with the Jacobian of tho three given flvu'voH. I (iall to mind that, by tho Jaeobian of the curves 7=0, V=Q, Tf = 0, is mount tho enrvo J(V t V, ^ ' Y> y , & d a U, tl a V, d a W, ~ 0, vis!., bhu imrvo obtained by equating to HOVO tho .Tuoobiun or functional determinant of tho fmietioiiH U, V, W, Hume properties of Hie Jiusobian, which arc material as to what follows, arn nuintioutid in tho Annex No, I, lj\n- tho (Mimiiloio statement of tho thooroin of the variable xomal, it would ho miooHwuy liu intei']ii-(!l, Hdoinnti-icully the condition a H- ','J + - => 0, tlioroby shewing how tho Hinglo HoriiiH of tlnj vai'iiibhj wnnal i Holodtod out of tho double aoriim of tho c.iirvoH nU + W+uW^Q i" involntion with tho givon ourvos. Huoh a geometrical iiilitn'pi'otatiou of (,\w ooiulition may bo nought for OH followw, but it is only in a liirti(iliH- (!umi, HH aittirwnnlH umiitionod, that a w>iivoniont geomotrifial intcrpvetation in th^itihy obUuned. 5-4. Uoimid.'i- tho lixed lino 0=^-1-^-1-^ = 0, and let it be proposed to find tho IOOIIH ol 1 tlio (r-'l)' 1 iiuloH of tho lino 11 in rogaitl to tho norioa of curves atf + l)V>Tr0, wltitnt '+7 + "0. Take (, i/, a) .w tho tsooHliimtoa of any one a o (> of tho noloH i i|UOHtion, thon in order that (a, ?/, s) may belong to one of tho (r-l)' J poloR of tho line <}-/' -I- wM'-s-O in regard to the curve ai/H-bF wo muHt havo or, what in tho namo thing, and filiOHo uquatioiiH give without difficulty n : b : c-,/-(K, W, O) : J(W t U, whonco, Hubntituting in tho equation ^ + ^- + 0"' wo Imvu as tho locus of liho (*-!) polOB in question. Each of tho Jaoobiana is & function of tho order 2r-2, and the ordor of tho IOCUB is thus =4r-4. A a tho givon curves y0 F0 TK0 bolong to tho single aeries of curves, it is clear that tho locus puna' through tho 3(r~l)' points which are the (r-l) pole, of tho fixed lins in regard to tho curves Z7-0, 7-0, W -0 respectively. 49G ON POLYZONAL CURVES. [414 55. In the case where the given trizoinal is = 0, s = r-l, that is, where the zomals + .M> = 0, + M> = 0, + lYcI> = are ouch of them curves of the order r, passing through the r intersections of the lino <!' = () with the curve = 0, then, taking this line *() for the fixed lino fl0, wo havo J(V, W, fl)=>J(& + M^, + m, <D) = <I>{j|/; JV}, Hi for shortness, (tfjn -'("-# Q, *)+*/(# fl <!>), anfl tho liko M to ^ othci . two Jacobians, so that, attaching the analogous significations to IN, L\ and \L W the, equation of the locus is ' l Jj where observe that each of the curves [M, N}^, {N> L} ^ ^ M] = Q ifi a flnrvo of the order 2r-3; the order of the locus is thus =4 r _ G and f aa bolbre) th[s locus passe, through the 8 (r-l)i pomts which are the (,-!)* ob B of ^ Uno *-0 m regard to the cnrves e+ii-O, e + Jtf*-0. 8 + JN&-0 respectively. 56, In the case r=2, the trizomal is where the zomals are the conies + i* = o, 6 + ^ = 0, 0- h TO = each naHsin f v through the same two points = 0, * = 0; the loaus of tho pole of 1 o b ~ m regard to the variable zomal, i s the conic l ' polos and second of tl e en V "' ^ ^ "^ ^ md tho (U '^ thereof, and it M y ", * ^ ^ jMobian c ic case in hand. (See as to 1 Au LT' T** " f ^ thaOTom ^ tho with the plan dopted i h L f lf 'n ^ "^ t0 ' } B '* if ' in acco ^ ce point, = 0, = are til ,1 ^ memoir ' TO at nce " that the . - any three circles, then in the trizomal ^ the ec l uafcloris a- i 414] ON POLYJ50MAL OUBVIflS. 497 coefficients I, m, n are nob given in tho first instance, but arc regarded as arbitrary, then tho hwli-montiouod conic in any conic whatever through tho three centres, and there belongs to mioh conic and the norios of Komalw derived therefrom an above, a Imomal urvo VWI" -I- Vm^}" 4. Vw(" = 0, Thin in obviously tho theorem, that if a variable oireli! linn itn centre on a givon conic, and cuts at light angles a given cirolo, tliou th(i envelope of thu variable) circle in a tmoinal curve V/?r + */inW -{- VV(, whore VI" " 0, s iV'=0, ($"=s() are miy throo circlos, positions of the variable circle, and I, lit, n an.) oniiHtaiiti i|nantitioH depending on thu soletttod tfu'co circles. 1,'AIIT II. (NoH, 57 to 104). iSUHHIDIAHY iNVEHTIQATIONft. Article Nits, 57 and f>S, Preliminary Memitrlca. 57. Wo luivo JHHt henn led do consider tin; ooiiHiH wliifslt piws bhrough two given s. Tlusro in no real IOSH of gunomlity in taking thenc to be tho circnlai 1 points ut infinity, or my tho puinls /, ./" vix,, ovory tbiioreiu which iti anywine explicitly or implicitly rolut.oH tu tht'Hc twn points, may, without tho noeuHsiliy of any change in the Htateiuont tbiM'oof, bo undor-stond IIH a thoorom rolating iiiHtoiul to any two points P, Q. I cull to mind that a oircln in a (ionic passing through tbo two points /, J, ami thai; Miuw at right angles to each otber aro lines harmonically related to tho pair of linuH from thoir intnrsiiotion Lo tho points /, ./" ruHpocbivoly, so that when (/, ( /) are ntplucud by any two given points whatever, tlie exproHHion a cirolo niUHt be understood to mean u onnid pnHHing through tho two jyivou points; and in upoalting of UHOH at right uugloH to (Mush other, it, ivmst be undtiwtood thub wo moan linen harmonically nilutod to tiho pair of linen from their interdiction to the two given points respectively. For iiiHtinico, the tluiorcm that tin.) Jac.oliiun of any throo circles ia their orthotomic oirolo, will mean that the Jaoobian nf any throe conies which each of thorn passes through the two given points IH tho orthotomio conic thnnigh tho name two points, that w, thn conio Hiioh that nb catsh of its intowoctioim with any one of tho three (WHICH, thu two tangnntH arc Imnmmiotilly rulated to tho pair of linos from this intor- Hootion to tho two given points rcHpeotivuly. Such cxtendod interpretation of any theorem in applicable ovon to the thooromH which involve distances or angles viss., "~ torniH "distaiuin" iviul "angle" have a determinate nignification when interpreted roferemio (not do bliu oircular points at inlinity, but hmteml thereof) to any two pointH wluitovor (HKO as to thia my "Sixth Memoir on Qualities," NOH. 22" Phil. !/ ! mwa. > v(l.cxT.TX.(lKfill),pp. (U~-!)0; HOC p. HG; [1C8]. And thia boingao, the theorem can, without change in tho Htatoment thereof, bo understood as referring to tho two given pointH. 58. 1 nay than that any theorem (referring explicitly or implicitly) to tho circular points at infinity J, J, may bo understood an a theorem referring instead to ^any two givon pointH. Wo might of course givo tho theorems in tho first instance in terms explicitly referring ta tho two givon points (viz., instead of a circle, apeak of a conic through tho two given points, and so in other instances); but, aa just explained, this is not really inoro general, and the theorems would bo given in a less concise and 0, VI. G3 498 ON POLYZOMAL CURVES, 4 | .( familiar form. It would not, on the face of the investigations, be apparent tlmt- in treating of the polyzomal curves &c. = 0, (0 = a conic, $ = a line, as above), tlmt we wore really treating of tho curve* zomals whereof are circles, and therein of the theories of fool and focofnci us ill to he explained, And for these reasons I shall consider tho two points <$ = (), <I> to he the circular points at infinity 7, /, mid in the investigations, &c., mako MH the terms circle, right angles, &c., which, in their ordinary .significations, havo hnp reference to these two points, The present Part does not explicitly relate to tho theory of polyzonml <jurvi% contains a series of researches, partly analytical and partly geometrical, whioli will made use of in the following Parts III. and IV. of the Memoir. tint Article Nos. 50 to 62. The Circular Points at Infinity; .Reotanyular and Circular Coordinates. 80. The coordinates made use of (except in tho cases where tho gonoml (riliu^r coordinates (,, y, ,), or any other coordinates, are explicitly referred to), will l,u ,i|,|,,,,- the ordmary rectangular coordinates K , y, or else, as wo may term thorn, tlu, nm.lur in for . it boing uBdoto,Kl tlmt, n ret i, ' ein eitteftr V d?f -i Th s r? " ft Mlcl ' (=:l) ' olse g> T/, .am 2^-lj. ihe equation of the line infinifv i - n n , r r by the equations (,, + i,/=0 ,- N !, t ^ T ! th P mtS J ' ^ '"'" 1V "" by the . nations (f4 /= fa dt-flTn' '""V! Wh ' ^ th MU " n coordinates the Jfdinaies of^ ^'.T^TS ^ T m the circular coordinates they are (1 00)11 m ' n ^ ', } r " the points /, J, are * respectively, or they are ec t ua ^n S of the lines through A to respectively These equations, if fo a ' } , (fl fl , flm . . equations of any two lines through the JL. 7 r arbl ^ry, will, it is clear, bo 6 "ie pomes 7, ,/, respectively. I -t I (IN |'(i|,Y/,t)MAL (MJIIVKH. lil, Wr liavc iruni I'ilhiT nl' iJh 1 1'iiiuit'ioiiK in (n\ //, 2) flml i'i. iliu diMluiH'c I'miii riH'h nl.lii'i 1 n|' any t-wn points (,', #, I), ami (<(, a', I) in a lint* lhriiii|'li / IT ./ in Ml. And in piu'liriiliu', il' 5 (I, llum nj 1 ( ;'/ J ; thub IH, t.bu ili^.lain-1' nl' tin 1 |i"inl> (n, i/', I) fmiu / ur ./ in in i-ni'-li CUHP-.U). li',1. ( IiMt'ihlri 1 i'T u fimnii'iil. miy llirrr {miiifs /', Q,A\ tlin |miq)uu<!i(Miltt.r (lintiui(!(t nl /' iVuhi (,?.! i-' -! Iviinifji' I\)A -;> ilhtimiv (,).-!; if (,J lin uny puinl, nn Uio Hun tln.mi;li rt hi rifht'i 1 nl' ill" |iuin|M /, ,/, anil in |wrUimliU' il' Q lin iiildim 1 ul" \;\w iioiiilH / t ./, llti'ii ilu< tiituujlt' I\)A i" liuilo. ttiil. tint (liM|.nntt<! (,)<'! in ^:0; iluUi IH, llui |it'ipcuiliriilitr tli'iliuit-c Ml' /' I'miii llif linn l.linnijfli ,-!. In niUitT nl' Uin (niinlH /, ,/, lliul iii. I'rnjii uny linr llimii[.;li i'illu-1 1 nl' l-licjio iminls, In . ; .xs , Hill., an jiuil, lal(!(l, (illr hiiuii;!*' /'(,M i'i (in ih', ii' miy tin 1 iriiuitflfM f'(A, I 1 JA urn i>mtli linil.n ; vi/,., ihn i'nniiliiMih<:i (riTtmi^uliii't "i /', ^l li"iiit; (*i\ tf, s - I ), (</, a', I) uv (riixuilar) (f, /, c I ), \ ! t. '', I), llu 1 i'\|Hi'^i"iri fur ith' ilnnlili'M ul' llii-sn l,riiuij.;lt'M ivspiMitivnly urn u, i'. I lira i'i, tlit;v in 1 .- (iv.'fMii-nlur rHui'tli (rin-iilur nti.i'ililiuh'Ml ^' fl.-, i/ ft',;. It.'pi.-.riiliH^ llin i|..til.l- inriiM 1 1) 1 /'/.'I, KM, i lit III.' pliilll'. .1. /', t'V VI, Wr IlllVi' i'l ,(.*-. us)' I {.V '"-' 1,1,0 > i ir, it, 1 ;r - i(~ I '/{//- u'c), w c/5 l <// - 't'*). *"' , mill tliu Hi|uun;tt 500 ON POLY201TAL CURVES. [4L1 whence the equations of AJ, A } J are f - as = Q, 7; - $z = 0, BJ, B,J f-&: = 0, 17-^ = 0. 65. Considering any point P the coordinates of which arc , ?;, s (- 1), |,it 91, S3, SI,, 53, be its squared distances from the points .4, 7?, A l} .B, respectively; klmn by what precedes 91 = -fl-'* i -, and thence SI. 23 = 51,. 33,; . that is, the product of the squared distances of a point P from any two points A, ft, is equal to the product of the squared distances of the same point P from thu U-,i antipoints A lt E } . This theorem, which was, I believe, first given by mo in llu> Educational Times (see reprint, vol. vi. 1866, p. 81), is an important ono in tho fclmury of foci, It is to be further noticed that we have if ' K t =(a- a ')(/3-/3'), be the squared distance of the points A, S, = -Hnunrod distal* of points A lt S lt Article No. 66. Antipoints of a Circle. 66 A similar notion to that of two pairs of appoints is as follows, via, if irom the centre of a circle perpendicular to its plane and in opposite HOUHOH, w,, m asm, off two distances each < into the radius, the extremitioa of bhao r cn ; , B is clear that the ****** f hef ^ (ln the lailG f ni Tt n e7or L .f " ded ^^ 6aCh ******* * the cen ^ of e on o he twn e ^ te crcle is * it- sect.on of the two cone spheres havmg their centres at the two antipomte reapootivoly. Article No. 67. Antipoints in relation to a Pair of Orthotomic Circles. *^ ^ ts c "' * these two circles cut .U if . i ,, Ah part cute fo , r n 6 Une AB V "-ogh the clebe a irde tci ia radii in the ratio 1 i and T * . bein ' lt la olear . concentric oirolos with thuir ^^r*^ r h f at > i , 01 say they are concentric orfchotomic circles. ON t'OUYXOMAL iJUttVKS. 501 Arl.idr NUM. ()8 to 71. Jfornitt of the Jfywtwn of u Circle, liH, In i vrli tubular I'.nm'ilmnl-iss tho liquation of a oirclo, coordinate!* of centra (it, if/, I) mid nuliiiH t-*u", !H VI" - O - (ts)" -I- (,y - it's? ~ rt'V - ; niiil in nmmlur I'tiniiliiiuUiH, Uio itonriUniititm of tho cimbiii being (a, a', 1), and radius II," us llrfni'i', (.lie. i!i|Mlll<Sini in (ill. I nliMnmi in t )ll ' w '"K' (i ' ial ' tiM1J oi'iK" 1 boinjj at) tiho oontro and tho radius lnMiig > ' I, tlii'ii wriUn^ iilnn of-il, Uu) (Miniition ol' Uio (JU'olo in ^==1, that ia tho riiv.iihii' riKmlinul.i'H nl' any \\n\\ik n!' l,lm cirelo, xprHHt!(l by IIKUUIH of a vaviablo para- ini-lrr O t uriv (/', j, 1 J . 70, (I.niHiilt!! 1 u cumiut, (Hiiiih jf } , Lhii HoordinntDH ol' which (vcotangular) are ..-. /. -(,- ! 1), uiiil (nrcuinr) aro f, 7,, ^ (^ L), Hum blui Ibrogoing ili>!in|.i-. il. in I'li'Hi-, tin; Hi|uiu'i* >!' tho tuiiKonMul iliHtuncu of bho point P from the pivrli. y|":0. 7 1. lint, Ilicrn in uimtlirr ml;t>r[inilntioii iC UUH Hanus function !?[", vin,, writing Ihi-ivin ^ I, iunl thru r^o) j -i*(i/-) il -i-('*v. v H.M- iluu. VI in l.lio H.|Uivil iliHl-iiiitH! 'f P fmiu oitlior of tho antipoiiiliB of bho .-iivlM (iinintrt yin K , il. will l. nusnllwtl.Hl. out f tho plnnu of tho circle), and wo have tln.M Mi.- HI...HVIII llmt ill" K'l 1 ""' 11 (tf |lll(> ^>K"nt,ial iliHtinico of any point 1 from tho itl t..i Hir Mi|iiurn .if its iH^tiuiCrt from uithnv ant.pomt of tho cirolo. t .,v,i n Arl.iolu Niw. 7!i to 77. On a Myntom of Sixteen Points. i- (.1, /A r? . .^) llli y ''" lir < 1 " )1( -y ulio point". ll11 ^ lot tll untipointfl of 502 ON POLYZOJIAL CURVES. [414 As regards the circle R, since its centre lies in BG, the circle passes through (Bi, C',); and since the centre lies in AD, the circle passes through (A lt D,), that is, the four points (A lt B lt G' n A) lie in the circle R. Similarly (A at B it G y , D 2 ) lie in the circle 8, and (A 3) B 3 , 6' 3| A) in the circle T, 74 The points R, 8, T are conjugate points in relation to the circle ; that is, 8T, TR, RS are the polars of R, S, T respectively in regard to this circle ; aucl they are, consequently, at right angles to the lines OR, OS, OT respectively; viz., liho four centres 0, R, 8, T are such that the line joining any two of them cuts at right angles the line joining the other two of them, and we see that the relation between the four sets is in fact a symmetrical one; this is most easily seen by coii.sifloration of the circular points at infinity 7, /, the four seta of points may be arranged thus : A t A 5 , A,, A,, A, S, B lt Bi, C, C : , Q , G s , A, A, A, D , in such wise that any four of them in the same vertical line pass through /, mid any four in the same horizontal line pass through J\ and this being so, starting for instance with (A t , B s , 3) A) we have antipoints of (A ,<U (4 a , A) are (A, C,), (A, A), (0., 4 t ), (A, A) u (G lt AJ, (B lt A), (^ A), (0,, A) (A, B), (G, LI and similarly if we start from (A,, B }t G lt A) or (A, t B tt <7 a , A). 75. I return for a moment to the construction of (A lt B lt G lt A); those arc points on the circle R, and (B t , 0,) are the autipoints of (7?, G); that ia, they are the intersections of the circle R by the line at right angles to BO from its middle point or, what is the same thing, by the perpendicular on W from 0. Similarly (A, A) are the antipoints of (A, 7)); that is, they are the intersections of tho circle J2 by the perpendicular on AD from 0. And the like as to (4,, B. t 6' 3> JJ a ) and (^ a , fi sj C 3) D s ) respectively. 7G. Hence, starting with the points A, B, 0, D on the circle 0, and consbi-ucting AB CI AC* and C nStrUCting also fche P^'Pendiculars from on U the pei-pendiculars on BO, AD meet circle R in (A, >)> (A t > A), GA > BD S >t (0tt Aa)> (B>| A)) " ^' ^ % >, (A a B) (G D) system is " 1 ' 414] ON POLYZ03IAL CURVES. 503 77. II' to fix the ideas (A, B, 0, D) are real points taken in order on the real circle 0, then the points II, 8, T are each of them real ; but R and T lie outside, $ inside the circle 0. The circles R and T are consequently real, but the circle >$' imaginary, viz,, its radius is =i into a real quantity; the imaginary points (A 1 , B t , G l} Dj) arc thns given as the intersections of a real circle by a pair of real lines, and the like as to the imaginary points (A S} B 3 , G 3 , _D 3 ) ; but the imaginary points (A,,, B. 2l G' a , A) lire only given as the intersections of an imaginary circle (centre real and radius a pure imaginary) by a pair of real lines. The points (6' 3) A z ) quA autipoints of (G, A) nro easily constructed as the intersections of a real circle by a real line, and the like an to tho points (7J 9 , A) quA antipoints of (B, D), but the construction for the two pairs of points cannot be effected by means of the same real circle. Article Nos. 78 to 80. Property in regard to Four Oonfocal Conies. 78. All tho conies which pass through the four concyclic points A, B, 0, D, have thoir axes in fixed directions ; but three such conies are the line-pairs (BO, AD), (CA, JW), and (AB, O.D), whence the directions of the axes are those of the bisectors of the tingles formed by any one of these pairs of lines; hence, in particular, con- sidering either axis of a conic through the four points, the lines AB and GD are equally inclined on opposite sides to this axis, and this leads to the theorem that tho antipoints (A a> B a )(0 B , D 3 ) are in a conic confocal to the given conic through (A, W, 0, J.))' } whence, also, considering any given conic whatever through (A, B, G, D), liho points (/L 1( S lt 0,, A), (A,, B,, O tl D a ), (A 3 > B a> O a , A) lie severally in three conies, each of thorn confocal with the given conic. 504 ON POLYZOMA.L CURVES. [414 ellipse we have the two points (G, D), then drawing the diameter OG conjugate to CD, and through its extremity G a confocal hyperbola, the antipoints (0,, A) will lie on the hyperbola. Suppose (A t B, G, D] are concyclic, then, as noticed, AB and CD will be equally inclined on opposite sides to the transverse axis of the ellipse the conjugate diameters OF, OG will therefore be equally inclined on opposite sides of the transverse axis and the points F and G will therefore be situate symmetrically on opposite sides of the transverse axis, that is, the points F and G will respectively determine the same confocal hyperbola, and we have thus the required theorem, viz., if (A, B, 0, D) are any four concyclic points on an ellipse, or say on a conic, and if (A a , B s ) are the antipoints of (A t Jl), and (G 3 , A) the antipoints of (G, D), then (A 3 , B 3 , 3> A) will lie on a conic confocal with the given conic. Article Nos. 81 to 85. System of the Sixteen Pointe, ike Acoial Case. 81. The theorems hold good when the four points A, B, G, D are in a line; the antipoints (#,, 6'i) of (B, G), &c,, are in this case situate symmetrically on opposite .sides of the line, so that it is evident at sight that we have (A lt B lt (7,, A). (Ag, #2, OB, A), C^BJ 83, G s , A), each set in a circle; and that the centres R, S, T of these circles lie in the line, The construction for the general case becomes, however, indeterminate, and must therefore be varied. If in the general case we take any circle through {B, G), and any circle through (A, D), then the circle R cuts at right angles these two circles, aud has, consequently, its centre R in the radical axis of the two circles ; whence, when the four points are in a line, taking any circle through (B, 0}, or in particular the circle on BO as diameter, and any circle through (A, D}, or in particular the circle on AD as diameter, the radical axis of these two circles intersects the line in the required centre M, and the circle R is the circle with this centre cutting at right angles the two circles respectively; the circles 8 and 2' are, of course, obtained by the like construction in regard to the combinations (0, A ; B, D) and (A, B \ 0, D), respectively. It may be added, that we have *) (extremities^) IB.O,A,D, S j- centre and -I of diameter S\- sibiconjugate points of involutions 40, A ; B t D, T) ^of circles y j \A t B' t Q t D t and that (as in the general case) the circles R, S, T intersect each pair of them at right angles ; and they are evidently each intersected at right angles by the line ABGD (or axis of the figure), which replaces the circle in the general case. 82. If the points A, B, G, D are taken in order on the line, then the points R, S, T are all real, viz,, the point R is situate, on one side or the other, outside AD, but the points S and T are each of them situate between B and 0; the circles M and T are real, but the circle 8 has its radius a pure imaginary quantity, 83. If one of the four points, suppose A is at infinity on the line, then the antipoints of (A, D), of (B, D), and of (G, D) are each of them the two points (/, J). 4 14] ON POLYZOMAL CURVES. 505 It would at first sight appear that the only conditions for the circles R, 8, T were the conditions of passing through the antipoints of (B t G), of (G, A), and of (A, B) reypectively, and that these circles thus became indeterminate ; but in fact the definition of the circles is then as follows, viz,, 21 has its centre at A, and passes through the antipoints of (B, 0): (whence squared radius = AB . AG). And similarly, S has its centre at 23, and passes through antipoints of (G', A) t (squared radius =BA.BG)\ and T has its centre at 6', and passes through antipoints of (A t B), (squared radius = CA , GB} ; these throe circles cut each other at right angles. As before, A, B, G boing in order on the line, the circles R, T are real, but the circle S has its radius a pure imaginary quantity. 84', That the circles are as just mentioned appears as follows: taking the line an axis of a, and a, b t G, d for the to coordinates of the four points respectively, then the coordinates of A^ t A arc & (a + d), $ (a - d) ; whence, m being arbitrary, the general equation of a circle through A lt A is a 3 -h y~ fyn&iz + [ f m (a -H d) ad] z z = 0, writing heroin m ~ d this hecomoH vix,, for rf co it w ^ which is a circle having A for its centre, and its radius an arbitrary quantity L If the circle passes through the antipoints of B, G, the coordinates of these are and wo find 85. Reverting to the general case of four points A, 3, G, D on a line, the theorem a* to the eonfocal conic* holda good under the form that, drawing any come whatever through (A lt B lt 6\, A), the points (4,, J5,, 0, A), and (A^ B a G A) lie in eonfocal conies, these conies have their centre on the line and axes in the dueotaon of and perpendicular to the line. When D is at infinity the confocal comes become any three concentric circles through (A, <& (ft, A), and (A SI Bj respectively. 506 ON POLYZOMAL CURVES. [414 where BCD, &c., are the triangles formed by the points (#, (7, D), &c, ; the analytical expressions are a : b : c : cl = so that 6, tf, 1 c, c', 1 d, d', 1 c, c', 1 d, d', 1 a, a', I a, a', I a 4b 40 4d =0, ^fl -L n/i -i. r*f .4. n/7 M tlLV Ut/ t^ ^J\J \ Ulu "^ \f aa' 4 b6' 4 cc' 4 dd' = ; this being so, it is clear that we have a, a', 1 c, c', 1 z* [a (a? a constant. - a"') 4 b (& 3 2 4- 87. I am not aware that in the general case there is any convenient expression for this constant If; it is =0 when the four circles have the same orthotomio circle; an fact, talcing as origin the centre of the orthotomio circle, and its radius to be =1, we have a a 4' 3 -a" a = l, &c., whence that is, if the circles A, B, G, D have the same orthotomie circle, then 51, 33, (", 55, a, b, c, d, signifying as above, we have and, in particular, if the circles reduce themselves to the points A, B, 0, D respectively, then (writing as usual 2T, S3, (, 3) in place of [", S3 , <', 2)) if the four points A> B, 0, D are on a circle, we have 88, This last theorem may be regarded as a particular case of the theorem a9r4bS3 +oS + d5> =K#= K, -viz., the four circles reducing themselves to the points A, B,' G, D, we can find for the constant K an expression which will of course vanish when the points are on a circle. For this purpose, let the lines BO, AD meet in R, the lines GA, BD in #, and the lines AS, CD in T\ we may, to fix the ideas, consider ASOD as forming a convex quadrilateral, R and T will then be the exterior centres, 8 the interior centre; a, b, c, d, may be taken equal to BOD, -GDA, DAB, -ABO, where the areas BOD, &c. ( are each taken positively. The expression a2t 4- b23 4- c( 4 d2> has the same value, whatever ia the position of the point P (to, y, g = 1), ; taking this point at R } and writing for a moment 414] then. ON POLYZOMAL CURVES. 507 BGD = (ROD - RBD) = RD (RG - RB) sin R = (7 - /3) S sin R, with similar expressions for the other triangles; and we thus have - = s 2 sin 72 (/3y - aS) (7 - /3) (S - a), that is, replacing a, & 7, 8, by their values, and writing also z = 1, we have a[ + b33 + cCS: -h d2> - ^ sin jfi . (KB , EG - RA . RD) BG . AD, whore ^smH.BG.AD is in fact the area of the quadrilateral ABGD; we have thus D whore it is to ho observed that &4, 50 being measured in opposite directions from S, must bo considered, one as positive, the other as negative, and the like as regards SB, SD. This expression for the value of the constant is due to Mr Crofton. In the particular case where A, B, C> .D, are on a circle, we have as before 89. If the four points A t B, G, D t are on a circle, then, taking as origin the centre of this circle and its radius as unity, the circular coordinates of the four points will be the corresponding forms of 3(, &o., being 91 = (f - o) the expressions for a, b, c, d, observing that we have & 0-', 1 1 1, & /3 3 S'vS 7> 7~ l 1 1, 7. 7 s 8, 8-', 1 1$ 5a , 6, 6 J if (/9y8>, &c. denote (/3 - 7) (7 - 8 ) ( s - 0). &0 " become which are convenient formulEQ for the case in question. 508 ON POLYZOMAL CURVES. [414 00. If the points A, B, G t JD, are on a line, then taking this line for the axis of , we may write 91 = (a - azf + f - 'V, &c. It is to be remarked hero that we can, without any relation whatever between the radii of the circles, satisfy the equation in fact this will be the case if we have a + b -t-c + d =0, aa +b& +cc + dd =0, a (a a - " 3 ) -f b (6 a - 6" a ) + c (c 3 - c" 3 ) + d (cZ a - fZ" 3 ) = 0, equations whicli determine the ratios a : b : c : d. In the case where the circles reduce themselves to the points A, B, 0, D, these equations become a +b + c + d =0, a +b& +cc + Ad =0, aft" + b& 3 + cc a + d# = 0, giving a : b : c ; d = (6cd) : ~(cda) : (dab) : -(o&o) ; if for shortness (bed), &c. stand for (6 - c) (c - d) (d - 6), &o. : and for these values, we have =0. 91. A very noticeable case is when the four circles are such that the foregoing values of (a, b, c, d) also satisfy the equation a2l the condition for this is obviously aa" or, as it may also be written, , V s '" " Article No. 92. On a loous -oomeated with the foregoing Properties, 92 If as above, A, B, G, D are any four points, and 2[, <B, K, 3) are the > cur vu f u foui ' P inte ^spootively, then the the foci of the comes wbch pass through the four points is the tetra Z omal posits r + p e+ M- +<tD n haa ' H has been aem - a (> t t TOI - * a " pos,Uon s of the pomt P; tatog P to be the other foe U8 , its squared distanoes are (k-Af, &o., whence for the first-mentioned focus we have 14' ON POLYZOMAL CURVES. 509 or rocollooting that a + b -I- o + d - 0, it follows that wo have for the locus in question a V?f -|- h Vi!3 -|- o V( -I- d V'33 = ; this IOOUB will bo discussed in tho sequel. I remark horn, that in bho ciuw whoro tho four points are on a circle, then (as mentioned above), tho nxiw of liho Hovoru.1 conies am in tho Hamo iixcd directions ; there are thus two sets of 1'ooi, thoKo on tho axln in ono direction, and those on tho axis in tho other direction; it might thoroforo bo anticipated, and it will appear, that in this case the totrammal broaltH up into two fcrissomul mirvoH. Article NOH. 9J1 to !)H. Formula* <w to iho two Sets (A t B, C, D), and (A lt B lt O lt A), cttoh of four Oonoyclio Points. Oil Umwidoi' tho four points A, tt t C, J) on a circle, then taking, as before, their uimilur ooordiuatoH to bo (, a', I), (/3, /3', 1), (7, /, 1), ($, ^ 1). ^ condition that tint jiointH may bo on a oirolo is 1, a, a', aa' 1 t t' 1, 6 , 6 , may bo writton vix,, thin o or if, for shortnoHH, wo tako ami fi I' thon tlio onuation i -A'-/, i .(. I -t- o S3 Oj tt' -1- &' + c' ~ 0, a/: (47 : ofe -ft/' : 6'ff' : c'V. f)4. Lot a, b, c, d, donoto as boforo (a : b ; c : d~B01) : thon wo havo a : b : o ; cl 7i 7'* ^ 8, 8', 1 8, 5 , 1 a, a', 1 a, ', 1 - , 512 ON POLYZOMAL OUEVES. [414 Article Nos. 9D to 104, further Properties in relation to the same Sets (A, B, G, D) and (A lt S lt G lt A). 90. It is to be shown that in virtue of these equations, and if moreover - + -,- H --- h^ = 0, 1 abed then it is possible to find /,, m lt n )t -pi, such that we have identically - m + w33 + g - p<$ + 1& - WjS, - !<, + p,3), = 0. This equation will in fact be identically true if only -ffl+yg'm + hh'n , -gh'^-g'hn^ = 0, cc'm + bb'n -ffp . 4- cb'in^ + bc^ = 0, gtfm-hVn + ffh -i-(/&S -ho'ih =0, cg'm - bk'n , + ch'm, + bg'^ +ff$i = 0. From the first and second equations eliminating m^ or ,, the other of these quantities disappears of itself, and we thus obtain two equations which must be equivalent to a single one, viz,, we have bc'ffl + o'g'afm + 6 h a'f'n + y'hffp = 0, b'cffl + cga'f'm + b'Vafn + gh'ffp = ; which equations may also be written and it thus appears that the equations are equivalent to each other, and to the assumed relation i+ + + =0 . abed 100. Similarly from the third and fourth equations eliminating m or f the other of these quantities disappears of itself, and we find cg'ff'l, - cga'fm, + a/G'g'n, - o'gffp, = 0, bh'ffl, - o/6'A'm, + 6/m/X - Vhff pl 0, equations which may be written /'/' W it' - 414] ON POLYZOMAL CURVES. 513 where we see that the two equations are equivalent fco each other and to the equation + + !!! +! = . ftj b, c, di It thus appears that the quantities l lt iti^, n l} p lt must satisfy this last equation. It is to be observed that the first and second equations being, as we have seen, equivalent to n single equation, either of the quantities m lt n lt may be assumed at pleasure, but the other is then determined; the third and fourth equations then give / 1( ^; arid the quantities l lt 7^, n l} p lt so obtained, satisfy identically the equation + T- i + + y 0. fti D! G! Q! 101. Now writing ff\ - - g (c'm + b'mi) + h (b'n + C'HJ), ffl\ = - G (r/'m - h'm^ + b (h'n - g'n,} t and ff'P = c ( G ' m + t '' wo find + frnj) (A'?t - </'!) + Q/'m - /t'm,) (b'n + o'n,)], thai) is .^' (^i^i ~ ^) = rt via,, this equation is satisfied identically by the values of J,, m,, n,, p : determined as above. 102. Hence if wii^amn, we have also Ifa^lp, and we can determine TOI, !. so that wiiTh shall m, vin., in the first or second of the four equations (these two being equivalent to each other, as already mentioned), writing m* = On, and therefore n t = g in, wo have cc'm + Wn -ffp + oft'n^ + Mm - Q = 0, which are, in fact, the same quadric equation in Q, viz., we have _ _ Wn-ffp cV bo' ' The final result is that there are two sets of values of l lt m^, n lt p it each satisfying the identity i - w,j + PII 0, 65 c. VI. 514 ON POLYZONAL CURVES. [414 and for each of which we have li , mi , HI , pi A , , - = wwi. 103. Consider, in particular, the case where p = ; the relation l + ^ + 5+^o a+b+c+d U) here becomes , __ cu/' a'h i ~~lf m -^f n - The equation in Q is . _ (co'm + Wn) & + eb'nP + bo'm = 0, viz., this is gvng tf-* m--- -- cw o ' Ml .- o ' ^-"T or else Since in the present case ^, = 0, wo have either ^ = 0, or else ^ = 0, and as might be anticipated, the two values of 6 correspond to these two cases respectively, viz,, proceeding to Bnd the values of l lt p l} the completed systems are e -\ ' *.=^K~H' --7- ' ! '=-f> ^=. s -n- r .-o '.-. <=-^, P-,. so that for the first system we have and for the second system n\ p\ f 5 6 fol ' egoing investi g ation would have assumed a more eimplo AMD v o o cncle AMD as ongm, and the radiu, of this circle been put =1; W e should then have <*=-, &c. ( and consequently -- f * he Cil ' leS AB01) mA *AA A * have been 414] ON POLYZOMAL CURVES. 515 I will however give tho investigation in this simplified form, for the identity (51 + m33 -1- w(S = l$\ 4- in$ + n$ ; viz., in this case we have I _ in (/3 7) (/3 S) (/3 7) (y S) - - J ft"~-"x ' S) ~ y ( mid tho identity to bo satisfied is writing foe, q^i*, wo find m,, and writing = as, y^-z, we find w,, and it is then oasy to obtain tho value of l t , viz., tho results are mi = ' a and therefore m,, = m?i; it may bo added that we have 8 a S \ 7 S/ ' / tf?) 7f vk, this is tho form assumed by tho equation :r + tr + ~ = Oi Hj MI \>\ PA.UT III. (Nos. 105 to 157). ON THE THEORY OP Foci. Article Nos. 105 to 110. Explanation of the General Theory, 105 If from a focus of a oonic we draw two tangents to the curve, these pass respectively through tho two circular points at infinity, and we have thence the generalised definition of a focus as established by Hiicker, _ TIB, m any or " focus ia a point 8 uoh that tho lines joining it with the two circular points at are respectively tangents to tho curve; or, what is -the same thing, i tho circular points at infinity, say from the points /, J tangents curve, tho intersections of each tangent from the one point with e the other point aro tho foci of the curve. A curve of the class n ntuj , - , nem ' fcd It ia to be added that, as in the conic the line joining the points nlact of the two tangents from a focus is the directrix ~_* M - to* in question, 516 ON POLYZOMAL CURVES. [414 106, A circular point at infinity / or /, may bo an ordinary or a singular point on the curve, and the tangent at this point then counts, or, in tho case of a multiple point, the tangents at this point count a certain number of times, say q times, among the tangents which can be drawn to the curve from the point; tho number of the remaining tangents is thus = n - q. In particular, if the circular point at infinity be an ordinary point, then the tangent counts twice, or we have g=2; if it be a node, each of the tangents counts twice, or q = 4 ; if it bo a cusp, tho tangent counts three times, or </ = 3. Similarly, if the other circular point an inanity bo an ordinary or a singular point on the curve, the tangent or tangents there count a certain number of times, say q' times, among the tangents to the curve from this point ; the number of the remaining tangents is thus =n~q'. And if as usual we disregard the tangents at the two points I, J respectively, and attend only to the remaining tangents, the number of the foci is = ( - q) (n - q'), 107. Among the tangents from the point / or J there may be a tangent which, either from its being a multiple tangent (that is, a tangent having ordinary contact at ^ two or more distinct points), or from being an osculating tangent at one or more points, counts a certain number of times, say r, among the tangents from the point in question. Similarly, if among the tangents from the other point J or /, there is a tangent which counts / times, then the foci are made up as follows, viz. \vo have Intersections of the two singular tangents counting as , r ' r Intersections of the first singular tangent with each of the ordinary tangents from the other circular point at infinity, as ... / , ,-. y ......... (n~q -r')r Do. for second singular tangent, ... f _ _ *. 1 / ' ' V* '1 ' J * Intersections of the ordinary tangents .... (n-q-r)(n- q' ~r') Giving together the ........ fr-ff) ("-flO fn " CaSG where ** tang^t" from each of the points I, J include more than one singular tangent. or r case to b6 considered; the line infinity may be an he I 1 f g "^ b * fche : Burning that it counts / tinJs among the tangents from either of the circular points at infinity, the numbers of the ON 1'OLYXOJIAH CURVES. 517 IK). (loimithsr any two foci A, 11 not in lined with either of the points /, J, Minn joining tlmHo with tho points /, ,/", anil taking ./In ./^ the iutorflootions of AT, HJ and tif ^li/, ///' (yl u 7J, being tlmroi'oro by u foregoing definition the antipolnts of (A, J3)), then A i, //, are, it !H clour, foni of the curve, Wo may out of tho ;i a foci soloct, and Mutt in I.2../) ilill'm'tmli wiiyn, a system of p foci such that no two of them Ho in linwl with either of Mm point* J, ./'; and this being HO, taking tho antipoints of rai'h nf thn i/i(;)--1) pati'H nub of tho p foci, wo havo, inclusively of tho p foci, in all /I-I-2. 4/>(/> 1), that is y/ 4 foci, tho ontiro Hyntom of foci. Arl.itiln Nun. I'll tin II 7. (;. tho Foot of Oonioa. Ill, A onnii! in u (iiirvn of thn (iliwa a, and th'o inunbor of foci w thus =4. Taking MM loci any two piiiiilw A, H, tho roinainiug two foci will bo tho antipoints J,, //,. In nvdiu- thud a givtsii point ^l may bn a 1'ocun, tho conic miwt touch tho lini'H ;l /, /I./; Hiniiliirly, in urdi'i' that a given point Jt may bo a fooua, tho conio iiniHi, l.uirh tlui linoH ,/{/, 7U; Mm cipiation of a conio having tho given points A, .11 Inr t'ici c'oiilninH tlii'i'cfuro a ningln avhitrury paranmtnr. 111!. In Mm CIIHO, liuwnvov, of liliu ]mmhola tho curvo touolioH tho lino infinity; Mmm in ruitKi!i|UiiiiL)y from nmh nf tlio points /', J only a wnglo tangont to tho I'.iii-v.', anil i:itiiHui|iuml.ly only cum fuouH: Mm juiraholtt having a givon point A for its liit-ii* in a i-uiii.'. Iniirliing Mio lino infinity and the liiuw AI t AJ, or aay tho throe nidi-it nl 1 Mm iriaugln AU] il.H liquation wmtimw Mioroforo two arhitrary pammoteiu i. Hutuniing to Mm gonoml ctinic, thmi aio c.ortain tnx.omal 1'orniH of tlio focal , mil, ul' any grnvb intnvHt, but which may bo mentioned. Using oirailar lrH, unil lukiiitf (, *', I) and (/3, /H 1 , 1) for tho coordinates of tho givon foci .-I, n vrhpi'p.tivt'ly, tlm conin bmiclum Mm lines f - M 0, ?/ - fit's = 0, f-^-0, i) "t-l's> ()-, Mm cqnatimi of a oonio Umching tho first three lines in Vi ("- ) -1- Vw (f -W) + ^ ('/-"a 1 ^) 0, vvln-ru /!, /, ui'o arbitrary, and it is any to obtain, in order Mmt tho conic may hitmh tin; fourth lint* j -#'*(), Mm eondiMou [U. In ihul, rt luiving thiH vttluu, tho otiimtion gives i (^ - M) 4- M (f - /?*) + 2 N/MF^^TF^) Iv (m ~ uml tiiking uvor tho lunn | , " ft , (? - 08' - ') *, (J8 - ) ( - *. # -a this givtw __ I (|? - #*) + m (f - ) -1- 2 Vim <-)!*" - /3*) - - IT? 518 ON POLYZOMAL OUHVES. [414 which puts in evidence the tangent y 13'z. It is easy to sec that the equation may be written in any one of the four forms ^) + (m - (* - '*) = 0, ( m _ viz., in forms containing any three of the four radicals Vf^T^ Vf-^, V?/-V^ v-n-fie. The conic is thus expressed as a tmomal curve, the Homals being ouch a hue, viz., they are any three out of the four focal tangents; the order of the cui-vo, as deduced from the general expression 2-V, is = 2 ; so that there is hero no dopraaHion of order. J-\ 5 '. But the ordinai 7 form of ^e focal equation is a more interesting one ; vix., ,$ being as usual the squared distances of the current point from the two mvon loci respectively, say then 2ft being an arbitrary parameter, the equation is the equation is here that of a tmomal curve, tho .omals being curves of the '' e s ( ^ o the ]in rttd'^v;' e rf s T ( ^ o) the ]in ^ ^ * -4 but in f', 7 PeCtlV t f y: the 8^ Pum 2-r gives therefore tho orclor -4, but m the present case there are two branches, viz., tho branches tline . the Ime mmt tw and onuttmg this factor the order IB =2, as it should bo. i ^ ^' ting tho ared distances of the current point from A lt J3, 414] ON POLYZOMA.L CURVES. 519 whovo k in tilio squared distance of the foci A, S, =4a a e 3 suppose: whence putting tt a (1. o 3 ) 6 a , tho equation becomes that is which is the required now form, It is hardly necessary to remark that the equation 2((S + V91 -I- V35 = 0, putting therein z~\, and expressing SI, 23 in rectangular coordinates measured along tho axen, is the ordinary focal equation 2a = V( - aef + f + V(aj + aej a + if. 117. I remark that tho aquation 2(t2+V2l+VS = gives rise to 4-a 3 but hero 91 - 93 = - 'kecflw, HO that the equation contains 2 = 0, and omitting this it buoomoH (tte - ow) + VSt = 0, a bmmial form, being a curve of the order =2, as it should bo; thiH iw in fiiot tho ordinary equation in regard to a focus and its directrix. Artiolo Now. US lio 12JJ. Theorem of the Variable Zomal as applied to a Conic. i:iH. The equation 2ib + V?T ! + ^W 1 = is in like manner that of a conic; iu Cacti, thin would Ins a curve of tho order = 4, but there are as before the two branches 2/^ + VVl"'Var0, 2/^-V9r + VffiT = 0, each ideally containing (s = 0) the line infinity, and tho onluv IH thus retluced to bo =2. Each of the circles 51 = 0, 58 = is a circle having double contact with tho conic (this of course implies that the centre of the eivcio SH on an oxiw of tho conic). We may if we please start from the form 2& + VW+ Vaj'0, and than by means of the theorem of the variable zomal introduce into tho oquHtinii one, two, or throe such circles. 110. It IH in thin point of view that I will consider the question, viz., adapting the formula to tho cimo of tho ollipao, and starting from the form 2s + V(;'^ae0) a "T]/ a + V(ai+w) fl + f = 0, the ufiutttion of the variable zomal or circle of double contact may be taken to be 4aV (<a - aas) B + f_ (<& + M*+ = Q -2" + ' 1-ff 1 + 3 whovo q IB an arbitrary parameter; writing for greater simplicity ,-1, and reducing, the equation IH 120. If q< 1, then writing q=-s\n9, we obtain the ellipse as tho envelope of the variable circle <>N* IMH.Y/tiM.VI. IT lili I . vix., of ri. mrdit huvimr th fcntn- mi iln> iihtj.>! ;i\\ M ., ,!,. , ,...,. ,/ , , i ., ... , , ' ;! * ' ''"'" Ulr t.tMHt.1'0, lincl ll,s I'lllllllH JT/H-IM//. (1 linllrn. Ill l-i-.tm;. ll^ n, ); . <,>,.: m |.|i.-[i t - vi'ry (ioiivuniiml KTH|ihinil nimilrurtinii nl itm *lli|.i..i h mm j, i..| lll( ,;., ,j ,(,,,, (t : : \ HIU"' 1 !!, l-llr f.'mlc. Il iiH in rlic riivln nl 1 riirvuiui.' nf ..H,' ..i ,.Hi. i .. ( w ; m^,,IVH,n (I , ,j h ttl , |, lp ,., IK Liu. n if,, illt(( ,, I.V I'tir (I | flu . Ih.. ii ( ,'|.-, ( | ,,. , iihM , .,, S4 , l!il. Ill thi' I'HH. f/ >|, Wl - IntlV M trill ,,.|->,... l-hii li^un^ vi/. llllVl! "I 1 , whnl. In Ilir HUH,, U,i nW) A' 1N ll , ( , minor HXIH of t! M , ,,|liu,. ,i V ' V * %iirw1 ' !tl n "' Iil M'hiH i. h, fun d,.;,;.' * "H" ' naH lh " - f, (JUHVES. ""'I,;' "" IVHllir ^' '> <' other extremity of the minor axis; from ' ""' ' /! ' "'" """ri-Unm, of tho consecutive circle, arc real, and give ,-.' tu *flO", tho circles are .till real, but ' !I 'I': '""" '/' --J:I,i,,-' !, K(!ll( , ml , illK (liralo wo interchange /, , o , 6, the - h?/ = HOG" ^, M I- i t-,|iu>f t ti. )1 |, tn |,ho Ilinnur oquation 1 ' 'to HIII ^) <J + y/ J 6 u cos fl, ^lJiMrrhol |y iUCUMH ttf Ulii (((|lialilOU *"'' , iiud Mun'oliiro Hiud^i tan <L kmO I !.!( i?| r i.t^tuiuy !uiii' ; r.tiuiutiun in tlu; thoory of Elliptic Functions. ','j- (,i |;Jn /*%;* f;/* tho ('ircular ('nine and the JBiairoiiiar Quartio. !iH' i (In- dlnws in in gwioral 6, and tho number of the hilly u<iinrNtiii^ (rtwu m thtib of a circular cubic, viz., a cubic jli- fin-ului 1 jininlH at indnity. Horu, at each of the circular ^nt u? lliiH point, i-cokoiw twioo among tho tangents bo the M- iHunltpr ul' tlui voinaining bangonts ifi thus =4, and the Ii., 1* IVnm any two pointa whatovor on tho curve tangents i!i-n ihts l-wi poucils of tangents are, and that in four ititttt^K ,*>. hMUiH^-Mi., h. .tfli othur, viu., if tho tangents of the firat pencil are (1. a, II. **/nnl U<^.. ,,f ttti> fM-c.iiitl in<:il, taken in ft propoi- order, are (!', 2', 3', 4'), tlint H,* I.iur U. '.*, :*. ** h.-iuMl-^ouH with ouch of tho avrangoments (!', 2', 3', 4')> (2 1 T 4' tti in 1 4 T , I i'(, il*. H", 2', 1'). And iii each ease the intersections of the IMUT *''U'j,,HihiiK i*.iirul li^' ', a conic piuwuig through tho two given points (;*1 i 1 '-^ t* * Mt. |...'i in -.:iu, littt u jt<tiii}* tlit'Oi'tlg Mi>'h |f.mu nf intiMit>. (hr run. ti-m ih' j^-"^ ti>n,.l:< i t ,i !h- jj,i r-< lh' v ON POLYZOJIAL CURVES. [414 125. Hence taking the points on the curve to be the circular points at infinity, we have the sixteen foci lying in fours upon four different circles that is, we have four tetrads of concyclic foci. Let any one of these tetrads be A, B, 0, D, then if Antipointe of (B, G)(A, D) are (S lt G } ), (A lt D t ), (G,A)(S, D} (0,,A 2 ), (B,, A), (A ) B)(G,J)) (A 3> S s \ (<7 a ,D 3 ), the four tetrads of eoncyclic foci are A, B, G t D- ^i, A, G lt A; A> -&, Cf,, A; ^3, J a , 0,, A- '^r^ '^ ^ ^' *' ^ -"i ""^ 'T""' and take fl ' ^ th r the rigina h t tf - , 6 TTf ^ SySt6mS is m tact that already discussed ante, No. 72 et seq. if, as ^ipoints of each pair, the four A ' *> G > D > of ointe on vi, 'the The hss s - at the h f altamtion f the the at f the quartic the 'pencils of tants h sixteen foci are related tf. IT circular ou ic T brea, in a binodal res P ecbivel y e homologous, the f ^ f ^ of Article No , m to 129 . 17 tangents in the case of and in the case of a to H * convenient to Qircular Quartic. is c i eal . ,,, > hey 1 T e * asymptotes. ^ ^ r tan from ^ he ly * to the ^ m P totea - Z ., in as y m P totea f ^e om've, intersection of the two imaginary 414] ON POLYZOMAL CURVES, 523 129. In the case of a bicircular quartie, the two tangents at I and the two tangents at J meet in four points, which (although not recognising them as foci) I call the nodo-foci; those lio in pairs on two lines, diagonals of tho quadrilateral formed by the four tangents (the third diagonal is of course the line IJ), which diagonals I call the "nodal axes;" and the point of intersection of the two nodal axes is the "centre" of the curve. The nodo-foci are four points, two of them real, the other two imaginary, via., they arc two pairs of antipoints, the lines through the two pairs respectively being, of course, the nodal axes; these are consequently real lines bisecting each other at right angles in the centre (with the relation 1 : * between the distances). The centre may also bo defined as the intersection of the harmonic of IJ in regard to the tangents at /, and the harmonic of this same line in regard to the tangents at J. Speaking of tho tangents as asymptotes, the nodo-foci are the angles of the rhombus formed by the two pairs of parallel asymptotes; the nodal axes arc the diagonals of this rhombus, and tho centre is the point of intersection of the two diagonals; as such it is also the intersection of tho two lines drawn parallel to and midway between tho lines forming each pair of parallel asymptotes, Article No. 180, Circular Cubic and Bicircular Quartic; the Axial or Symmetrical Qase. 130. In a circular cubic or bicircular quartic, tho pencil of tho tangents from / and that of tho tangents through /, considered as corresponding to each other in some one of tho four -arrangements, may bo such that tho line IJ considered as belonging to bho two pencils respectively shall correspond to itself, and when this is so, tho four foci, A, 1$, G, D, which arc tho intersections of tho corresponding tangents in question, will lio in a lino (viz,, tho conio which exists in the general case will break up into a lino-pair consisting of tho lino IJ and another line). The line in question may bo called tho focal axis ; it will presently be shown that in the case of the circular cubic it passes through the centre, and that in tho case of the bicircnlar quartic it not only passes through the centre, but coincides with one or other of the nodal axes, vise,, with that passing through the real or tho imaginary nodo-foci; that is, the curve may have on the focal axis two real or else two imaginary nod' The focal axis contains, aa has been mentioned, four foni t.ho vnmainine- twelv 524 ON POLYZOMAL CURVES, [4 1 4 the points 7, J is a cusp, There remain then for the circular cubic and for the bicircular quartic the cases where there is a node or a cusp at a real point of the curve; and for the bicircular quartic the case where each of the points I, J is a cusp in general the curve has no other node or cusp, but it may besides have a node or cusp at a real point thereof. 132. I consider first the case of the bicircular quartio where each, of the points J, J" is a cusp. The curve is in this case of necessity symmetrical ( J ) it is in fact a Cartesian ; viz., the Cartesian may be taken by definition 'to be a quartic curve having a cusp at each of the circular points at infinity. But in this case, as dis- tinguished from the general case of the bicircular quartic, there is- an essential degeneration of all the focal properties, and it is necessary to explain what these become. The centre is evidently the intersection of the cuspidal tangents; the iiodo- foci (so far as they can be said to exist) coalesce with the centre, and they do not in so coalescing determine any definite directions for the nodal axes; that is, there are no nodal axes, and the only theorem in regard to the focal axis or axis of symmetry is, that it passes through the centre. Of the four tangents through the point /, one has come to coincide with the line IJ\ and similarly, of the four tangents through the point J one has come to coincide with the line J7; there remain only three tangents through / and three tangents through J, and these by their intersections determine nine foci viz,, three foci A, B, G on the axis, and besides (J5j, C,) the antipoints of (5, Q)>. (0,, A a ) the antipoints of (G, A) and (A a> B 3 ) the antipoints of (A, B), 133. The remaining seven foci have disappeared, viz., we may consider that one of them has gone off to infinity on the focal axis, and that three pairs of foci have come to coincide with the points I, / respectively. The circle (as in the general case of a symmetrical quartic) has become a line, the focal axis; the circles R t S, T (contrary to what might at first sight appear) continue to be determinate circles, via., these have their centres at A> B, G respectively, and pass through the points (B lt G\), (0i, A z ) t and (A 3> B a ) respectively, see ante, No. 83. But on each of these circles wo have not more than two proper foci, and it is only on the axis as representing tho circle that we have three proper foci, the axial foci A, 8> Oi in regard hereto it is to be remarked that the equation of Jfoe curve can be expressed not only by means of these three foci injhe _form Vj2l + VmS + V^ = 0; but by means of any two of them in the form Vm + Vm53 + K = 0, where K is a constant, or, what is the same thing (2 being mtroducedjbr homogeneity in the expressions of [ and S respectively), in the form Vj8[ + VmSJ + K& = 0. 134. Using for tbe moment the expression "twisted" as opposed to symmetrical Jit Witt appear ^f Ho, 161-164, that if storting with three given points as the foci of a bioiroular ciuarho, we impose he condition that the nodes at I, J shall be each of them a cusp, then either the quartic wrtl be the on-ole through the three points taken twice, in which case the assumed focal propeL ~" or else "" 414] ON POLYZOMAL CURVES. 525 (viz,, the curve is twisted when there is not any axis of symmetry, but the foci lie only on circles) then the classification, ia Circular Cnbics, twisted, symmetrical, Bicircular Quartics, twisted, ("Ordinary, symmetrical, -! [Bicuspidal = Cartesian, and each of these kinds may. bo general, nodal, or cuspidal viz., for the two last mentioned kinds there may be a node or a cusp at a real point of the curve. 135. In tlio case of a node, say the point lY; first if the curve (circular cubic or bicircular qnavtic) be twisted then of the four foci A, B, C, D we have two, suppose B and G, coinciding with N ; and the sixteen foci are us follows, viz, B > 0, A , D are JV, N, A, D; 8 lt O lt A,, A N, iY, Antipoiuts of (A, D); G' a , A a , S 3 , A . Antipoints of (N, A), Antipoinfcs of (iY, D)\ A at A, Oa> A Do. do. viz., we have the points (A, D) each once, the node N four times, the antipoints of (A, D) once, and the midpoints of (JT, A) and of (N, D), each pair twice. But properly there arc only four foci, viz., the points A, D and their antipoints. The circle subsists as in the general case, and so does the circle R(BG, AD}, viz., this has for centre the intersection of the line A'D by the tangent at N to the circle 0, and it passes through the point JV, of course cutting the circle at right angles : the circles 8 and T each reduce themselves each to the point N considered as an evanescent circle, or what is the same thing to the line-pair NI, NJ, 136. The case is nearly the same if the curve be symmetrical, but in the case of the bicircular quorbio excluding the Cartesian: viz,, we have on the axis the foci ]3, G coinciding at JV, and the other two foci A, D\ the sixteen foci are as above and the circle R is determined by the proper construction as applied to the case in hand, viz., the centre 11 is the intersection of the axis by the radical axis of the point N (considered as an evanescent circle) and the circle on AD as diameter; that is Sfficx&A.RD. And the circles S and T reduce themselves each to the poi"" * r considered as an evanescent circle. 137. Next if wo have a cusp, say the point K; first if or bicircular quartie) be twisted then of the four foci A A, B, G, coincide with K ; and the sixteen foci arc as follows, S , , A , D are K t K t if, D, B lt 6',, A lt A &> K> Antipoints of O tl A,, B,, A Do. do, A, t B 3t G a> D a Do. do. , , , ut tin* Lw i,,,,| t . r,.H,,rLivHy. Mu- r.pmlMm will , ON POM/MMAI, ruuvr i, |'.| |.| vi/., wit havti tlio puinl. It HIHV, MM- pninf- i\ nim- i;m> . ;uM Hi-- .s^r^^inf < ,,| A" /i Mint<! l.imi'H. Dud pi-nperly l.hc puinl /) in flu- MH|\ t-rii'i Tin- i-ur|, o P , r ,t u.mM uppcur, (HJ// (tiniln Minnigh A', /', 1ml pn'-iilily I|H< pmhrulu * n- !.. i\l tillHpiiltll illll^'llli limy Itit M tn-IIPl- lT|H'i".rlililMV,- M| flu. . n,'!, " ,-!' Hi Mil' ririilcfi A 1 , iS', '/' rciliici- tlu'iiini'lvt-t riti'li h. \\\< p-.in! /i" , .,i s ->i'i. p J poinl. INN. The liliit is ihr ni'u- if MM- rum- li.- i.\imn--iii 1 ',,!. h-K ; ., hi^iiTiilai- (jiiiulir t>xi > linlin/<; Mm t ';u-h",mi t ; ilir . ,-!. u >, h.-i- f! ( . ,. llU.il. Mli> rllHJiiilill ttili^rlil.. l.'ll), Km- the (!iirli*iiiiui, it' MI.IV |.i it n,.,!,- .V, M<m ..( Hj. Mit. l.wn, Hiippittiu /f mill ft, niiin'iili' uiili A'; tin' ninr !' i MI, ,1 . . fl , ., furl nrc A" nine linn'.'i; Imi in |i (l 'i tli>j< \,> t ,, t pi,,|,,i |,., M .< I'M). A eiruultir eultin euiuiul, liuv.- tw. h<I. <i m,i. ., ,i l,,,.^, rmiln; itiid Miinilarly u lijrjrriilur ipmrfje inn,., 1 | til v. )V,M t ,. !, , K nl Mm piuntH /, ,/J iinlt'i^t ii Im-aK up iut-t fv.. ((*].-!, s!i- ?!).->? Ins (UtiiMiileri'il in Mie inrrpifl in ri'lei-fin'e In lh.< |M.I)]..I M >( in, ii^.n . f ' - ---- r..r.r.Tt r.;a-t^-A B :^- in Arllolu N, w , , b, IH ,I,,W, /( ,W 7W, / w rt, r fm ,,,, ,,,.,, ' ..... ' ' "- l ,., T,, ON POLY2OMAL CURVES. 527 is fche same thing, &l (P$ + W + <&) + z" (at; + b*} + eg) = 0, "3. (, v), z) being any coordinates whatever, this is fche general equation of a cubic passing through the points (f0, 2 = 0), (, = 0, = 0), and at these points touched by the hues f0, *? = respectively. And if ( T,, 2=1) be circular coordinates, then we have tho general equation of a circular cubic having the lines f = 0, t}=Q for its asymptotes, or say tho point f=0, <? = for its centre; the equation of the remaining asymptote is evidently p% + qy + ez = ; to make the curve real we must have (p, q) and (, &) conjugate imaginaries, e and c real. "143. Taking in any case the points /, J to be the points = 0, z = and ^ = 0, 2 = respectively, for the equation of a tangent from I write p%=Qss\ then we have dtj (Oz + qri + ez) -\- z that is 2 a (ad + op) -h ^* (0 and the line will be a tangent if only (6 3 + e0 + ftp)* - 4jtf (a^ + qp) = 0, that is, the four tangents from / are the lines p%=Qz, where is any root of this equation ; similarly tho four tangents from J are the lines gy = fa, where < is any root of the equation cq) = 0. + bpy + cpa) = 0, 6p) + rf . ^ = 0, Writing the two equations under the forms 0, e 9 + 2aj - Seay - the equations have the same invariants; viz., for the first equation the invariants am easily found to bo /= 3(e 3 -4&j)-4ag) a + 72(ce~2a&)p2, J- = _ ( e s _ 4Jjj 4 ft!? )3 - 36 (ce - 2al)pq (e a - % - 4a(?) - 2 1 ft " a " 9 - and then by symmetry tho other equation has the same irm invariant! I 8 *- / 9 has therefore the same value in the two ei equations are linearly transformable the one into the other, which is the before- mentioned theorem that the two pencils are homographic. 144. The two equations will be satisfied by # = </), if only ljp aq\ that is, if p j- , q j'j putting for convenience j in place of e, the equation of the curve is then 0. 528 ON POLYZOMAL CURVES, [414 In this case the pencils of tangents are a% = kdz t li) = k&3, where 6 is determined by a quartic equation, or taking the corresponding lines (which by their intersections determine the foci A, S t G, D) to be (a% = k6tf, bi) = k9tf) t &c., these four points Ho in the line al--bi) = Q, which is a line through the centre of the curve, or point 0, T) = Q: the formula) just obtained belong therefore to the symmetrical case of the circular cubic. Passing to rectangular coordinates, writing #=1, and taking y~0 for the equation of the axis, it is easy to see that the equation may be written or, changing the origin and constants, tcy- + (to - a) (a - 6) (a - c) ~ 0. Article Nos. 145 to 149. Analytical Theory fm- the ^circular Quariic. 145. The equation for the bicircular quartic may be taken to be k (f - V) (^ - /9V) k <& T,, ) being any coordinates whatever, this is the equation of a r?, f f I h in f; T th S am; li r g o th ' an ot the same me 2 = m regard to the tangents at ( = (), *o) If we o h r the = for one pair, and the lines -fl, v-' rr- tho point -* we , h.ffl I' " dal . aXe& In OTder that tha and 7-0 ,m . *>. aol n f to MffMri". *, 4 o real. The points (f- itdVlf^r The "i: 16 /;* J ' ^ " "7' the of a^biousida, o ^ be meS & ^ ^ fol * h,cle 8 ourvo of of n -O fni- real 0, -0) ^B a tho case we thati ^uation of the curve (.* . and the condition of tangenoy is ON 1'OLYtfOMAL CURVES, 529 via., the tangontH from 7 arc -*, w hei-o is any root of this equation. Similarly, if wo have tho ttuiguntH from ,/ iiro ^-c/,/30, wlioro 4, in any root of this equation. 14-7. Tho tiwo (!i[un(ii<niH may bo written which () ([ iuifci(.im huvn lih (! HIUMU iiivarianfcHj iu iholi for tbo first equation tho invariants iu'o fcmml tu l)o IIH Co I lows, vix., if for Hlnu-tnoHH (7= ~tikW{3*- 4Ao + fl a , thon J ((t"a a + 6 8 /3 H ) - 21 OJt^fiS and Uum by Hyimiudry bliu othor (!(|ati(iu HUH tliu Hanio invtirianUi. Tho ftbaolufco invni'iuiit 7 il ^,/ a IUIH thiiH thi) sinno vuluo in tho two oqiuitimiH, that i, tho oquafcions iu-0 linoarly traiiHfunnahln (Jin one into tlui oliliov, which in tho before-mentioned thoorwn that tlio iiMic.ilw urn 14S. Tim (iqiiatiiniH wilt ho Hutinfiud hy 00 if only a = ft/9, thnt i, if (t t &m& m; or ^hy 0*z~~<l> if only rta-ft^, that IH, if , ft^wi/9, -,; tho oquation of tho curve is in tlumu two WIWJH 7/) -(- o^' 1 0, ^ (f ~ a'^' J ) (T;' J ~- ^*s") + 85 1J ^ -I- ?HS" (/9f - mj) -(- ca j - 0. 1C to fix tho idww wo ntluiid In tho (imt OIIHU, thuii tho otiuation in ia wo may take OH corresponding tangents through tho two noclew respectively %~ Qftz\ tho foci ^1, 7A 0, D t which aro tho intciwjotions of tho pairs of lines (k ss 0. VI. 67 530 ON POLYZOMAL CURVES. "414 y=d,(3z) l &c,, lie, it is cleav, in the line fti-ct'tj Q t which is 0110 of the nodal axes of the curve. Similarly, in the second case, if & be determined by the foregoing equation, we may take as corresponding tangents through the two nodes respectively %=Q'i.z, i)~ Q$z\ the foci (A, B, G, D), which are the intersections of the pairs of linos (^-Bjftz, t) = -d$z\ &c., lie in the line /3 + ?; = 0, which is the other of iho nodal axes .of the curve. In either case the foci A, B, 0, D lie in a line, that is, we have the curve symmetrical; and, as we have just seen, the focal axis, or axis of symmetry, is one .or other of the nodal axes. 149. In the case of the Cartesian, or when a = 0, /3 = 0, via., the equation am = b{3 is satisfied identically, and this seems to show that the Cartesian is symmetrical ; ib is to be observed, however, that for = 0, /3 = the foregoing formulas fail, and it i.s proper to repeat the investigation for the special case in question. Writing a. = 0, /3 = 0, the equation of the curve is AV + ez-fy + 8 s (of + by) + cs 1 = 0, and then, taking f-0fo for the equation of the tangent from /, we have + i)x.b (e6 + 3 and the condition of tangency is viz., we have here a cubic equation, Similarly, if we have i } = 6az for the of a tangent from J, then 60 + c) - (4 + 1)" = 0. Hence 6 being determined by the cubic equation as above, we may take A = 6, and consequently the equations of the corresponding tangents will be g=*$bg y = 0az viz the foci .A, B G will be given as the intersections of the pairs of lines (f-^fil' V-toM\ &c. The fod he therefore in the line of-6,-0; or the curve is symmetrical the local axis, or axis of symmetry, passing through the centre. Article Nos. loO to 158. On the Property that the Points of Contact of the Tangents /rom a Pair of Qoncydic Foci lie m Gircle, (A GA , four cc ses . Ci. A, a.. A), (A B 2) C t , A), (A tl B a> C s> A), that is, A, B, G, D are in a e rclo rzt 414] ON TOLYZOMAL CURVES. 531 151. Consider tho caso of tho bicirculnv quartic, and take as before (f ~ 0, = 0), and (^ = 0, s=s()) lor tho coord i nates of tlic points J, J" respectively. Lot the two tangents [Voin tho focus A bo oar = (), ?; '# (), say for shortnosa # 0, p' = 0, thon tho equation of tho cuvvo is expressible in the form pp'V= V s ( l ), whore tT= 0, V0 aro ouoh of tliom a oirolo, viz., V and V arc each of them a qtiadric function containing tho torniH &, etj, z% t and ?j. Taking an in dc tormina to coefficient X, the oqnation may bo writ ton pp' ( 7 + 2X K + Tfypt ) = ( V + Xp/) 8 , and thon A, may bo HO determined that U"-h SXV-l-X-^y - 0, shnl! bo a Q-cIrcle, or pair uC IhuiH tlirougb / and ,/'. Ii w titwy to HUO that wu have thus for X a cubic equation, that is, bhuro aro throo vahiOH of X, for each of which the function f/H^xK + X^/ aHHUinoH tho form (% - &z) (y - p'e), = ttf Buppoao : taking any one of thiiHO, and dianging tho value of V HO aa bhat wo may havo Fin place of V+\pp' t tho (iquation in pp'qq' + V' J , whero 7=0 in an boforo a ctrclo, the equation whowB that tho point* of (sontiict of tho tangents -p = 0, ^>' = 0, f/ = 0, </ = () Ho in HUH circle V^= 0, This (.urcnmHtanco (ihut X fa dotoniiiinid by a cubic oipiutioH would auggoHt that the fwiUH v = 0, (/'^O IH 0110 of tho throo foci JB, C7, D ooucyc-lic with A\ but tins is tlio vory tiling whioh wo winh to prove, and tho investigation, though HomowLmt long, IH an intuvoHting oni!. 1fi2. Starliing from tho form pp'qy' V'\ then introducing aa boforo an arbitrary onoftlciout \, thu oijuabion may IMS writton and wn may dotonninn X HO that yr/ -1- 2XK-1-X'V nlmU be a pair of linos. Writing V 77 f 7; - ,V - 7,'f* -1- Jlfa 8 , and Hubatitntiug for ;y/ and ryr/ their values ( j) (ij - 's) and (?-/3)(i;-^), tho t;([uabion in quoHtion iu (I -|- 2X// 1 -I- V) ft; - (jS -I- 2X/, -I- X Q ) 7;^ - (y3' + 2X// -I- Xtf ) f * -h <^/9' 4 2XAT + XW) = 0, and tho required condition in (1 -I- 2X/r -|- X B ) 03)9' -I- 27Jlf + V/) (/9 + 2X7, + X 9 ) (^ + 2X77 or ntduoiug, this IH + X (( - 0) (a' - /S') 4- 4/f Jf ' - 27/ - 2/V) - 0, vix., X is dotorminorl by a quadrio equation. Calling its roots X n and X-,, tho for o<|iiation, Hulwtitubing thoroin fluccoHsivoly those values, bocomos (f - 7*) (o? - /*) 8>) respectively, Bay H-' = and **' = 0. i ThU InvoiiUgatlon is similar to Hint in Salmon's Wglw Plane Qurwt, p. 100, in ngntd to tli tixngontu o( n qimvllo ourvo. G 532 ON POLYZONAL CURVES. [414 153. WG have to show that the four foci (p = 0, #' = 0), (<? = 0, q' = Q), (r=0, / = (}), (s = 0, s' = 0) are a set of concyclic foci; that is, that the lines p~Q t <? = J r=0, 5 = correspond homographically to the lines p'~Q t q' Q, r' 0, s' ~ ; or, what is the same thing, that we have 1, a, a', act' =0, 1, A /3'>./3/3' 1. 7. y 1 . 77' 1, S, S', SS' or, as it will be convenient to write this equation, a-$ <y-^ _ a-S j9- cc'-^'y-S'^a'-S' jS'- 154. We have V ' 8 = The expressions of a-&, &c., are severally fractions, the denominators of which disappear from the equation ; the numerators are for - 8 , = a (1 -i- 2\,# + V) - 03 + 2\ a ) _ for /9 - % for 7 - S, .and it hence easily appears that the equation to be verified is , + a V) (1 + G8 + 2^ + a\*) (1 + + X a 3 ) + V), -fi 155. This is if for shortness B=_ a- the equation then is , ' = -2 (a' -' 414] ON POLYZOMAL CURVES. 533 156. Calculating Aff-A'B, GA'-C'A, OD'-C'D, BD'-B'D, those are at once soon to divide by {(a0'-a'0) #-!-(</ - /3')~ '(a-0)| ; we have, moreover, - {(' - 00') 7^ - L (a' - 0') -'(- 0)} K - ') # + L (' - 0') --'(- viz., this also contains the same factor ; and omitting it, the equation is found to be {(a - 0) (' - 0') - 4 (01? - 1) (&B - ') } -2 {(*'- 00') # -i( + { ( - ft) (a 1 -00 + vix., substituting for \i + X a and \\ their values, this is {( - 0) (a' -00-4 ((3H - L) - {(art 1 - 00') /f - L (a' - 0')) {( - + {-(- 0) (a' - 0') 4- 4 (ff - i) (a'Jf - L 1 )} [M + Hffl - Lj3' - Z'0} == 0, which should bo identically true. Multiplying by H> and writing in the form + {_ (a - 0) (tf - 0') + 4 (off - i) (a'J5f - L') } (m - LU + (0/f - L) ($B - L')) = 0, wo ab onco see that this is so, and the theorem is thus proved, viz., that the equation being pp'q(f= V, the foci (p-O, p'-O) and ( ? = 0, ^ = 0) are conoyclio. 157. By what precedes, \ being a root of the foregoing quadnc equation, we may Wlt tho focus r = 0, r' = 0_ is concyclic with the other two foci; but from the .equation of tho curve V^Jffqq', that is we have qq' or, what is the same thing, viz., this is a form of the equation of the ourve; substituting for p, p 1 , q, ?', r, r' their values, writing also ' *nd changing the constants X, K (*. X ^ : Z-M : ^ : ^ the equation is ws 4 Vwi = o, 534 ON POLYZOAIAL CURVES, [414 viz., we have the theorem that for a bioiroular quartio if (f - az - 0, i) - a'z ~ 0), (f fiz = Q,- y-ft'z^Q, ( 72=0), t] f/z Q) be any three concyclic foci, then the equation is as just mentioned; that is, the curve is a trizouml curve, the zomals being the three given foci regarded as 0-circles. The same theorem holds' in regard- to the circular cubic, and a similar demonstration would apply to this case, 158. It may be noticed that we might, without proving as above that the .two foci (p = 0, p' = Q) t (y-0, </=Q) were coucyclic, have passed at once from the form Pp'<tf=V*, to the form \V^ + t/tg+K Vr/ = (or Vm Wm33 == vV( == 0), and then by the application of the theorem of the variable zomal (thereby establishing the existence of a fourth focus concyclic with the three) have shown that tho original two foci were concyclic. But it -seemed the more orderly course to effect the demon- stration without the aid furnished by the reduction of the equation to the trizomal form. PART IV. (Nos. 159 to 206). ON TRIZOMAL AND TETRAZOMAL CURVES WHERE THE ZOMALS ARE ClKOLES. Article Nos. 159 to 165. The Trizomal CurveThe Tangents at I, J t So. 159. I consider the trizomal where A, 2, G being the centres of three given circles, S[, &c. denote as before, viz., in rectangular and in circular coordinates respectively, we have 51 = ( - azj + (y - dzf - a'V, = (f - a z) (y - dz ) - a' V, S3 = ( - Mf + (y - v s y* - y v, = - - _ yv By what precedes, the curve is of the order =4, touching each of the given circles twice, and having a double point, 01- node, at each of the points i, /; that is, it is a bicircular quartic: but if for any determinate values of the radicals V Vm, >Jn, Wif ii ii . , , " + + =0, then there is a branch containing (, = 0) the line infinity; and the order is here =S; ., the curve here t,r is an"" 7, f t ?. P t ta 7l J mCl """^ an0ther P int * infini; (lala theie is an asymptote), and is thus a circular cubic. ' * points 6 / 1 T Z m t M 1 ^ inVeSt ; gaU "? the e " atioM f *1 odal tangent, at the poms 7 , / respeotwely; using for this, purpose the circular coordinates (fe ,, ,-i) -0 -O^eTaf, TV" "f ed l'n- * ^ding the tangents a (f-0, ,-0} we have only to attend to the terms of the second order in (f ,) and 414] ON POLYXOMAL OUBVKS. 535 mmilavly lor finding tho tangents at (?> ~ 0, z Q) wo have only to attend to the terms of t)ho Hooowl order in (?;, z), But it in easy to see that any term involving a", I", or o" will bn ol' l;lm third ordor at loa.st in (, 2), and mmilarly of tho third ordor at leant in (?/, z}\ honeo for finding tho tangents wo may reject tlio torniH in question, or, what in tho Hiunn thing, wo may write <!,", h", G" each = 0, thus reducing the three oir<;loH In thuir mnpootlvo centres. Tho equation thnn becomes z) (y - a's) -I- ^n (f ~ Js) (7, ~^9'0) -!- V^T(f^)Oy*) - 0. , Fur linding tho tuiigtnitH at (^--O, -e - 0) wo liave in tho rationalised equation to ubkoiitl only to thn tornm of Lhu wmoml order in (^ f 0); and it i easy to BOO that any twin involving ', #', 7' will Iw of tho third order at least in (, 0), that is, wn may rc-duoo a', $', 7' (ih to x.ovo; tho irrational equation thon booomcs divisible by V?;, and throwing out thin factor, it in V; (f - ff ) -|- Vw (f ->) + VB "(f - 7*) = 0, VIK., thiH (iuation whicli ovidently lielongK to a iwiir of linon passing through tho point (gcsO, -()) givotn tho tungontH at the point in question; and similarly tho tangents ut tho point (j) .= 0, s <=. 0) aro given by tho tHpiatiou ^(ij- ^)-l- ^1(7; -^) -I- Vw'ft "- V*) -O- 101. To complotu Mm nolutinn, aUending to tho tangents at ( = (), ^ = 0), and putting for HhnvtiiuwM \ sa I in - n, /t K3 - i "1- 'HI ~ it, Jt S3 J '/?t 'I' | A =~: /' J -h wt u -I- ;J tlu^ rationaliHod nqiiutiiHi in oiwily found lio be - ; iiurt it IH to bo notinod tlmt in tlio uuwi of tho circular cubic or when thon A0, w> that Iho (!qunliiini contaiiiH tho factor 3, and throwing this out, tho munition givo* a Hmtflu lino, which in in fact tho tangent of tho circular cubic. 162. Hutuming to tho bicircular quartie, wo may Book for tho condition in that tho mulo may bo n oiwp: the required condition w obviously -h wvy) a 0, or obHoi'ving that ft A - A + ^i; a - 2iX, &c. tbia is 536 ON POLYZOJIAL CURVES. or substituting for X, /*, v, their values, it is or, as it is more simply written, I m n n j_ _L n -t- T~ \J, [414 163. If the node at (?j = 0, 2 = 0) be also a cusp, then we have in like manner I- I I i Now observing that a, a', 1 A ', 1 7. 7'. 1 = 03 -7) (7- ')-(/?' -7) (7 -4 = fi suppose : the two equations give I : m, : = fl (/9 - 7) (/3' - 7') ; fl (7 - a) (7' - a') : il(a- or if fi is not = 0, then - 164. If or, what is the same thing, if A # 7. y. , =0, 6, &', 1 c, c', 1 = 0, the centres A, B t G are in a line; taking it as the axis of ta t we have a = '==, /3 = ' = &, 7 = 7 ' = c; an d the conditions for the cusps at /, / respectively reduce themselves to the single condition 1 so that this condition being satisfied, the curve is a Cartesian; viz, given any three circles with their centres on a line, there are a singly infinite genes of Cartesians, each touched by the three circles respectively; 414] ON POLYZOMAL CURVES. 537 the line of centres is the axis of the curve, but the centres A, B t G are not the foci, except in the case a" = 0, &" = 0,o" 0, where the circles vanish. The condition for I, m, 11 is satisfied if I : m : n~(b~c) 5 : (c ~ ft) 2 : (a 6) a j these values, writing VZ" : Vm. : */n ~ b ~ c : c : a b, give not only VT+ Vm + Viz = 0, but also a yT+ 6 Vm + c v'-n = ; these are the conditions ' for a branch containing: (z* = 0) the line infinity twice ; the equation (b - G (c - -f (tt - 6) is thus that of a conic, and if a" = 0, &" = 0, c"~0, then the curve reduces itself to 2/ 9 =0, the axis twice. 166. If O is not =0, then we have I : m : n = (/3 - 7 ) (/3' - 7') : (/-) (7 -') = ( - /3) (a' - #), viz., Z, ??i, ?i ai'e as the squared distances 7^C' a , ffjl 8 , AB* t sa}' as / 2 : g* : k s ; or when the centres of the given circles A, li, G are not in a Hue, then f, g, h being the distances BO, CA, AH of these centres from each other, wo have, touching each of the given circles twice, the single Cartesian which, in the particular case whore the radii a,", V, c" are each = 0, becomes viz,, this is the circle through the pointy A, B, G, say the circle ABO, twice, Article Nos, 1G6 to 169. Investigation of the Foci of a Gonic represented by cm Equation in Areal Coordinates. 166. I premise as follows: Let A, J3 } be any given points, and in regard to the Jiriangle ABC let the areal coordinates of a current point P be , v, iv\ that is, writing PEG, &c,, for the areas of these triangles, take the coordinates to be uiv. w = PBQ ; PGA : PAS, or, what is the same thing in the rectangular coordinates (a 1 , y, 2=sl) ( if (a, a', 1), (6, &', 1), (G, o' ( Ij, be the coordinates of A, S, respectively, take . y> z : j y . z fa, I, b', 1 G, C' , 1 a, c, c', 1 , a, 1 b, C. VI. 538 ON POLYZOMAL CURVES. or iu the circular coordinates (, y, 2 = 1), if (, a ', i), (fr p> t 1), ( % y, ].) bo tho coordinates of the three points respectively, then : ^ MO = & ?}> z 7) 2 Si '/i * , fl? > ft ', 1 7. 7. 1 a, a', 1 7. 7. 1 a, ', 1 ft /9'. 1 167. For the point / we have ( ,, ^) = (0, 1, 0), and honco if its aroal coordinates be (, v a , w 6 ), we have U : and hence also, (, y, w ) referring to the current point P, we find Vrfw - w a y = ( 7 - a ) [(a' - /3') (f - u) - ( - whence and in precisely the same manner, if <, <, w ; refer to the point 168. Consider the conic i 11 ta ^XV^T which ar ( X ' w j as is well knoivn (4, -B, a, j, G, consequently for the conic w, v t w) a = 0, ;: hate : er; and then f01 ' ten 8 flnta poiat the ooordimtten of this P^* *o tho eonic i a and that of the pair O f tangents from / i s W. i, 0. J, ff, these two line-pai,, in t eraeotin& rf = 0) rf 414] ON POLYZOMAL CURVES. 539 169. In particular, if the conic is a conic passing through the points A, B, G t then taking its equation to be the inverse coefficients are as (I 3 ,, m 3 , n*, - 2mn, -toil, - 2Zni), and wo have for the equations of the two line-pairs " 0, = 0. Article No. 1.70. The Theorem of the Variable Zomal 170. Consider the four circles 5[" = 0, 33 = 0, <T = 0, ) = (91 - (m - fl*) + (y - aty* - a'V, &c.), which have a common orthotomio circle ; so that as before a2! + b5i + cS + d3) = 0, where a : b ; c : d = CW : - OD.d : DylU : -yIJ?C'. I consider the first three circles as given, and the fourth circle as a variable circle cutting at right angles the orthotomic circle -of the three given circles; this being so, attending only to the ratios a : b ; c, wo may write a : b : c =DBG : DGA : DAB, that is, (a, b, c) are proportional to the nrool coordinates of the centre of the variable circle in regard to the triangle AHG, 171. Suppose that the centre of the variable circle is situate on a given conic, then expressing the equation of this conic in areiil coordinates in regard to the triangle ABO, we have between (a, b, c) the equation obtained by substituting these values for the coordinates in the equation of the conic; that is, the- equation of the- variable circle is aSr + b33 + ctr ==0, 540 ON POLYZOMAL CURVES. then the equation of the envelope is (l\ w a , ', -mn, -nl t -Imfy 31, S3 , (S) 3 = 0; that is, it is or, what is the same thing, it is [414 172. It has been seen that the equations of the nodal tangents at fcho points /, J respectively are respectively 0, a*) + Vm (-)&) + V*T(f cte) + Vm (7; - 0'*) + V (^ and that these are the equations of the tangents to the conic Ivw + mwii + nnv = from the points /, / respectively. We have thus Casey's theorem for the generation of the bicn-cular quartic as follows :-The envelope of a variable circle which cutfl at right angles the orthotomic circle of three given circles 91 = 0, 33 = 0, S = 0, and 1ms its centre on the conic tm + mwu + miv = which passes through the centres of the three given circles is the biciraular quartic, or trizomal m8 + VjiS = 0, which has its nodo-foci coincident with the foci of the conic. 178. To complete the analytical theory, it is proper to express the equation of the orthotmmc circle by means of the areal coordinates ( U> v, w ). Writing for shortness a a a -* = a -a,* = a\ & Cl) and therefore then if as before &o., > y, z : at, y, z : x, y, z 6, 6', 1 G, C', 1 a, a\ 1 G, C', 1 ft, of, 1 6, 6', 1 and therefore : y * the equation of the orthotomic circle is ffl-a*, y-a! 2) aa + afy-tf* ffl-te, y-Ve t Iw + Vy-Ve a-cz, y- G 'z> ca +c'y~ oV -viz., throwing out the factor *, this is 0, 414] ON FOLYZOHAL CURVES. 541 or, what is the same thing, it is (au + bv + ovf) 10 + (a'u + b'v + c'w) y - (a\i + $v + c\v) z = 0, viz., it is (aw + bv + ow) 9 -l- (a'tt + b'v + c'w) a - (a v zt + b"v + c'w) ( + y + w) - 0, that is, substituting for .\ 6\ C N their values, it is tt'V + 6" 2 y a 4- c" a (o a 4- (6" 3 + c" 2 - (6 - c) a - (&' - c') a ) wo + (c" a + a" 2 - (c - ) B - (c' - a')- )wtt + (a" 2 4 i" 2 - (a - 6) 2 - (' - t') a ) uv = 0, aiirl it may be observed that using for a moment a, & 7 to denote the angles at which the three circles taken in pairs respectively intersect, then we have 2&"c" cos a = &" a + c" fl -(6-c) a -(&'-c') 3 f &c,, and the equation of the orthotomio circle thus is (1, 1, 1, cos a, cos/3, cosyJa'X V'v, c"w) s = 0. 174. We have in the foregoing enunciation of the theorem made use of the three given circles A, B, C, but it is clear that these are in fact any three circles in the series of the variable circle, and that the theorem may be otherwise stated thus : The envelope of a variable circle which has its centre in a given conic, and cuts at right angles a given circle, is a bicircular quartic, such that its nodo-foci are the foci of the conic. Article Nos. 175 to 177. Properties depending on the relation between the Oomo and Circle. 175 I refer to the conic of the theorem simply as the conic, and to the fixed circle simply as the circle, or when any ambiguity might otherwise arise , then as the flrthotomio circle This being so, I consider the effect in regard to the tnzomal curve, of the various special relations which may exist between the circle and the come. If the conic touch the circle, the curve has a node at the point of contact. If the conic has with the circle a contact of the second order, the curve has a cusp at the point of contact. If the centre of the eirele lio ou an a=ds of the eonio, then the four intersections He in pat syletrioally in regard to this axis, or the curve has tins ax, S a, an axis of symmetry. eirc.es intersecting in the, t point, 542 ON POLYZOMAL CURVES. the circular points infi ' 7 ! " , thfl general case there are at each of of the one pan ^ [ h "2 T "' "' ^P '*- of the * sections, the four foci of the lie H ^ ""' there are thu3 fo r * of circle., the two tangent T U 1^' T *" "" COT JS the two foci on tll f axis " " ^ "T^ * . h are each of them the intersect O f ,' 7 , 7 f C '' r a " t; P oi nte of these, the other circle. mtosM " of a tangent of the one circle by a tangent of f th i. a circn.ar cubic having the of the conic), then the c at ^-"1 ""T""^ "* "* ""^'y f a " axi circles touching eaoh other Id TL^ "c He", ^' ' ?' bre!>lB " **> centres at the two fo ci situate on t,,at a $ *- "* I7n Tf j-t *ur int^secti^ - *~ of the P ara.o,a H the eirde touch the paraoda, the e, has a node at the ^int'of contact. a cusp at^l^f coif * * ^ <* -one, order, the curve ha, TJ* -4-1 intersectioL Te'itlfi^te^^*:; "" ^ / the P a ^ol a , then the four 1 this axis for an axis rf s X '" '^ * " lis 1>xis ' md tho the contact, , the curve , and the c,c,e f the P oi t3 vertex), then the curve has a ta e T f ''I'" 1 ' ' ^ >uahll the P arab la * oh other and the parabola at h 've to tt T^ * ?" ^ ir le parabola at its vertex, and the ch' e h ^1 , "! M i8 the tan S ent to the for its centre, and pa s in throu.hl , haVmg * he feous of the P*ol >i of the seLa^s ^ ^ * fe - 'hing, haLg its ", bicirc 1 qnartio such that its && irc1 ^ - " curve is a !' ""* ^ ** * he ^<* llrteraect 'o of the conic with the ', ^ f " n - ax!al fooi of the * he ial fooi ' ^he third axial Tf +1-, four nodo-foci oeindae Cartesian having the ce of of the cuspidal tangent, of the Cassia other droJa, or say with the orthoZ Cartesian; vk, the antipoint ^ heT . , focus i, the centre of the'o tLoli 414] ON POLYZOMAL CURVES. 543 Article No, 1*78. Gase of Double Contact, Casey's Equation in the Problem of factions. 178. In the case where the conic has double contact with the orthotomic circle, then (as we have seen) the envelope of the variable circle is a pair of circles, each touching the variable circle ; or, if we start with three given circles and a conic through their centres, then the envelope is a pair of circles, each of them touching each of the three given circles ; that is, we have a solution of the problem of tactions. Multiplying by 2, the equation found ante, No. 173, for the variable circle, and then for the moment representing it by (a, b, c, f, g, h$w, v, w) a = 0; then attributing any signs at pleasure to the radicals Va, Vb, Vc, the equation of a conic through the centres of the given circles, and having double contact with the ortho- tomic circle, will bo (a, b, c, f, g, h}j>, v, ?y) a -(Va + ?jVb + wVc) a = 0, viz., representing this equation as before by hw + mwu + nuv = 0, wo have _ _, _ I : in : u-f-Vbe : g-Vca : h - Vab, that is, substituting for a, b, c, f, g, h their values, and taking, for instance, a, b, c = fi"V2, fc"\/2, o"V2, we find ' -67, that is I, on, n are as the squares of the tangential distances (direct) of the three circles taken iu pairs, and this beingjo, the_equation of a pair of circles touching each of the three given circles is Vm + VmS + VS' = 0. It is clear that, instead of taking the three direct tangential distances, we may take one direct tangential distance and two inverse tangential distances, viz,, the tangential distances corresponding to any three centres of similitude which lie in a line; we have thus in all the equations of four pairs of circles, viz, of the eight circles which touch the three given circles. This is Casey's theorem in the problem of tactions. Article No. 179. The Intersections of the Come and Orthotomic Circle are a set, of four Qoncyclio Foci. 179 The conic of centres intersects the orthotomic circle in four pc each of 'these the radius of the variable circle is =0, that is, the points are a set of four coneyclic foci (A, B, C, J3) of the curve. Regardmg the the circle which contains them is of course the orthotomic circle; and singly inHnite series of curves, viz., these correspond to the singly mfii conies which can be drawn through the given foci. As for a given cur 544 ON POLYZOMAL CURVES. [414 four sets of concyclic foci, there are four different constructions for the curve, viz., the orthotomic circle may be any one of the four circles 0, R, 8, T, which contain the 'four sets of concyclic foci respectively; and the conic of centres is a conic through the corresponding set of four concyclic foci. We have thus four conies, but the foci of each of them coincide with the nodo-fooi of the curve, that is, the conies are confocal; that such confocal conies exist has been shown, ante, NOR 78 to 80. Article NOB. 180 and 181. Remark as to the Construction of the Symmetrical Curve. 180. It is to be observed that in applying as above the theorem of the variable zomal to the construction of a symmetrical curve, the orthotomic circle made use of was one of the circles R, 3, T, not the circle 0, which is in this case the axis- in fact, we should then have the conic and the orthotomic circle each of them coinciding with the axis. And the variable circle, qua circle having its centre on the axis, cut* the axis at right angles whatever the radius may be; that is, the variable circle is no longer sufficiently determined by the theorem. The curve may nevertheless be constructed as the envelope of a variable circle having its centre on the axis- viz writing 51 =(tf_^) 2+ ^ a 'V, & c> , and starting with the form then recurring to the demonstration of the theorem (ante, No. 47), the equation of thejanable circle BSr + bS' + oC'-O, where a, b, c are any quantities satisfying iT" ' r ' What is the same thin g, Baking q an arbitrary parameter, and writing S - - 1 J. m 1 n a -H-<r, ^-i-ff, - n i = -4 the equation of the variable circle is Compare MOJL 118-188 for the like mode of construction of a conic; but it is proper to consider this in a somewhat different form. 181. Assume that the equation of the variable circle is 3> = 0-^) 3 + y a -cZ% 9 = 0; we have therefore identically ar + W + or + dfc'-O, viz., this gives a +b +c =-d , aa +bfi +cc=-d<2, a (a? - a"*) + b (6 a - V*) + (p _ c "*) = _ d ^ __ d ^ and from these equations we obtain a, b, c equal respectively to given multiples of d; substituting these values in the equation ^^ = 0, d divides out, and we have an UJN i J O-LYZOMAL CURVES. 545 equation involving the parameters of the given circles, and also d, d", the parameters of the variable circle; viz., an equation determining d", the radius of the variable circle, in terms of d, the coordinate of its centre. I consider in particular the case where the given circles are points ; that is, where the given equation is The equations here are a 4-b +o -d, a + b& 4- cc = d", act 3 -I- b6 a + cc 9 = - d (rf a - d"), and from these we obtain U? 1? so that the equation - + + - = becomes 1 I (a - 1)) (a-o) m(b -jO(& Z^L + " ( c ~ a ) ( c - & ) = ^b) (d - c) - d" a + (d - t>T(d - ) - &** "*" (d - a) (d - 6) - d" 8 ' or, as this is more conveniently written, I 1 ___, * + -J!i * y/ ^ Q t T^"o (d - 6) (d '- c) - d" 8 c - a (d - c) (d - a) - d" 2 a - 6 (d - a) (d - b) - d" 2 vk, considering d, d" as the abscissa and ordinate of a point on a curve, and repre- senting them by w, ?/ respectively, the equation of this curve is I 1 _m 1 , _^ i - o, which is a certain quartic curve ; and we have the original curve as the envelope of a variable circle having for its diameter the double ordinate of this qnai'tio curve. I m n T M A Write for shortness ^ , ^^, ^Tfi* 1 " ' of the quartic curve may be written 2i [(- )'(* - fiXffl-o)-^^- ) ( 2i viz., this is . %L[co(a>-ct)(e>-'V)(>-o) - (a + b + c) cc + (06 + o + i 0. VI. 546 .ON POLYZOMAL CURVES. or what is the sanle thing, the equation is ' - (La + Mb + No) (x - a) (a - b) (a- - c) + f {(La + Mb -H jYc) + Lbo + Jfoa + #aft) = 0. In the particular case where L + M+N = Q t thai is, where 4. r 7 ' ' / - vi, . 0' C C (i ft (J the quartic curve becomes a cubic, viz., putting for shortness S = the equation of the cubic is ai-S" viz,, this is a cubic curve having three real asymptotes, and a (linmotor ab ri angles to one of the asymptotes, and at the inclinations + 45, -'15" to fcliu other two asymptotes respectively say that it is a "rectangular" cubic. Tho rohition + a^6 =5 impli08 that tho ourvc ^+Vwii6 + Ve'0 is a OavliOMuin, and we have thus the theorem that the envelope of a variable circle having for dtamotoi- the double ordinate of a rectangular cubic is a Cartesian. I remark that using a particular origin, and writing the equation of tho ruofciuiguliu- cubic in the form j sa ^-2ma! + + ^ > the equation of tho variable circle in ft 1 d that is where d is the variable parameter, Forming tho derived equation in regard to d, wo n have ^A and thence nn P - known form of the equation of a Cartesian. = ( which is a 414] ON POLYZOMAL CURVES, 547 Article, ,Noa. 182 and 183. Focal tforimikQ for the General Curve. 182. Considering any three circles centres A, B, G t and taking 91, &c., to. denote as usual, let tho equation of the curve be then considering a fourth circle, centre D, a position of the variable circle, and having therefore tho same orthotomic circle with the given circles, so that as before bho formula No. 47 (changing only U t V t If, T into 91, 2T, (T, 3>) are at mice applicable to express the equation of tho curve in terms of any three of the four circles A, B, G, D. ' , - : In particular, the circles may reduce themselves to the four points A, B, G, D, a act of concyclic foci, and here, the equation being originally given in the form the sarao formulas are applicable to express the equation in terms of any three of the four foci. 183. It is bo bo observed that . in this case if the positions of the four foci are given by moans of the circular coordinates (a, -, lj, &o., which refer to the centre of tho circlo A BOD as origin, and with the radius of this circle taken as unity, then tho values of a, b, e, d (ante, No. 90), are given in the form adapted to the formal* of No. 49, vis;,, we have a : b : c ; <U(j8yS) : -/3( 7 Sa) : ?(S0) : -S(#y). , / if*t fi , - where W) = ( ~ 7) <7 - S) (S - /5), &o. The relation - + ^+- = 0, putting therein I : m : n p(/3 -7)" : r/3(7-) : T 7 (-/3) a , (or, what is the same thing, taking the equation of the curve to be given in the form 03- vw. thie equation, considering p, a, r, a, ft, 7 " given, determines the position of the fourth focus D, or when A t B, 0, D are given, it is the relation which must exist between p, <r> T; and the four forms of the equation are vi,, the curve is represented by means of any one of these four equati each of them three out of the four given foci A, B, U, V. 548 ON POLYZOMAL CURVES, [414 Article Nos. 184 and 185, Case of the Circular Cubic, 184. In the case of a circular cubic, we must have Va/3 OS - 7) + V/3~<r (7 - ) + V<yr (a - /3) = 0, which, when the foci A, B, 6, D are given, determine the values of p \ a- : r in order that the curve may be a circular cubic, We see at once that there are two sobs of values, and consequently two circular cubics having each of them the given points A, B, Q, D for a set of concyclic foci. The two systems may be written viz., it being understood that V^S means Va.VS, &c,, then, according as VS has one or other of its two opposite values, we have one or other of the two systems of values of p : ff : T. To verify this, observe that writing the equation under the form Vap : Vjfo- : V^=aVg- Vo^y" ; ySVg-Va^ : ryV'S-Vo/g-y, the second equation is verified ; and that writing them under the form where M = the second equation is also verified. 185. If we assume for a moment = cos + i n = fl fa & c>( viz<| if ffl 6 c rf t^nchnat 10 ns to any fixed line of the radii through A t S. C, iJ res^eedvely then and thence or else : cos | (c + rf _ fi _ sn : sin i (6 + d - o - a) sin J ( -a) Putting in these formula, : sin ^ + d "~ " " 6 ) sil1 4 ( a ~ & )- ^ ^ then we have B - = (6 _ ) (7 - J. = ^ ( c _ a ) f 41<L] ON POLYZOMA.L CURVES. and for nitlusr sob ol' values tho verification of tho relation 549 V (/3 - 7 ) + V/3^ (7 - a) -H VTT (a - /3) = 0, will dupond on liho two identical equations tiin .4 Hiu (;j - 0} -I- Bin 7^ mi (G - A} + sin (7 sin (A - 5) - 0, COM A Hin (./i - 0) 4- OOH Ji sin (0 - A) + cos a .sin (A-B)^Q: although tho forogoing Holution for tho caso of a circular cubic is the most elegant ono, I will prcwjutly rotui-n to the quostion and give the solution in a different form, Artiolo No. 180. *W Form-alee for the Symmetrical Curve. 180. In this nynunobrical cfwo, whoro the foci A, B. G, D are on a line, then if, ns UHiuil, a, 6, o, d .lonoto the dirttancoa from a fixed point, we have the expressions i)l' (a, 1), (!, d) in a form adapted to tho formulas of No. 4-9, viz., -&)M^ tlmli, assuming : m : thu - -r -i -r a 1) o ,vud tho .^nation of tho curve may bo presented under any one of theourforms - V?&-d V(o-&) (VSt.^.V Article No. 187. tttsfl of the Symmetrical Circular Cubic. 187. Kor a circular cubic we must have */- ' Theno oquatiunfl givo Vp equations), or G!HO Vrl ' 1 : 1 . VT- 1 . 1 . which obviously satisfy the two 580 (IN I'lU.v/MMAi. iTuvr/i, In (noli, Miomt valucH ulivimifily .''iiti-'ly (In 1 -<-rMn>i t.puh.m. ,1,^1 p,, f , , ,, ,t . f HiitiHly tlui Ih'Ht, (iignuMon, we liavi< niily I" wnh< Hn in HU<|M ijj.t (..,, />:<;: T' /]/-l(/i !'')(" I </): .V -Mr * H in'. * <?* If .(,,.*/.,,.> ;. ' ' ? * ' P * f f I/'^(<H /).(<' .|. ( /y. The- lir:.[, ; i..|. -jivr. i-r !.. MIK.- (//r|\'VI | (r ,n\\*\ i in /*IM .M, ll t'.lllH (Illlll.llillS lln> lilir ,? I) |i,,[. |y, ), U f r ,-.(,,-, jr ti , in liwirn; tl il\- j.rnu..|' >'!, vvilli lli>- ),-. i v ) /; / / , " ' " '' * (A-(j)('/.-l-(/--/i--)\ / yi I (-... r)</. I ./ r ,n\'W * ,.j C(||iaii(lll ill' Wllittll i,'l. Hi' .'Mill;,.., r\|.|,--.iM.r |,, >;,,!, ,,j (f. Artirlc INN. in A', anil >"" N """ '"' "'"" <.-",,,,, ,:,,<,.,;;,;,; : i ;;...'ir 1 :.:';;:; 1 anil tlutf, I.,, micl Oho Hko HH rogftixln ON 1'OLY/OMAL UUUVJ58. .551 INI). ThoHO valuuH of V? : Vwi ; VK givo tho oquationR of the two circular cnbica with' tho foiii (A, 1i t <!, ;j), the equation' of oaoh of tlitjin undor a fourfold form, vix, ( wo Imvu mid c'i it, , ], (I, ~ CJj ,, 6, - ! <2, -I- &, , - 6, -l- d (6| (t| j (Jj (t| (first OUI'VO), />, = (HOCOUtl OUllVO), HJO. Similurly 6VI and 7l/> niunt in 8, "mid if wo dcmoto by a u , & 3 , j, c?j tho irs fnun iS T nf l,ln; I'onr points roHpoclitvoly, HO that <y&j M^ ~ nwl, B S' (obnorvo bhub if UN UHual /I, ,/J, (7, .1) avo talccm in owlor on tlio cit-clt) 0, thon A, MO on opptmHid HidoH of W, rind mmilurly Jl, D ai-o on oppiiaito Hidon ul' V> HO that leaking "JP ftu lmitivo <?.j, tl t will Ins iid^itbivcOj w liavo a ; b : o : d (ft,, - rf.,) : rf a (6- a -(( a ) : - , (&s ~ tl^ : -^(y a -f(,), and Uion thu (ii of vnbniH and -I- , : -|- : 0, Vi + Vm -I- V;i --- 0, uro mitinfiotl by tbo two sotn It I) I* ?1 /. .,.__/. / J. /i - (/,J * (jy , [(,j {(jj . IJj "|~ (/.J.J and wo havo tlui mpiutioiiH of tho wain** two cubic. iwi'WH, ouch utpiattou undor a fourfold lorn i j vix,, UU-HO am :Q and 6, - ff a ) (V?f, VS, -h il tl H- t'a . (uoooud 101. And again ^17i and C'D moot in T, and donating by ^, 6 a , c dl rf 3 the tlistanooB from T of the four points roapootivoly, ao that ajb tt = c a rf a = md, *%\ wo havo u : b : c ; d = & a (c a - <i) : -ft,(c,-(Q : -f4(s-^) : o a (fi Z^); 552 ON POLYZOMAL CURVES. [414 ami the equations - + ^ + - = 0, VJ + Vm + V = 0, then give for */i t 1 a b c of values, viz,, these are two sots and and we again obtain the equations of the two cubics, each equation under a fourfold form, viz., these are and _ /* J~ ft -mm ft JLm /I I "3 T l<-3| t( 3 -f 1/3 , "3 ~ &3 I fy ~ MS > Bg 3 , 63 - a. = 0. 192. The three systems have been obtained independently, but they may of course be derived each from any other of them: to show how this is, recollecting that we have MA, MB, HO, then to compare the similar triangles and the similar triangles j.^ , j.jj =a a> 3> eg, (jive Ji G RAG give 0,-Cs : Oj a,, RED =b- - using these equations to determine the ratios of ,, 6 a , Ca , ^ a we have _ ~' that is 414] ON POLYZOMAL CURVES. 553 and henco & a (- &iC! + fl! 3 + chdj. - di 2 ) + c a (- hdi + Oidi + ^d - CjriO = 0, that is Z> a (d a &) + c a (OiO, - 61 c4) = 0, but O 1 c l -t 1 d l |-(e 1 --d,) l or the equation gives & a +jC 3 =0, or say 6 a : c a = &, : -h, and this with ^r^ = ^ = ^' gives 'all tho ratios, or we have a, : & a : c a : d a &, (ffl! - d.) : 61(61-0.) : -^(a.-dO : -di(6 1 -c l ). We have then for example & a -c a : c 3 -ff 3 : a a -6a6,-Ci : GJ-O, : d-ft,; &c., showing the identity of the forms in (d, &i, c,, d,) and (a a , 6j, c a> cZ a ). Article No. 193. 2Vons/ormo*ion to a #ew Set o/ Conoydio Foci. 193, Consider the equation which refers to the foci 4. 5, 0, and taking D the i^w/^i l \?ii: tliafc we can find i,, w u such that identically ~Wt +m95 + = -Zi and that m lttl == mn. The equation of the curve gives -$ wo have therefore _ that ia> vi,, this is the equation of the curve expressed in terms of the concyclic foci Article No. 194 2fe Itonuomol OHT^ JfcwmpawW- or 194. I consider the tetrazomal curve where the zomals are circles described about any gi a vi. 554 ON POLYZOMAL CURVES. [414 There is not, in general, any identical equation a9P + b33 + c( -H d3> = 0, but when such relation exists, and when we have also ~+b~ + Q+3 ==0 > tne11 tne ourvo breaks up into two trizomals. When the conditions in question do not subsist, tho curve is indecomposable. But there may exist between I, m, n, p relations in virtue of which a branch or branches ideally contain (^=0) the line infinity a certain number of times, and which thus cause a depression in tho order. of the curve. Tho several cases are as follows : Article No. 195. Gases of the Indecomposable Gurve. 195. I. The general case ; I, m, n, p not subjected to any condition. The curve is here of the order =8; it has a quadruple point at each of the points /, J (and there is consequently no other point at infinity); it is touched four times by each of the circles A, S, G, D; and it has six nodes, viz., these are the intersections of tho pairs of circles __ v?w + Vil 3 = o, VM + Va>" = o, = o, ?r + = o ; the number of dps. is 6 + 2 . 6, = 18, and there are no cusps, hence the class ia = 20, and the deficiency is = 3. II. "We may have there is in this case a single branch ideally containing (#=0) tho line infinity; tho order is =7. Each of the points /, J is a triple point, there is consequently one other point at infinity ; viz,, this is a real point, or the curve has a real asymptote, Them sire 6 nodes as before ; dps. are 6 + 2,3, = 12 ; class = 18, deficiency = 3. III, We may have m = 0, */n + Vp = ; there are then two branches each ideally containing (a=0) the line infinity; the order is =6. Each of the points J, J is a double point, and there are therefore two moro points at infinity. These may be real or imaginary; viz., the- curve may Imvo {besides Jibe asymptotes^at /, J) two real or imaginary asymptotes, Tho circles V/[ + Vm23 = 0, V*iS + Vio==0, each contain (*0) the line infinity, or they reduce themselves to two lines, . so that in place of two nodes we have a single node ab the intersection of these lines; number of nodes is =5, Hence dps. are 5 + 2, 1, =7. Olaaa 16, deficiency =3. IV, We may have Vr : Vm : V ; V^ = a : b ; c : d 41-1 I ON I'O&YKOMAL CURVES. 555 l,h*'Ui in hcin it hiu^li) brannli containing (^ J = 0) bho lino infinity twice; the order is . . (i. Kuril nl' llic points /, ./' IH a double point, and there are therefore two more pninlM ul. inlmity, t.linl. in (bimidtsH tlm asymptotes at /, J), there are two (real or imaginary) miyiiiplntni. Tim mnnbui 1 of iioduH, as in the general case, is -6, Hence . I- 11.1. : >N; I'liiHH in "K; dullciuuuy =2. lun I tunic" Mm imiliidud partimdai' WHO whuvo the circles reduce themselves to their rn; vi/., wr liavu hern llw tnivvn wl.i.-li (M.T ti/i/ii N. Id) in in IUl tho (iiirvo whioh in tho locus of the foci of the ri(M l ( . H whirl, ,HIHH ilmMiKli thn lour 1M nfcH ,L, ., (J. .. It in at present assumed that ,,, ,.,. .,,,., lin , ,,ul, a rirnlts tliw tuuso will bo ooiwidorad jrort No. 199. 11 we huvn /', . I/' m,.-li IlK in /,:; (,M. 7W iu ^ W d ^ (W> in 2- f then those pomt. If. N. 7 1 iav l-luv.' "f I'bn nix nudcK. In i'aot, writing down the equations of the two im( , n|(hi . rvi| ||M| . whp|l t ,| l(1 IJlireoilt point in taken at ll t we have 23 : - "< = ^^ ; ^(ABOf.VB I 1 f .-I I I \' ' ffr'llM"!iL<" i ll,llllil MI* 11 i*iv j -i - - ' . tt !'?" Vul ,th nl- Urn tw mroh* im bhnmgh tho point * or tins pomt is a ,,Mrlr. Mimilurly, UM- ),niulH N and '/' aro uwh of thorn a node. V. II 1 (|M.| ( . in.' IMMV Lluvt' bmiu'lu-i P L ! f U'.t >ili ill Mil 1 KM IlLH i. i </ *" ' v * -"-"----- ~nf j. _ b , "" U "' ' < .,""" < .";. in Ul val, or ono col and two imaginary; that ,,, 1,,,,,1,-H MI n,l....ly """ !"'";' J, \ lu ,, , wympt otos, all real, or one real and it. ,,!.!,, i 1 1m iitVIllltllMfN ai I, >' ) uu'ii* i"> -i i .,../.. ,! it. ! -. Km-li i.f Li"* "ir'JH V9H-VSU0. &o., contains tho^hno ^^^ Q 1S llniH n'lniT'i i' ' li' rlitHH ! - !'l'", di'iii-it'iiry - . to 556 ON POLYZOJIAL CURVES. [414 IV. VT+Vm + Vtt+N/p^O, a Vr+fiVm + cVtt-f d Vp-0; corresponds to IV. supra, viz., there is a branch ideally containing (z* = 0) the line infinity twice. But, observe that whereas in IV. supra, in order that this might be so, it was necessary to impose on l t fii, n, p three conditions giving the definite systems of values VT : Vm : */n : */p = a : b : c ; d, in the present case only two conditions are imposed, so that a single arbitrary parameter is left. V. 'Ji~*Jm = >Jn = *jp\ corresponds to V. supra. VI. V^-H Vm = 0, V?i + Vp = 0, a*Jl + &Vra4- c V + d Vp - 0, or what is the same thing, *Jl : </m : V?i : Vp = c- rf ; d - c : b - a : a - b ; the equation is thus (c - fZ)(Vf - V3T) - (a - 6) (VST ~ VgT) = 0. There is here one branch ideally containing (s 3 = 0) the line infinity twice, and another branch ideally containing (2 = 0) the line infinity once ; order is = 6. Each of the points I, J is an ordinary point on the curve, the remaining points at infinity are a node (21 = 53, ( = 3)), as presently mentioned, counting as three points, viz., one branch has for its tangent the lino infinity, and the other branch has for its tangent a line perpendicular to the axis; or what is the same thing, there is a hyperbolic branch having an asymptote perpen- dicular to the axis, and a parabolic branch ultimately perpendicular to the axis. The number of nodes is =5, viz., there is the node 31 = 33, S = !D just referred to; and the two jKrire of nodes ((c - d) VSP - (a - b) V(T = 0, - (c ~ d) V$T 4- (a - 6) V^ = o) and ( c -rf)V2[ + (a-6)V3) = 0, (c-^V^ + ^-^Vr-O), each pair symmetrically situate in regard to the axis. Hence also dps. = 5; class =10; deficiency = 1. And there is apparently a seventh case, which, however, I exclude from tho preaonb investigation, viz., this would be if we had a , b , c , d , a 3 , & , G\ d* , a"\ b" s , c" 3 , d"*, that is, a, b, c, d denoting as before, if we had VZ : Vi : V : >jp = a : b : c ; d, and act" 3 + b&" a 4- co" 3 + drf" a = 0. For observe that in this case we have dSy-O, and + + - + == Q- abed ' that is, the supposition in question belongs to the decomposable case. Article No. 197. The Decomposable Gurm, 197. We have next to consider the decomposable case, viz,, when we have 414] ON POLYZOMAL CURVES. 557 ftco ewifo, Nos, 87 et seq.ii there appears that (unless the centres A, B, 0, D are in a lino) tho condition signifies that the four circles have a common orthotomic circle; and whan we have also d formulas for - the decomposition are given ante, Noa. 42 et seq. Writing therein , 38, (T, 35 in place of ^> K W > T respectively, it thereby appears that the tetra- our Jo V&T + VmW WW + V* 3 = 0, breaks up into the two trizomal curves = o, vpp + v^ + vis 5 = o, whovo A=^ + , ^=vr + 5, c 111111 whmo w , J . 2i . + . "b + o ' a b Artiolo Wos. 198 to 203. Oases of 0. Dec m po S Me Ou,;e, Centres not a K. 198 I assume, in tho fivst instanco, that the oenta. of the cu,les ave not m a llno! wo have tho following ca.cs: ^ ^ Mnud ^ =g . I. No furthc. relation between J , - , P , ^ '^ j.^^ qim , ic . tho ordov of oaoh of tho tmonuda as =4, that - /- /- , ^-n. rtip order of the tetrazomal is =7, that ot one II. \fl + Vi + V + Vj) = ; me omu tho trizomals must bo = 3. To verify this, observe that we have - - i _4.. VI VI V bod for Vl+ VS+V5 the valne __ H^^- 1 ' ,Ti n the siga condition, , b y . 558 ON POLYZOMAL CURVES. [414 and hence one or other of the two functions V + Vw, + Vj, V + Vm^ + V?^ is = ; that is, one of the trizomal curves is a cubic. III. Vr+ Vp = 0, Vm + Vn = ; order of the tetrazomal is = 6 ; and hence order of each of the trizomals is = 3, To verify this, observe that here which since a + b+c + d=0, gives -?S so that, properly fixing the sign of the 1 1 if Lid /aH radical, we may write Vi + jJ =r- Vm = 0. Wo have then A = ^TT V V^ + V = A /~~ (b + c) Vm ; a V abc which last equation, using A/r~ to denote as above, but properly selecting the signi- fication of +, may be written ,/ . ,/ . b + c /ad / Vni! -H vi = + T A/ r- vm. Hence V + (VS + V) = i {(a + d) VI + (b + o) A M w ( V DC a-f-d f /r viz., V^TCVflh+Vw,) with a properly selected signification of the sign + is =0; and similarly VJ> + (Vw^ + V 3 ) with a properly selected signification of the sign T is =0; that is, each of the trizomals is a cubic, 199. IV. V^ : Vm : Vw : ^ = a : b : o : d (values which, be it observed, satisfy of themselves the above assumed equation l - + + * + = Q] the. order of the tetra- f* U u Cl / zonml is_ = 6; and the order of each of the trizomals is here again =3. We in fact have Vt = a + d, V^ + Vn^b + c, and therefore V + V^ + V^ = 0; and similarly +VH 3 = 0; that is, each of the trizomals is a cubic, I attend, in particular, to the case where the four circles reduce themselves to the pomts 4, B, 0, jD; these four points are then in a circle; and the curve under consideration is 414] ON POLYZOMAL CURVES. 559 in tihe gonoral case where the points A, B, G, D are not on a circle, this is, us has boon soon, a aextic curve, the locus of the foci of the conies which pass through tho four given points; in the case where the points are in a circle then the sextic breaks up into two eubics (viz., observing that the curve under consideration is V^T + Vm& + V?i + V = 0, where VF : Vm : Vn : Vp = a : b : o : d, these values do of lihomsolvos satisfy the condition of decomposability - + ^ + J; + d ==0 ' ) ' that 1H ' the looua of the foci of the conies which pass through four points on a circle is composed of two circular eubics, each of them having the four points for a set of concychc foci Tt is easy to .sec why the sextic, thus denned as a locus of foci, must break up 'into two eubics; in fact, as we have seen, the conies which pass through the lour Jmoyolio points A, B, G, D have their axes in two fixed directions; there is con- H0 ( ,ucntly a locus of the foci situate on the axes which are m one of the hxed tUroutioi*. and a Repavate locns of the foci situate on the axes which He in the. othu of tho flxod directions; viz, each of these loci is a circular cubic. 200. Adopting the notation of No. 188, or writing .(and thovoforo 6,0, = o,d,) we have a . b : e : d *ft 'Movcovor and wo have ugroeing with the formute No. 188. The tetrazomal curve 560 ON POLYZOMAL CUBVES, is thus decomposed into the two trizomals (6 - o,) VI + (c, - V33 + (a, - 60 Vg = 0, (6j - cO VI + (c, + VS - (a, + 60 VI = 0. 201. Observe that the tetrazomal equation is a consequence of either of the trizomal equations; taking for instance the first trizomal equation, this gives the tetrazomal equation, and consequently any combination of the trizoraal equation and the tetrazomal equation is satisfied if only the trizomal equation ia satisfied. Multiply the trizomal equation by - Oj -f d\ and add it to the tetrazomal equation ; the resulting equation contains the factor c^, and omitting this, it is where observe that ^-d is the distance BO, and 0,1 dt the distance AD. But in like manner multiplying the second trizoraal equation by ct^+di, and adding it to the original tetrazomal equation the resulting equation, omitting the factor a it is (6, - cO (- E + 2>) - (a, - d,) (S8 - S) = ; viz., it is in fact the same tetrazomal equation as was obtained by means of tho first trizomal equation, The new tetrazomal equation, say (6, - cO (- VIT+ V) + (a, - d,) (VS - VS) = 0, is thus equivalent to the original tetrazomal equation; observe that it is an equation of the form VJ2I + VwS + VM 4- Vp5) = 0, where and where consequently Vi + Vp=0 J Vm + V?i=0, that is an equation of tho form (198) III., decomposable, as it should be, into the equations of two circular cubics. Writin where 8 is an arbitrary parameter, the curve is obtained as the locus of tho inter- sections of two similar conies having respectively the foci (A, D) and tho foci (, G) (see Salmon, Higher Plane Ourves t p. 174): whence we have the theorem, that if A, B, G, D are any four points on a circle, the two circular cubics which are the locus of the foci of the conies which pass through the four points ^1, B, (7, J), are also the locus of the intersections of. the similar conies, which have for their foci (A, D) and (B, G) respectively; and of the similar conies with the foci (, D) and (G, A) respectively; and of the similar conies with the foci (a, D) and (A, B) respectively, 202, V, Vr=Vm = V7i = Vp. The order of the tetrazomal is =*5 t whence those of the trizomals should be =3 and -2 respectively. To verify this observe that the ON 1?OT,YHOMAL CURVES. 561 .-.mut-iim ' ' + '" \ M I 1 '.- () irivtiH L -I- ; l + l + \ = 0, and combining with a + b + <H-d = 0, 1 u I) c d n a b (s d l|u"HM iivc <nlv nitUfitii'd liy ono itf tint HyHtoran (a + b = 0, c-t-d = 0), (a + e 0, b + d==0), la |, .1 ,0, It I (^ -0), SulniiUnK U> lix the ideas the first of these, or writing (a, b, o, d) = (a, -a, c, -c), >tn thiil wi> ,.,. n.iuiitinn whirh HiKhilicH l-hat thu radical axis of the circles A, B is also the i m.|i,-nl \i< -I' ill" iiivnlt-H (h J>\ tlum, writing Jus wo may do, Ul' lltU'M 1 - - V, -I -1-1, -2, V;l-l, 0. ,., A-i-Vw,.-^,:.!). whuh iv !H ono of bho tmomals a cubic, viz., this is the T ....... 1,,-r ,n, ....... I B - ..... Wr U, U. UH, ;;:;: ;:: , - - - . // iiu"l "I 1 l-lm i-iirlrH C 1 , A tlte ,*r ..... u Hit*'; Hi" , . . . , , " A" d" Wo establish as before the relation - c d i.\vr lr thn mUu it , , o lUV HM ininnv". . , . of toti'azomal = o, 01 wia I. " * unit 4, i 17. nt tvi7omals =4 and 3; same J j. V i II j order of totvazomal = 7 ; oi tuzoraaia . " tr'^W^o, V.H-VS.O., o^ or -., - B - s ; nro iw III. /. 71 C. VI. 562 ON POLYZOMAL CURVES. [414 204, IV. vT+ Vm + Va + Vp = 0, a V/ -f b Vm + c Vra + cl Vp ~ ; order of tetrazomal = G ; this is a remarkable case, the orders of the trizomals are either 3, 3 or else 4>, 2. To explain how this is, it is to be noticed that in the absence of any special relation between the radii, the above conditions combined with - + - r -I --- h T - give abed vT : Vm : V?i : Vjj = a : b : c ; d( l ); when I, m, n, p have these values, the case is the same as IV. supra, and the orders of the trizomals are 3, 3, But if the radii of the circles satisfy the condition = 0, 1 > 1 , 1 , 1 ft , & , C ) d a', &' , c 3 , rf a ft" 2 , 6"-, o" 1 , rf" a then the two conditions satisfy of themselves the remaining condition - + ~j --- uii = o, -_._ abed and the ratios Vf ; **/m : Vw : arbitrary parameter. "We have instead of being determinate as above, depend on an and between I, m> n, p only the relations Vr+ Vw + Vn + Vjj = 0, a^l + b V?n, + c Vw + d*/p~ 0, We find first ^ = Vi + 4.VPJ5 fa (b v " o?, i/ a , a 3 , tn 9 In place of \/f, , \/a, \^, wo have to find *, y, a, w ftow the conditions X + IJ+ & + Ws=0, +rfiu=0, where tlio oonatanta are connected by the relation It readily appean that tha lino represented by tho first two equations towhe s _ tho quadrio aurfaco in tho point x:y:xi w=>& : b : o : d, so that the B e are in general the only values of *Jl; Vw ; VJ: v In the oaso next referred to m the text the line lies in the surface, and the values are not determined. 41 4"| ON POLYZOMAL CUKVES. 5G3 and thou _ (cl - a) Vi = (6 - d) Vm + (c - d) Vn, (d a) Vp = (a - 6) Vm + (a - c) V, wlu.m(!0 d \T- a VjJ = -j (b V?t - c Vm), Cti Cl mid wo hiwo thus (ol,,rv that itL the ca^not__ under consideratioa bVS-o^-0, ami thevefove \//, -I- VwiT-l- Vwi O, V+ Vms + Vw^O). In liho proHont ctiao wo have thouoo ad_(6-c) , mly 0110 of th wo havo V/-l- "Wo havo then hub wo find and thence P = : Hence in virtue ot A/ b c d-; triaomal ls a qua* Bonding Vernal is , co^ but the othe. 564 ON POLYZOMAL CURVES. [414 205. V. \ll = *J-m= Vtt = Vp; order of tetrazomal is =5; orders of fcrizonmls =3, 2; same as V. supra. VI. ^+p = 0, m + m=sO, a\+&w + oVw + d = 0; order of liotriizomal = 5 ; orders of trizoraals are 3, 2. We have here Vm, = Vm + A/ r^j b Vw, Vjij = Vm + A / V r~i bed or wilting the values of Vm,, V^ in the form ./ j~ , /ad c Vn, = ~ Vm + A/f- -T V be d ad g cnau as oeiore t = equations are V/j = .-_ ,\/; anc j guniiaviy then observing that as before l=~m, if to fix the ideas wo assume v7= A -V^ fcho DC V be; ' , a whence We have moreover and thence so that the 414] ON POLYZOMAL CUEVES. 565 VII. If we have 1,1,1,1 = 0, and ( 1 , 1 , 1 , 1 ) (VF, 'Jm, \ ! n, Vp) = 0, a t I , o , d ft , & , c , d a? , & 8 , c a , d 3 a a , ft 2 , c a , tP n "a /."a "s ,7"a r,"a 7,"a n "3 ,J"s (0 , U , C- , tt II j O , (/ , to tho totraaomal has a branch ideally containing (s 3 == 0) the line infinity 3 times ; order is = 5 ; orders of the trizomals are 3, 2. We have here Vr : Vm : */n : Vw = a ; b : c : d, and thonoo = a + d = a + d . bed / . , = b - --- , Vnifl - b bed which give Moroovei 1 bcd ] a bed V^ + V?H t + VTIJ = 0, V^ a + V?ft a -f V?l. a = 0. a V/i -l- 6 VM&! + c V?ij = ft (a + d) + &b -h cc 0-o) bed bed and similarly whence in virtue of one of the two expressions is ^ bo "(5 -a)" 1 ; and the trizomals are thus a conic and a cubic. Article No. 206. The Decomposable Oww f Transformation to a differen Conmjclio foci, 206. Consider the decomposable case of Vm + Vm23 + VftS + VpJD = ; vi,, the points A, B, G, J) Ho here in a circle, and we hav ( Taking (A, A) the antipoints of (4, D); W. ft) the antipoinf 566 ON POLYZOMAL CURVES. [414 VdDj = 9135, 93^ = 186: (No. 65) and referring to the formula, ante, Nos, 100. et scq., it appears that we can find Z,, in l} n ls p t such that identically and moreover that ^j = ^j 1] mn The equation of the curve gives which may consequently be written 2 = ; viz., this is = ; that is, the two tmomals expressed by the original tebrazomal equation hivolviiyr Un- set of concychc foci (A, B, C, J)) are thus expressed by a now totra.omal oqukfciou involving the Afferent set of concyclic foci (A lt A, 4, A); and wo might of coin-so m like manner express the equation in terms of the other two set* of oonqyolio foci a, #*> 9| A) and (A 3 B 3 O a> $ 3 ) respectively. It might have boon anticipated that ^ Uld naralv 7 - ^ Uld " "^^ ^ f b ' le ^P^"" ^HlH epazately pass fro.n the original set to a different set of concyclio foci, and the two s M wouicl {i raiht be J d to - into a smgle tetrazomal equation; bnt the direct transformation of the equation is not on this account less interesting, ANNEX I. On the Theory of the Jacobian. Consider any three curves ZT- , 7=0, F=0, of the samo order r, then writing J(U, 7, F)= *, z , , we have the Jacobian curve J(U, V t F) = 0, of the order 3r-3. A fundamental property is that if the curves U~Q F-0 W n t common point, this is a nnint- *U T i.. f ' U> ^~ lmv o any double pit, that L P I 1 i, ':"' "^ "? nl ^ S ' but ifc k, - 0,<V=o ? qUeSt W have /= . ^ have also oo , e f roY 0+z<i r- m0+ . i = 0, Jf.fl, > = each f V' = ' " l = ' ' l = each of th each of the order ,) which have in common the 414] ON I'OLYXOMAL OURVK3, 567 (r ft') (r ~ H) points of intersection of tho curves = 0, = 0, each of these points is a nodo on tho Jaoobian, and houco that the Jacobian must bo of the form <I' 3 = 0, whom obviously tho degrees of A, B t must bo i + 2s'-8, r -h s -h s' - 3, respectively. In tho particular ease whore '=0, that is whore I, m, n arc constants, wti have A. ; tho Jaoobian curve then contains as a f'uotor ( ( I> = 0), and throwing out, thu (iiirvo in ./^) -|- G'<D ~ 0, vix., thin is a ourvo of tho order 2r + -s 3 through tiiuih of r(r ) pointu of intomxition of tho curves W = 0, = 0. Li partionlar, if r = ii l ,v 1, that in, if the cui'voH are tho conicH B + M ) = 0, M -|- JI/0) o ( () H- jVtj) ~ 0, pasHing through the' two points of intersection of tho conic HsaQ by tho liim <J>~0, thon tho Jonobiau is a conic passing through tlicso sumo two points, vin., ltn liquation IH of tho form $> H- il<I> = 0. This intorsects any one of tho givon (ionioH, Hay H-I-M' O in thu points (), <!' = (), and in two other points ( H H- iitp 0, ii .=(>; at mo/* of the last-mentioned points, the tangents to the two oiirvoH, and tho linos drawn to tho two pointn H = 0, <1> ~ 0, form a harmonic pencil. Although \,\\w in, in faot, tho known thoorom that tho Jooobian of three circles is thoir ortliotnmie oirtilo, yet it is, I think, worth whilo to give a demonstration of tho theorem as ahovo stated in reference to tho conies through two givon points. Taking (s 0, //; ~ 0), 0? - 0, y - 0) for tho two given points <"> = 0, = 0, tho gouoral ittpiation of a c.onie through the two points is a qundrio equation containing terms in 2 1J , SM, zy, tvy ; taking any two Much conies os' J H- 2e -|- 2i/^ -h 2/(,^ == 0, thoso intniwwt in tho two points (a> 0, 5-0), (y0, ^ = 0) and in two other points; hit (n;, y, s) bn tho coordinates of either of tho last-mentioned points, and tako (X, Y, %) as (jurnmt (soordinateH, thu is(|uutioim of tho linos to tho lixod points and of the two tangents are h A'*-#w0, \ r e-Zy =0, (hy -I- gz ) (Xz - %v) H- (A +/s ) ( F* - %) - 0, (//// -1- Cte) (.A^ - &;) + (//* + Fz) (Yz - Zy) - 0, whonco tho condition for tho harmonic rotation is (hy -I- yz) (Hat -I- Ity f (Jus +f*) (Hy + Gz) - 0, 0, but from the otiuatioim of tho two conios multiplying by &//, Jft and adding, wo have | (o/jf -h 7t(7) z* + (W+fH) yz vix., tho condition in thus reduced to 568 ON POLYZOMAL CURVES, [414 so that this condition being satisfied for one of the points in question, it will bo satisfied for the other of them. Now for the three conies C2 a + 2/ yz + 2g zee + 2/t an/ = 0, cV + 2/' yz + tym + Way = 0, cV + 2/ V + V^ + 2A"y = 0, forming the Jacobian, and throwing out the factor 0, we may write tho equation in the form Of + 2Fyjs + 2 to 4- %Hmj = 0, where the values are G = g (fo" -/V) 4- ^ (/" C -ft' ) + ry" (/c' -/''), tf - (A'/" - h"f) + </' (/."/ - A/') + cf (hf - A'/), 2F = A (fo" -/V) + A' (/' C -fo" ) + A" (/' -/ ), 2ff = A (<ff - cV) + 4' (fy ~ off" ) + h" (eg' - o'fl- ) ; and we thence obtain off + AC = - viz., the condition is satisfied in regard to the Jacobian and tho first of tho throo conies; and it is therefore also satisfied in regard to the Jacobian and tho other two conies respectively. I do not know any general theorem in regard to the Jacobian which given tho foregoing theorem of the orthotomic circle. It may be remarked that tho UHO in this Memoir of the theorem of the orthotomic circle is nob so groat as would ali first sight appear: it fixes the ideas to speak of the orthotomic circle of throo givon oivoloH rather than of their Jacobiau, but we are concerned with the orthotomic circle loss aH the circle which cuts at right angles the given circles than as a circle standing in a known relation to the given circles. ANNEX II. On CASEY'S Theorem for the Circle which touches three gwen.Circles, The following two problems are identical : 1. To find a circle touching three given circles. t<n,eVen p" sVI^of ^ ^ "" "^ f ^ '* = 0) *>* bp Jjf, f ' ^ v 6 fll ' St r b ! em if We use * to de te a given constant (which may be -0) then takmg a , * and {(*-,") for the coordinates of the centre and for the radms of one of the givon circles ; and similarly fc, 6', J(f _^. c> $_ 414] ON POLYZOMAL CURVES. 569 other two given circles; and 8 t S', i(-f") for the required circle; the equations of the given circles will bo ? + (*- a") 2 = 0, and that of the required circle will be ( fl! _S) a + (y-fl l ) !l + (*-O = 0. In order that this may touch the given circles, the distances of its centre from the centres of tho given circles must be i(S"-a"), i(S"-b"), i(S"~c") respectively; the conditions of contact then are (8-ay + (8'-ay + (8"-a! f y > = V, (S - 6) 3 + (&''- 67 + (S"-&'7 = 0> (ir we have from these equations to determine S t S', 8". But taking (a a', a"), (b, V, I"), (c, c', c") for the coordinates of throe given points in space, and (8, 8 , b ) (or tho coordinates of the centre of the cone-sphere through these points, we have the very aiuno equations for the determination of (S, S', S"), and the identity of the two problems thua appears. I will presently give the direct analytical solution of this system of equations. But to obtain a solution in the form required, I remark that the Aquation of the cono-sphoro in question is nothing else than the relation that exists between the coordinates of any four points on a cone-sphere; to find this, consider any five points m Bpacc, 1, 2, 8, 4, 5; and lot 12, &c. denote the distances between the points 1 and 2, &c.^ then 'wo have between tho distances of the five points the relation = 0; 0, 1, 1, 1, 1, 1 ..a o 2 2 1, 0, 12", 13", 14, 16 g 9 2 1, 2f, 0, 23, 24, 25 2 2 3 1, 31, 32, o, 34, 35 ^_ o a M 2 2 1, 41", 42, 43, 0, 45 1, 5l", 52", 53", 5' whence taking 6 to be__the centre of the cone-si we havo 15 = 25 = 35 = 46 = 0; and the equation b. o, Ti a , i?, I 21, 0, 23, 2 ii a , 32 a , o, s 2 a 2 41, 42, 43, C. VI. 570 ON POLYZOMAL CURVES. [414 which is the relation between the distances of any four points on a cone-sphere; this equation may be written under the irrational form 28.14 + 31.24 + 12.84 = 0. Taking (a, a', a"), (b, b\ b")., (c t c', c"), (#, 2/, *) for the coordinates of the four points respectively, we have 23 = V(6 - c)' -I- (&' - G J + (ftTT^, ii = Vfo - a) a + (T/ - ')' + (* - '?. 31 = V(c - a) 9 + (c' - a') 2 + (c" - a") 5 , 24 = vfy- &) + (#- &')+(*-.&''), 12 = Vfa^)' + (a 1 - &7 + (of" - ft' T )T 34 = V(fl- C )* + (y- c '> + (- C ' ; ) a , or the symbols having these significations, we have 23.14 + 31. 24 + 12. 34 for the equation of the cone-sphere through the three points ; or rather (since the rational equation is of the order 4 in the coordinates (, y, z )\ this is the equation of the pair of cone-spheres through the three given points; and similarly it is in the first problem the equation of a pair of circles each touching the three riven circles respectively. In the first problem the radii of the given circles were {(*-<&")> *(-&") *'C*-o") respectively; denoting these radii by a, & 7 , or taking the equations of the given circles to be a') a -a 3 =0, (-&)H(2/-&') 3 ~/3 3 -0 J O-o} a + (z/-c') a -y = ( the symbols then are 12 = (o^. o/ + (' - 6') a - ( - jS)', ^-^"c and the equation of the pair of circles is as before 23.14 + 31.24 + 12,34 = 0; where it is to he noticed that 23, 3l, 12 are the tangential distances of the circles 2 i and S, 3 and 11 and 2 respectively; viz., if , ft, 7 are the radii taken positively, then these are the direct tangential distances. By taking the radii positively o Z^T'f ^ P , lmUr ' btain in ^ fo - equations-the tangential dices big all direct as above, or else any one is direct, and the other two are Inverse: we have thus the four pairs of tangent circles. The cone-spheres whioh pass through a given circle are the two spheres which th" he'for U ^ ^ antiP intS f ^ given Cil ' cle ; d ^ ise ^ to that the foregoing mvestigatzon gl ves the following (imaginary) construction of the 414] ON POLYZOMAL CURVES. 571 tangent circles; viz,, given any three circles A, B t G in the same plane, to draw the tangent circles, Taking the antipoints of the three circles, then selecting any three antipoints (one for each circle) so as to form a triad, we have in all four complementary pairs of triads. Through a triad, and through the complementary triad draw two circles, these are situate symmetrically on opposite sides of the plane; and combining each antipoint of the first circle with the symmetrically situated antipoint of the second circle, wo have two pairs of points, the points of each pair being symmetrically situate in regard to the plane, and having therefore an anticircle in this plane; .these two antioirclos are a pair of tangent circles; and the four pairs of complementary triads give in this manner the four pairs of tangent circles. I return to the equations " ' (a- - gy + (y- 8J + (z - 8J = 0, (a-S)* + (a'-S') 9 + (a"~S") = 0, (b - Sy + (&' - S')' + (6" - 8"? = 0, ; (c - S) 2 + (c' - SJ + (o" - S'J = ; by eliminating (8 t 3', 8") from these equations we shall obtain the equation of the pair of cone-spheres through the points (a, a', o"), (&, If, &"), (c, c', c"). Write to -8, y-8', z~8" = X> Y, Z, then we have Z" + Y* + Z* = 0, and, putting for shortness then, by means of the equation jusfc obtained, the other three equations become SI -I- 2 [(o - a) X + (a' - y) Y+(a" -z}Z} = 0, Those last otiuations give X : 7 where X = b'o" - 6V + (c' - V ) e - (G" - 6") y t p = O 'a" - c"a' + (a' - G' ) - (a" - c" ) y, v = a'6" - a"6' + (&' - ft' ) z - (b" - a") y, V6"o -60" +(o"-6'>-(o -6 )*, /*' = o"ct - ca" + (of 1 - c") so - (a - c ) 2, y ' = a "b - al>" +(!>" - a") as - (b - a ) *, X"=6c' -fc'c +.(o -6 )i/-(c' -6'), fj,"=ca' o'a +(a c )y~(a! o' ), y"=:a6' -a'6 -H(6 -ft }y-(b' -a!}\ 722 572 ON POLYZOMAL CURVES. [414 and the result of the elimination then is (Xffl + + & ) + (X'Sl + P'% + !/($)' + (X"9l + ^"33 + i/'<)' = 0. But aubstituting for 21, 8, 6 their values, and writing, for shortness, - i = VG" - b"c' + cV - c V + a'&" - o"&, -j = &"c -6o"-l-e"a -ca" + a"& -a 6", - & = 5c' &'c + c ft' - C'B H- a 6' - a' 6, A = a (6'c" - 6"c) + a' (b"c - bo") + ft" (bo 1 - 6'c), -^ = (6' C " - &V) (f* 2 + a' 2 + ft" a ) + (c'a" - cV) (& 2 + 6' a + &" a ) + ('&" - a"b") (c + c" J + c" 3 ), - q = (i"c - 6c" ) (ft 2 + ft' J + ft" a ) + (c"ft - cci" ) (6 a + i' 2 + 5" 2 ) + (a"6 - alt" ) (o a + o' a + c" 3 ), - r = (6o / - b'o ) (a 2 + a' s + a" a ) + (eft' - o'a, ) (6 9 + 6 /a + i" 3 ) 4- (a6' - ab' ) (c 3 + o' a + o" 3 ), -Z =(c -6 )(a a -fft /3 + fl" 2 ) + (a -c ) (& 3 4 b'^ + b" 3 ) + ( 6 -ft ) (c a + c' a -I- c" a ), -7rt = (c' -6 1 )(ft a + a /a + a" 2 ) + (ft'-c' ) (6 2 + 6' 2 + &" 3 ) + ( &' -ft' ) (o^ 1 + c' a 4- o" u ), - n - ( c" - b" ) (a 2 + a' 3 + ft ") + ( ft " - c " ) (^ + ji + &" 3 ) + ( &" - ft " ) (c'- 1 + c' 3 -I- c" 3 ), we find 4- z* + 2i (a? + f + tf) - %x (KB +j _ with similar expressions for X'Sl + /*'33 + 1/(, V2E + /53 + y'% and the result) I ny - ms-p}* Iz ~q } a viz., this is (ffi 2 + 2/ + ) (4A (it, +j + 4 A 2 - 2 (ip +jq + fo.) + (^ a + m" + n n )J (te + my + s) 9 + 4 (ins + jy + Jb) (pa + qy + rz] no) + r ( maj i n of fche P-k of ""wphoim The norm of " be 8iWe t0 a numei ' ical ^ .| I /i I ON VOLYXOMAL CURVES. 573 t.hn numuricul l'iu',t,(H' ol' tins (^pimmm in question is in fact =-4, that is, the norm in K= - -I. (/ J -|- f + 2 1J ) 3 (# -V-f -I- /.; 3 ) -I- &c. ; MO l.liiit aU.nndhig only Lit tin; higlumb powci-H in (i, y, z) wo ought to have (/>o)' J -l-</'' ..... ( 0'M-(//WriV( ( ;-< t W .....< laM'jM-n 1 1, l'i i-tisy Ui Him t.hai, Uui norm is in fuot conipoHcd of the torms 2(//-o')M (ft-o) a -(tf-rt) a -('t-W. I- -3 (</ - ')' J 1- (b - c) a -I- (o - <0 a ~ ( ft - W> .,. o ( (i - _ h'Y ( (6 - o) 3 - (o - (i)' J -I- (<t - &)"], llll( ) ul' |,hn mnuluv U.rnw (, A, o), (i^ //', o"), and in K f, o')- (", ^ Oi 'ho above writlcn lijriim urn := !< inio (//7(ft-/0('t-o)-l-('-'OMft-o)(6-tt) + (tt l -6')'Co-fl)(o-&). ,,,,, w | I,!,, vnln. ..I 1 Uu. nnnn is tluu. KC+J' + ftO. i' al "> 111 ANNKX HI. ,, rto W.m ,,/'(/, ..-.)V(--,-(o Tho iinnii <( V(/.|-VK-|.Vir in , ,i , i- Jn i /P.I, \l\f-\- V -\* Vlf-l 1^ J vhi-nnt tliiili ol V a -I- (/ ! v i i i i " ' a 'I 'I 'I __ 'I ., L, 1, -li " L > wluM-u tho laHt tonn i ^2 into is obviously composed anil tho norm in n similar manner, 574 ON POLYZOMAL CURVES. [4-14 Now, applying the formula to obtain the norm of (b - c) */(7+0 + a + (c- a) */+8^& -I- (ft the expression contains six terms, two of which are at once seen to vanish ; and writing for shortness () in place of (1, 1, 1, -1, -1, -1} tho remaining terms fti-o + 2 () {(6 - c) 3 a, (c - ) fr (a - + 2^ Q ((6 - C ) , (o _ a). A ( fl _ ft + 2^ () ((& _ C ) a ( c - of 1>\ (a - 6)- c ^(& - c ) 3 , (c - ) 9 , (a - Z>)' ) ; the first of these terms requires no reduction; the second, omitting tho factor 2, (6-c) a a [ (6-c)a'-(o-a)>&'-( ft --6)v;] + (o - ) 2 /3 [- (6 - c) 3 a 3 + (o - a) 9 6 3 - (a - 6)2 *j + (-i)'7h(6-o) a a-(o-o)fi + ( a -6) atf .-i. which is ' J ' n Similarly the third term, omitting the factor 20, s which i Bonce, restoring tho omitted factors, and oollootmg, we find ON" I'OIVY/OMAI, OIMIVKH. 57f) oiiuo, fh'Hf, writing -w, h~~m, ti-m in plow of </,, /, c ; (Jinn //* lor ifc" a , - ft" 3 , o"' J ) I'm- (, #, -y); uml linally hitrndiioinj.,' s lor hniiingnmiity, wu Fl Norm ((ft - o) vV' ~. ,,4' -|- j/' ~ "" ^ -I- (o - ,/,) ^ -|. (^ ~. A) v* j - s' J into S' J ((6 - ())' tl"' ] (I! )//'* -I- (fi 6V (/'* -- 2 (d - ((.y ( - /0 1J /'" a o'" j - a (i/ - fty (/. - *i)' j <;" y ( f," u - 2 <ft - (t ) ij ( (J - ( '*//' (6 -- d) (; - ./,} (if, ft) [ (A - n ) u" .| - (d . - .) },"> -\. ( lt , ... ft) (I "J - ') (ft - o) (o - ./,) (( - l>) | (ft - f i) <(," ;l (z'> hn - w (h -|i c ) -l- (*;') O t iind Flint HU na l,li) cqnuti.it. (t>- -V) \W-\- (o*- ) \\>\' (<i,.- -!>}\ f (*".( ' J :-!(!) l,hn linr inliniLy twii:n, unil llir <iur\'o in UHIM a i< Until tihn tix|>n^Hii)ii ul' thn nonn ! vix,, wlimi Mm rlinr niittli'K Jntvn nniOi of Miriu rJn ttituui nuliim //', flin tiurvn IN Uin |iuir nl' |iurallrl limi //^ -/:" a" ;.! j atid in |mrl-ii:iiliir wlirn /*;" 1 0, nr Mm oimli'H ntilmto ihiMHMi-KvM uiMili lit a |inini., Lhuu Um mimi in // J :M(, Mm uxis t.wioo. ANNKX IV. o. //IH Tristmwt. ttttrw* Jltf*\>\ ( itiV-\ -v'||'. ,. 0, WV/i Aiuw ft ('/iw/j, Tilt! l,i-ixoinnl citrvn V///4- V/l / '+ ^ulK^O, him nnl in ^ninjrul nny nnctnn v (!impn; in ilui paHimiliu 1 im-nc whom l.hd ^mnul CIUTOM unj firrl<% \vn liavn hownvor wuin how Urn mlititt I : m \ tt, nmy hi) iltitisnninwil MU Mint thn HIU-VU Mltull uctpiim a notlo, (i\vo nixli'H, or a miHjt; vix,,, n^itrdin^ ti, h, r. IIH iwmmt uivul fiinnliiiut^H, wu havn lutrn n - 'I-,'-!' M ^0, Uui lortiH ul' i.lu.i (H'liUvH tif tlu; viiriahli! ninih), anil Lho Holution it t-i (.* on (.'HtublwhiiiR a rnlation hoiwrun UHH <:ituio luid tlm nrLhuLoiiiui oinilis or Jnnobian of Liu) limits givmi oiniluH. X 1m vo in my pitpur " Invt-Htixutiuiw in (uimiuaUnn with Ciwwy'H K.mution," Quirt. Mulh. Jnur. vol. vnr. (1HOT), pp. HIM- -IM.2, [:)()] ^ivon, uftor , a xtilution tf tho xni.'ml (piittitiuu in Htitl Lho nuiubur of Lho curvtw O, whitih liavu a cimp, or \\-hiith Imvn twct itodnM, and I will horn i-oprocluoo thu loiulinx jiuinUi ttf thn mvun%niiim. I ivumrh, that allliouyh ouo of tho loci involved in it is tho Hivinu iw Umfc oceumiijf in Lhu wwo of tho throo oiroluu <vix,, wo Imvo in wutli ctiaa this Jnoohiiui of tho givon ciirvw), fchu other two looi S and A, which prutMiil thomaolvoa, nuoin to Imvo no voliUiuii to fclio ctmio <ti" c which ia nmdo uso of in tho pavtioulw caso. ,T T * f>7() ON I'nl \Vn havo Urn curves V I), I' 0, II' I), nii-h ..| I)M' ,,,,. ,.,,(,., r , U1(r j Hidwiiitf a pninl. tlm i nlimilrx wlmrml 1 mv (/, m, ). ur r,i;anl -. ( ,,| i, ',ii..ntlin L r i,, lliiK poinli Urn niirvtt V//A-I- ^m I' | v'/< II" (I, ),,iy l..r nti..| m. -, *. ih, t . mv ., (i | lrilu ,' ,,., uliovo a liiirvi! u|' Urn nn| ( >r :!r, linvin ? { ; ; ruuiu.-h, mil, ...,), ,,| ||,., nvi (l ,-u'ivi' .' any iiodn, and in nrdcr Dial, thi't mm' miiy lm\.< n nu.tr, u i- ., -, ,, ; ., MV ,!(.,, ( j to Htm, lii mi Uic mirvi' ./, (hi- Jan.Iiiiui <.)' || ( >- i| n ,.,, i/ and A will t'uin!M|iniid In ciuili M|||IT puiu! tu i (l ,i,ii puinti wlialii'Vt'i- mi llir niivi 1 A, (In- dim' ,ti will Imv of ,/; and rnnvi'i-'ii'ly, in nidrr tJial, UK. n Tim (itirvtt A IIIIH, Imwi'ViT, iiMih<ri and riij.n-, ; ( -n | t i,..,|,- ,,3 \ nl' ,/, vi'/., llir (/, JK. ) al, a iindt. i.C ;\. i|,,, ,-nn,- If p, tt I.hn.-i.a 1 1 ,- .-,!. " .,,, M im ., ,. A. HUH |' n ,|i- Wllir ( . n . ll|llim ll , ) . 11|H|l|i , l|l , | m , , (( . n ^ MM|ft A inlu rnmM'x ' nl. IN ihafi lur Um ,'imv ih \ " ' '""i'"'-! |,,., , , , m<1H | , 418 415. rnllltKCTKiNS AND ADDITIONS TO TIIK MKMOIH ON THK TIIKtillY OK UKl'IIMKX'Ah SUHKAOKS (Phil. .'Avow. vol. cux, IHttll, |-ll I |). n- /*AtVt,v*yt/nVfi/ Vniii.v*fi7iW i J M/* //n 1 MJ/<I/ NfictWi/ i;/' London, vol. tn.xil, (fur tho ynir iNYli). |tp, MM HV. Knvi\vrl July i!i!,--Ht'inl Nnvtunlinr Hi, 1H7I,| I, ! AM iiuMtMl hi l)r Xt'tilliKii Ilir ill' 1 iviuurli: I,luil. ull.lmuj,(I) ilin " tilV-poinlH 11 iitiil "nil' jtlitMf'V '' fV|liiiil iii tile iiit'iiM'ii', am rt'itl hin^iihiriUt'w, Ihoy urn nul, Iliu Midgiilniiiit''! In uliii-li Ilif if', /' lit 1 tin 1 Itn'imilm ri'lnr, Tin) IIIOHI, couvtuiitMili \vny of i'liiTf-cfiuj,; (Kin i 1 * I" H-iiiin nil lln< titnniihi* with /?, /)' UH tlny uliuitl, hul< l,u wri in, w* IMV (In* nniuhiT tii "Mll'-|i'"iiil!i" iui"l "nlV-plniH'H 11 niHjn'rtivnly; vix, u - t> tlnm IH ninl M', (iir-|i!iini', ^', uii-xplniiii'il nii ihc litriiniltM UN thry Nhiml, tnkinx ntrtiiint !' flu; iiiirxplitiiiitd Hiii^uluntiDM itml /" hul, nut; tnktnx' ">" iicntmil nl iill !' tin* ull-|HiiulH untl ufl'-plaucH o>, r./. Tlio in whii'h iht'M' n- luki-ii ini" wciunt niv: i? (n - 3) Her -h *$ + 7 -I' (? -H w, CD - 2) ( - M) a (S f ." - :iw) -I- M (HO - Mff - * - :f&>) + ii if, ( g) ( - ;i) 4JL- 4' (6 - 2p - j) + '1 c ( - 3) ( - MJ OA + (o 3tr - x " ilw ) *!" 2 which roploco fialnntn'ii nrixinnl fomuilw (A) und (B). C. VI, H AND AIUUTHINM Til Till; MKMOIK 2. In Mm '''"' lh " " ......... ...... ...... '-ml = " '"".V - .. ( . .,.,,,). ..,. ,, 7 , ('-*l>~M('fl)(H..:\).. ( R..<!.. i ..... , Ht . |MA . ,..,, inn! Hiili.HriMiriu K th.Mfn in n 1 : <ii,i . n /. M, i "' "<" H J ('// I U!,-| , -I/,' , H/ ( i M,,- , | |V --H/- ..SA I 1N/7 ! |-; v , |.j,'. (i, "^.'.'i/f...M; mi icnii iit r, fl ....... '-'- ..... - '- .,, .,,-,,, 7 .. a,- i)'"' HA- -...'IhV..,^ ; ,, 7 , ^ - aiS! (*-//, f , 4 * ..// . 415]' ON THE TH130BY Off 2U50IPKOOAL SUBFAOMS. 579 Tho reciprocal of tho first of those ia " a-' = - n + ' - 2/ - 3^' - 26" - 4/f - 3a>' ; vk writing a = u(n- 1) - 26- 3o, ami 3(^- 2)- 06 -80, tins in <T' - 'in ( - 2) - 86 - llo - 2y" - S^' - 26" -, Mi' - 8<a' ; and ifc thun appears that tho order a-' of the apiiiodo OUL-VO w roduood by IJ for enuh off-piano a)', 4. AH fco tho otliov two o<|uationH, writing for. p, a- lihoiv valuer, bluwo bocomo J-K10+3Z + 5/34-67 = ^(2^- 4) -23, 2^ + 3<o -h 4i + J H/9 -|- 7 = ( j (fin - 1 2) - -I- 8 0, o(iiuiti(m wluoh admit of u gtiomolritsal inturpi'oliatioii. In fiiot, whou tlunx3 is only n nodal cnrvo, tho fiwt oqnabi(H) in j + .--= ft (2- 4)-2</, which wo may verify when tho nodal ourvo IK a oomploto inliui-Hcotion, Jf'^0, Q0; tor if fclio otiuution ol 1 tho mirAioo is (A, ]j t 0$2> t Q)' J = 0, whc;ru Uio do^rooH of A, Ji, 0, 1\ Q (mi w-2/ n-f-fj, n-$f/,f 9 g rcwpeotivoly, then tlui piiiwh-poiutH iiru givon by tho equatioiw P0, ^ = 0, y!.6 v -;i j = 0, IUK! blio nuinbor ,/ of piiioh-pttiutH i/ - 2) ; but for tho cnrvo ./ J =0, Q wo havo 0, and i(,H ordor mid O!OHH aro tf"/^/ (/-!-//" 2), ov tho formula JH thiiH vuriiiod. Hiniilivi'ly, whou bhoro i only a ouHpidtil ourvo, tho Koenml oquabinn IH 2 X -H !l w o (r, H ~ 12) - (h- -I- 30, wluoh may bo vorifiod wlion tho oiiHpidul ourvo IH a nomplobo intornon(iion, J J 0, tho oquation of fcho Hurfnco w lioro (vl, J?, OJP, (3J 9 = 0, whoro /lf?-..// 9 Mj , and tho pointH ^> "' givon IIH tho iiitoraocfciouH of tho curvo with bho aurfaou (A, X, Now AO-& vanishing for P 0, Q wo mufifc Imvo ^1 A 3 -|- yl/, 7 A^ + G', whoro ^.', .//, 6 V vanih for P = 0, Q = 0; and thouoo M = Ajlf -I- Jlf ' + JT, whoro jlf, JV" vanish for P-0, Q-0, Tho uquabiou writing thuroin P = t Q = 0, bhua booomoH A 3 (jY' - jlf fly ** Q ; and its itiboracotionH with the ourvo P-0, Q-0 aro tho points P0 > Q=.0 > A0 oaoh bhroo thnos, and bho points ^ = 0, (3 = 0, JTa-jfa0 oaoh twice; viz. bhoy are bho pointH 2x + 3w. But if bho dogroo of A is -X, bhon bho degrees of JV", j)/' s a a , a/3, ^ aro 2?i - 3/- 2r? - X, 2n-2/-8^-X, n -2/-X, n~f-g-\ t n -%g~\ whouoo bho dogroo of A a (N'a is ^Sw-G/'-G^ aiid tho number of points ia =*/? (6?i - (i/- G.</), vix. this ia or it is =fl(5)t-12)-Gr; so that being 0, the equation is verified. 7S 2 580 CORRECTIONS AND ADDITIONS TO THE MEMOIR [415 5. It was also pointed out to me by Dr Zeuthen that in the value of 24tf given in No. 10 the term involving % should be 6^; instead of + 6%, and that in consequence the coefficients of % are erroneous in several others of the formulas. Correcting those, and at the same time introducing the terms in , and writing down also the in 6 as they stand, we have 4i =...- 2# + 30- So), m =...- Q x + 90- 9w, 2o- = ... - e~ to, 8/j = ...+ GX- 90 + Qai, c' = ...-% + 100- 20u. The equations of No. 11, used afterwards, No. 53, should thus be ** + 6r = ( on - 12) c - 18/3 - 5 7 - 2# + 30 - 3o>, - 24- 89 + 18r = (- 8w + 16) & + (lfi - 36) o- 84/3 + 9y + 4j ~ 6^+ 90- 9 and from these I deduce 6. In No. 32 we have (without alteration) = 16; but in the application (Nos. 40 and 41) to the surface J?P* + G&Q* = Q we have = 0, and there are ^~f n off-poiute, F=0, P = 0, Q = 0, and x = gpq close-points, Q = Q t P = Q } Q^Q, The new oquatimin involving w are thus satisfied. 7. I have ascertained that the value of {3' obtained, Nos. ,51 to 64- of the memoir is inconsistent with that obtained in the "Addition" by consideration of the duficusimy', and that it is in fact incorrect. The reason is that, although, as stated No 53 the' values of two of the coefficients D, E may be assumed at pleasure, they cannot in conjunction with a given system of values of A t B, 0, be thus assumed at plotvmra: viz. A, B being =110, 272, 44 respectively, the values of D t E are really deter- minate. I have no direct investigation, but by working back from the formula in fcho Addition I find that we must have D = ^ = 315; the values of the remaining coefficients then are " ., - , . or the formula is - hfff - g'B' - M - \y _ M y _ v < & _y v . butj^have not as yet any means of determining the coefficients /, /' of the torma 415] ON THE THEORY OF RECIPROCAL SUBFA.CES, 581 From the several cases of a cubic surface we obtain as in the memoir; "but applying to the same surfaces the reciprocal equation for jS, instead of the results of the memoir, we find h 1 -- 4, t/' + lQv =-198, </' + 2/A = 45, g + (/' = 18, X =5 (so that now X + X' = - 2, as is also given by the cubic scroll), And combining the two seta of results, we have h - 24, X = 5, /* = ^~ V = -3f - h' = 4j 9' = 18- X' ___ 7 hut the coeffioionts (/, ft', *', /, /' are still undetermined. To make the result agree with that of the Addition, I assume = -8G, a/=-l, ,9 = 4-28; whence we have - 24(7 4- 40' and if we substitute herein the foregoing value of 44</ 4- ^ >', we obtain 4- (- 93 4- 252) o 4-158^4-937 + 66; -24(7 -285 -^-27j-33x +-' + 4(7 + 103' + ;'+ 7/+ 8 X '- which, except as to the terms in to, en', the coefficients agrees with the value given in the Addition, Dr Zeuthen considers that in general i' = i ; 1 1 pref verified it, 416. ON THE THEORY OF RECIPROCAL SURFACES. rt as an Addition by Prof. OAYLBT in Dr SALMON'S Treatise on the Analytic becmelnj of lime Dimmsiom, 4th Ed. (8vo. Dublin, 1S82), pp. 592601.] 620. IN further developing the theory of reciprocal surfaces it has been found necessary o take account of other singularities, some of which are as yet only imperfectly to g!re * ***** -** "i < n, order of the surface. ' a, order of the tangent cone drawn from any point to the surface. S, number of nodal edges of the cone. , number of its cuspidal edges. p, class of nodal torse. o-, class of cuspidal torse. b, order of nodal curve. Je t number of its apparent double points. /, number of its actual double points. *, number of its triple points, j, number of its pinch-points, g, its class. c, order of cuspidal curve. h, number of its apparent double points. 416] ON THE THEORY OF RECIPROCAL SURFACES. 583 6, number of its points of an unexplained singularity, X, number of its close-points. a, number of its off-points, r, its class. /3, number of intersections of nodal and cuspidal curves, stationary points on cuspidal curve. 7, number of intersections, stationary points on nodal curve. i, number of intersections, not stationary points on either curve. G t number of cnicnodes of surface. B, number of binodes. And corresponding reciprocally to these : n't class of surface. ft', class of section by arbitrary plane. 8', number of double tangents of section. ', number of its inflexions. p', order of node-couple curve. a-', order of spinode curve. b', class of node-couple torse, k', number of its apparent double planes. /', number of its actual double planes. if, number of its triple planes. /, number of its pinch-planes. q', its order. 0', class of spinode torse, h', number of its apparent double planes. $', number of its planes of a certain unexplained singularity, ^ t number of its close-planes. to', number of its off-planes. r', its order. /3', number of common planes of node-couple and spinode torse, stationary planes of spinode torse. 7', number of common planes, stationary planes of node-couple tor i', number of common planes, not stationary planea of either torse O' t number of cnictropes of surface. B' } number of its bitropes, In all 46 quantities, 584 03S T THE THEORY OF RECIPROCAL SURFACES. [416 621. In part explanation, observe that the definitions of p and <r agree with thow given, Art. 609: the nodal torse is the torse enveloped by the tangent pianos along the nodal curve ; if the nodal curve meets the curve of contact a, then a tangout plant! of the nodal torse passes through the arbitrary point, that is, p will bo fcho uumbur of these planes which pass through the arbitrary point, viz. the class of tho torse. So also the cuspidal torse is the torse enveloped by the tangent planes along fcho cuspidal curve ; and a- will be the number of these tangent planes which pass through tho arbitrary point, viz. it will be the class of the torse. Again, as regards p' and a-' : tho node-couple torse is the envelope of the bitangent planes of tho surface, and l;ho node- couple curve is the locus of the points of contact of theso planes; similarly, tho spinode torse is the envelope of the parabolic planes of the surface, and tho Hpinodo curve is the locus of the points of contact of these planes; vi 1 /,. it is tho ourvo UIl of intersection of the surface and its Hessian ; the two curves aro tho rociiprooalu of the nodal and cuspidal torses respectively, and the definition!* of p', a' correspond to those of p and o% 622. In regard to the nodal curve 6, we consider k tho number of its apparent double points (excluding actual double points); / the number of its actual double points (each of these is a point of contact of two sheets of the surface, and fchoro in thus at the point a single tangent plane, viz. this is a plane /', and wo thus linvo /'=/); * the number of its triple points; and j the number of its pinch-points these last are not singular points of the nodal curve per se, but aro singular in regard to the curve as nodal curve of the surface; viz. a pinch-point is a point at which the ^ two tangent planes are coincident. The curve is considered as noli having any stationary points other than the points 7, which lie also on tho cuspidal ourvo; mid the expression for the class consequently is q = ty-l- 2/c 2/- $y Qt. G23. In regard to the cuspidal curve c we consider A tho number of its - double points; and upon the curve, not singular points in regard to tho ourvo per so, but only in regard to it as cuspidal curve of the surface, certain points in number 0, x, <o respectively. The curve is considered as not. having any actual doublo or other multiple points, and as not having any stationary points except tho points (3, wliich he also on the nodal curve; and thus the expression for tho class is r = c 3 - c - 2/t - rt/3. 624. The points 7 are points where the cuspidal curve with tho two ahootH (or say rather half-sheets) belonging to it are intersected by another shoot of tho surface; the curve of intersection with such other sheet belonging to tho nodal oiirve of bho surface has evidently a stationary (cuspidal) point at the point of intersection. As to the points ft to facilitate the conception, imagine tho cuspidal ourvo to bo a semi-cubical parabola, and the nodal curve a right line (not in tho piano of tho curve) passing through the cusp; then intersecting the two ourves by a series of parallel planes any plane ^ which is, say, above the cusp, meets the parabola in two rea points and the line in one real point, and tho section of the surface is a ourvo with two real cusps and a real node; as the piano approaches the ousp, these 4lfi" ON' TIIM TUK011Y OV K1SOI l.'UOOAFj HTJIU'ACJISS, 585 approach tn^oUior, and, \vhim tin 1 , piano pawoM through the nun]t, unite into n Hingular poinl, in the imknro of. 1 a triple point; (node -|- two <SUHJ)H) ; anil when tho plnno pannes below l-lio (!iwp, fclui t.wo (IUHJIH of tho Hookian boeomo imaginary, and (ilia nodal lino on iVoni minodal to mmodul. (i2i"i. At it | mink i bho initial oiii-vo OI'OHMOH tlin miHpidul onrvo, boing on fcho nido awuy from Llio two liiUf-HlinolH of klus mii'laco inniodnl, iviul on t-liti Hides of kho two liiiH'-Hhi.!Ol,n nvuiiodal, vix. lihn two liall-Hluiotn iutoi'Hiiiik oatili oklior iiloag bliin portion of tlm nodal oiii'vu. Thorn in ub Uio point a mngln tun^ont platiOj whioh IH a plains i' '; and we tlnw have i-j-t'. (iiit). AH uliTiuly inoiitinnrd, a cnionodo (J in a point wlinro, iimtciul of a lilium, wo have u tiinj^mli i|imdriroii(9 ; mid itl; a binodti H lilio ipiiulricono (liig inko a pair of plunnH, A. enii;kro])i! (' in a pliinn toiuihin^ tho HurJiino alcm^ a oonio; in klm oivmi of u bitropo l t tlus donio dogonnmU^ into a flab omiic or }iir of points, (Iii7. In llii! original Ibrmuhu for tt, (n-~ a), ft(?*-2), o ( - 2), wo have hi wi-ito *- It iimU'iul nf , and tlm fonnnlin aru furthor ivwulillutl by VOIIHOM of (.ho Hin^iiliiritioH and w. Ho in dim original lorninlm for (/ .(n 2)(n JJ), />(< 2) (it 8), ( 2) (- -'1), wo have iiiHltnwl o(' R to wrikn fi - (V - MM ; uwl to milwliilnito now (ixjutJHHionH for [((/], [f/fi], [lit)], v\'/.. lihemt aru ltib'\ ~nb %p~j, [ltd] s= rfo Jicr ^ , Tho whole HorirH of ((|iiat.ioim thiiH in (]) a'^-M/. (3) /'/ (!i) '*. (4) (* --u (tt- 1)~ 26 --Mo, (5) '.( -li) -lift -Ho. (It) S' - J?i (n - 2) (H U - i) - (n a - 3t - 0) (26 -I- lie) + 26 (6 - 1) -I- (\bo -I- I!G (o - 1). (7) tt(- 2) -,// H- ^ H- 2<r + Hra. (H) &(w-2) p + SS/3 + S7'l-^- (0) o(M-2) 2(r -h 4/9 -h v + ^-l-w. (10) (t (n 2) ( - II) 2 (B - - w) -I- Jl (c - .V - % - flw) -I- 2 (al> - 2/> - j ). (11) 6(H-2)(H~n)*U- -h (6-2p'/ ) + a(6o-8^-27-t> (12) c (n - 2) ( - ) = /t -I- (c - 3ff - # " 3ffl ) -I- 2 (60 - 30 - 2? - 0- 74. C. VT, ' A 586 ON THE THEORY OF RECIPROCAL SURFACES. [410 Also, reciprocal to these (15) '= ?>'('-!) -26' -So'. (16) * = 3'(tt'-2)-6&'-8o'. (17) S = ' (' - 2) ('' - 9) - (' - ri - 6) (26' + 3o') + 26' (6' - 1) + M'o' + jjo' (o' - 1). (18) a'('-2)~*'-.B' + / 3' + 2ff' + 8iu'. (19) &'('- 2)= p' + 2/9' + 37'+3i'. (20) c' (' - 2) = 2<r 7 + 4/3' + 7 ' + 5' + w '. (21) ffl ' (' - 2) (' - 3J = 2 (8' - f?' - SM') + 3 (aV - 3d' - #' - Sw') + 2 ('&' - 2/a' - j'). (22) 6' ( B ' - 2) (' - 3) = ^ + (a'6' - 2^' -/) + 3 (6V - 3/3' - 2y - ") (23) c'(n'-2)('-3)= 6A' + (aV-3tf'- x / - 3') + 2 (6V ~ 3/9'- 87' -O- (24) r/ = 6'^ - 6' - 2tf - 2/' - 3y - Gi'. (25) j-' = c' a -c'- 2^-3^9', together with one other independent relation, in all 26 relations between tho 46 quantities. G28. The new relation may be presented under several difforout forms, oqmvalonl; to each other in virtue of the foregoing 25 relations; these are (26) 2(-l)(n-2) (27) 26^-I2c-4(7 in each of which two equations S is used to denote the same function of tho accontod letters that the left-hand side is of the unaccented letters. ( 2S ) /3' + 0'= 2(ft-2)(lln-24) + (- 66tt + 184) & - 24(7 - 285 - 27j - 33^ r 11 + 4 ^' + 10^'+ r? v + 8v' Or, reciprocally, v T + (-98w' + 262)o / + 22 (2^ + sy + 30 + 27(4)8'+ y + ^ ' - 285' - 27 - ON TIIK THKOUY OK HKOHMlOOAf, HUHFAOKB. 587 inj.;' i!(|iuitinii (S>(i) iu (mil, oxpi'OHHtm blind (Jin sui'limo ami iln rooiprminl Imvn Hiinio tlnl'minncy ; vi/. l;lm oxproHHimi for trho dulioiuiwy in (ItO) ;i Mlcimiuy - \ (,, - I )( - 2)(w - I)) - ( ~ H)(6 -I- o) -I- ~i (f/ -I- r) + 21 H- 5/9 -I- 57 -I- 1 - fttf, (^7 ) in a mttil.ioti for ilrH vahiti n(n (i I- 2 '2 . Tilts n(|uiil.inii (2S) (diitr do Prof, Oiyltsy) in Uus nmTiMil; Itirin of mt Hi-Mi; obl-iiincd liy him (widh wnnti isrnirn in dh tiuiuorutal oonil , linl; whirli in btuil. obl-uimsd by niciutM of l;ho injuiilJtHi ilwill 1 in tint invtiMl-igaUon. lu liml; t cuimidnrin^ i--;to ( \vd lutvu from (Jui liivdi 2fi ttipiiitioim _.. v i; K. - 1 ~ ninl ntiill'IplyiiiK Utcw* i!(|iutliiiiiH Uy l\w ituinbnm wil n|i|wiLo (in Uioin ii tuliHii- 1 , wt- liinl IVoin (2tt) ; rt UH , unct itn 1-lioi'i'tn (2(1) wn Imvc !l* ('i|iuil.iiiii (2V); iuid from thin (2H). or by a liko , (2H). w olthiiiicil witliuuli inudi ililliimlty. AH Ln Llin H SJ-tjquivUuiiH or HyiiiiimbrioH, ' Uiul. (lie lir.M., ihin], loiirtJi, and litlh iu' in I'mit iimlurliMl tunoiix llin original (lii- itti t'X|nvHMJnii wliioli vuniHlirs in in llust il); wn htivn I'nun Minin K-o^M' ..... ', and llii'imi' !t;t (J - < Ma' ---', wlvich in , or wn hnvi* lliiiM lilt! Mi-mind n|imtiim; but, tlio Mixl.lt, wvuiitli, and uiglitli ttquuLiniiH Iiavn yt to IIP cibtiiiiuid. liW), 'IMm n|nn|.ion (!">). ("'). (17) tfivo ftwm (7). (), () (a- ft (a - 2ft 2) (K fl) - ('* a - " - ) (28 -I- H) H- 28 (S r '!) + 8* -1- U (* - 1) J liuvu )(-B) -Jtt -fi/3- 4y-.8i-flH.aM. w - 2) ( - M) 2 (fi - 0') - Hi' - 1HA ~ flfto + 1/9 H- I2y -i- Gt - Ow, 742 588 ON THE THEORY OF RECIPROCAL SURFACES, [41(5 and substituting these values for K and S, and for a its value =n(n ~ 1) 26 3c we obtain the values of ri, c', V ; viz, the value of ri is ri = n (n - I) 3 - n (76 + 12c) + 46 a + S& -f 9c a + 1 oc - 8k - ISA, + 180 + 12? + 12t - 9rf -22-35-30. Observe that the effect of a cnicnode G is to reduce the class by 2, and that of n biuode B to reduce it by 3. 631. We have (n - 2) (n ~ 3) = H S - n + (- 4n + G) = a + 26 + 3o + (- 4?i + 6), and making this substitution in the equations (10), (11), (12), which contain (-2) (ji-8) these become 6 (- 4n + 6) = 4i - 26 2 - 9/9 - 67 - Si ~ 2 p -j, c(-fe + 6) = 6/i-3c a -6^-4 7 -2t~3 t r~ % ~Sa) ) (the foregoing equations (0) Salmon p. 536); and adding to each equation four time* the corresponding equation w:th the factor ( B -2), these become Writing in the first of . means of the values of "* ^ ^ Othm ' 3r + c which give at once the last three of the 8 S-equationa. The reciprocal of the first of these is 'o- + * / -2/-8 x '-20'-4B'-3', or writing herein a = ^-i)_ 26 _ 3c aud ^(n-q-U-Bo, thia is 2) - 86 - llo - 2/ - 3^ _ \ < . 451 ON TIM'l TIIIOOIIY OK KI'lOUMUtUAL HlWtfAdM. 58D (IN2, IiiHlniul (if nbtainhitf the, mwnml and third ripmtintiN an above, wo nmy to ihn vn.hn) nf /> (- <\<)i. -I- (i) add Uvi(! thn vulini of 6( 2); und to liwiuo bhu vtiluo of o (-- >ln. ( (i) mid tlirno l.hncH tho vahtn of o (u 2), tlnin obtaining <!([iiuU(niH frets from {) and <T ri'Hpnsf.ivnly ; UIUHU uptatioiiH an; ft ( 2u -I- a) :-* 4/,: - 2// 1 - fiyj/ - III -I- W -,7, (- fin -I- (J) - m - ({' J - fi 7 - <li - % -I- M w, wliitth, iiiti'oduning lihornin l;ho valiutH of (/ and r, may also bu wribtuu (i (f/ - J 1 2) -I- Hfl ^ (ir -h 1 H/3 -I- 5 7 -h 'h' -I- 2^ -I- Sw. iii^ IIH /(ivnn, llm nrdrr of Mm mirlmn! ; Mio iiuilal mirvn with Jin mnjj h, A',/, /,; l.lio I'.iinpidid cui'vn mid its niiif{ulanUi's <\ h\ uuil iJit! unauliil-idH /if, y, ?! wliiuh ri'latn in tin! inl.mwd.iiniH of t-hr nodid tuid tsuspidal C.III'VUH; thn lirnl, of lihi! t.wo rijiial.iunti I^'IYI-M /', I, ln i iniiiihin 1 nf pinch-points, Imin^ Hingnlnril-itis ol' lihn nodal curve tpinad Uin mirlHcn ; and Ihn Hrnnnd ciiualinii I'Hhililwlmn a viiltttion ImUviuni 0, ^ ( w, thn nf Miujjiiliu 1 pDinlii nf tint c,nnpidal c.urvd tpioiul thn HUi'liKU). In thn ram 1 of a nodal r.urvn only, if thin tin a mmpldiM intiTHi'iiiioii ./'-(), (^ 0, (Jin i'lpiatii'ii n|' l,lu' mirl'an- w (A, tt, f'O/', Q} 1 <-(), and l.lin lirnl. npiaiion iw in 1 , iiH'tiiining I i), nay j $(n ... l)/i- 2// 4 -|"W 1 , wliich may Iw vrrilird ; ami HO in Iho rn.'M' nf a riifipiilal tMirvr niily, wlit'ii t.liifi in a rnniplrtu itiiiuwt'tion V :(}, f j *.-: 0, l\w r.pmlinn nf UK- Hinfaw in (A, li.HQr, <$*>(), wlinm A<! H^-.tiU* ] NQ\ and Li to ^rrond ripial.inii in ,1 (. . fin -I- (1) r. = 1 2A - liu' J -- 2% -I- :U? - Hw, nr. miy ^1 - Mot ; '.(r.H. -|})(i.-G g -l-J2yi .-I-JW, whit-li may II!MH 1m vnrilii'tl. liMII. \\*<! may in thn 11 ml innlanir mil, of Ihn -Hi tpmnl.itiw cimmdoi 1 UH givon ihn !! t|intnli[it'ft ; /', A-,/, t\ o, A, /', x ; ft 7. ' : : '' ^i ilim nf tin! ail ntlatiiuiH, 17 iH.Tinini! tlio 17 fpmiitities u, fi, /c, p, ff\ j, 7 ; r, w 1 ';*', S', *' J A'./ i a' : '. and thciv rrniitin thir {) u'pHttinim (IK), (I!)), (20), (SI), (22), (US), (24), (25), (2H), eliH ihti la ipiiiutitinrt ^ cr'j Jf, t', j', v'; /', '. x. '. '*'; j8'. 7'; ^ ^. 590 ON THE THEOHY OP RECIPROCAL SURFACES. [416 Taking then further as given the 5 quantities j', x ', a', G f , B', equations (18) and (21) give p', <?', equation (19) gi ves 2/9' + 3 7 ' + &', ( 2 ) 4/9'+ 7'+ ff t ( 2S ) /9' + i0', sojhat taking also t' as given, these last three equations determine /3' ; y, 0'; and equation (22) gi vea ', (23) /,', " (2*) </, (25) M r ', viz. taking as given iu all 20 quantities, the remaining 26 will be determined. o = (-!), S = J w ( W - l) ( ?i - 2) ( - 3), a= ' = 3 (n ~ 2), 6' = fa ( n _ i) ( w _ g) (, - n 8 + - 12), ^ = ^ (M _ 2)(? p^ G , B + 16? ^^^ 164w ^ miB + 647; ^ io ^ t ^ (7! - 2) (r _ - 2) (IGn* - G4m 8 + SOtf - lQ8n + 156), 2) (ft - 3) ( - the remainiag quantities vanishing. I Hi I ON TIIH TIIKOUV OK HK(?I IMKJOA li HUlirAOKH. 5!M fittn. The (|ii(>sl,ioii of mn^nlaritie,4 lian bnon oinmiilernd inulor 11. nmns ^onoval point, nf view liy /million, in Mio memoir "Kiwhnrdlit! den HititjuliivHiJs ipii null mpjioi'li i\ mu! druHo nndliple d'uim mirlime," Mtilh. Amittlcn, (,. iv. ]>p. I- -20, 'IH7I, Vfe nUiibnUw l,u A ninnbm 1 of Hiiigulur pnini,n, vix, points nl. any oun of whidi Uin iiui^nnlH form a rcHin nf tin 1 (H'ddi- /t, inn) nliiHH M, wil.li // ]- 1) douhlii liimn, nl' \vhi(;li ;/ iLrn hint((?nls In tinuinlinH ill' Miii nndal rui'vc Lliroiitfli Mio puinl,, and ^ !," Hlial-iuiinry lines, wlidntnl' ^ MIX tinij^i'iil-H lit Itriinolicn ill' UHI i^mpidal diirvo l-hroii^h Llm puini, and wiMi . ilnnUIn plani'H and '/t Mlaliiuiniry pliincs; innntovitr, Mnw poinl.H liiivo unly (Jut prnptti'i'.inn wliioli ui'i' |.lni IMOH(. KViirml in Ilin CIINII nf a Hin'l'urr n^'tirdifd nn u Inimn tH' puinis; and ^ iliwiil-t'ti H fiiun I'xl.i'iiditijif i.o all mid i |iiiiul.H. |Thn Inn^iiiiff jfi'iinnil didiniliimi indndivH l-lit- i'lliMiiiidiiH (/,- i l' ; ; tl, )/ < ; i) :;;'. t,*,- !//.: 71 = = ()), iviill |'alnn, bill, nod prtijiul'ly] Hut ItinndiiH (ft, "'%, i;- ' I, '','/ "fen. : ()), [id innlinlis ulsn (,iu> {^tV-pniuis (/t ^ = c !J, j - n I , //. sij:-i (--^ 0>'|,j And, liirUinr, ti innnhiT nl' nin^ulfi.1' jilanrH, vix. jilaiu-H any mm of \\-liinli a riirvn ol' the cliiHM ^ uiul nrdnr i/, wil.li //*!?/ dnnlilcs (.itii^tmtH, ul' whinh //' ure ^I'lii'riil.in^ HIICM of |.ln< iiiiiltM'unplii lomi!, ,o'-l- f' ntul.innary tan^t'iil.M, of wluiili c' arc !joui'Vn!.iii|,' liui'M u|' iho Mpinuilit forno, u' duulitn poinln und )/ eunpn ; ii, \n t inorenvrr, uiipjui^i'd l.liiil, llu'si' plnin'M Inivn only ( li (1 jinipriiit!^ wluiih are 1. 1 in IUOH). Lfunenil in 1 lln riiHf nf a HiirfiKin rr|.;jinlcil UM nit eiivi'lnpu u|' J|M liiiij^cul, planen ; nml ^' iltMinl'prt a. Kiilit Msileiidiliff in all Hiirh plannn. ('IMm ilH'mUiuii innlutlcM lln^ oiuolFropPH (// - - v i!, // ; i/ ' ; g' ; ! (,'' ; it'.. -ii' 0), and julHM, hu(, not, propi'vly'l l,hn ItiirnpnH (;t' : - - i' 1 i/'^fer.. . (I), fit. inrltidi'N ulno MM; (itV-plniK's (fi. 1 t>' =.'t, ;;' ; ^/^"l, >/' /ri T ll.'lti. ThiH In-ill^ HO, uinl wriditix flic dpiiLUniiM (7), (H), {!)}, (10), (11), (liJ), cnutaiii in ivnpur.1, ol' ilm nt;w uilditiiKiul lii-rniH, vix. Uii-sr urn /*( a) :-.... i -s [//(/- ay). 2)(n aiitl lUartt avc nf wmwc Mm reciprocal tcnnn in Llm reripronil eipiutioiiH (1H), (I!)), (20), (21). t^iJ), (:0. Thenu loviiiuliu art! tfivm willitml. ilitinniiHirutiitn in Um nioiium' jnnl- rofiarrwcl Ut: Un principal nltjiict of tlui mnmmr, tw Hhown liy iln l:iLl(.', i tlw roiiHuler- uliuil mil nf Klieh Kingilliir puhitH untl ]tliilHjA, hill "if l.lle, umlti]ilu i-\H\\\, HHOH of n Murfiicc; nml tu \v^iw\ tit thuwf, bhu muinnir Mlnmld bn NOTES AND REFEBENCES. 384, THE conclusion arrived at Nos. 27 30 that the transformed curve of the order D + l depends upon 4D G parameters is at variance with Kiemann's theorem according to which the number of parameters is 3p-3, (p Kiemann -D Cayley), = 3133, and this last is the correct value, My erroneous conclusion is referred to 111 the preface to Clebsch and Gordan's Theorie der Abel'schen Jfimctwnen (Leipzig, 1806), "Unter den von Riemaim behandelteu Theilen der Theorie haben wir die Frago imch der Anzahl der Moduln einer Klasse von Abel'soheii Funcbiouen ausschliessen m mltsson geglaubt. Diese Frage ist durch die achaifsinnigen Betraclitungon des Herni Cayley Gegenstand der Controverse geworden : sie ist iiberhaupb wohl zmiftohst iwr durch tiofe algebraische Untersuchungen endgilltig zu entscheidon, fllr deren Sohwioiigkeifcen die gogcn- wai-tig bekannten Methoden nicht mehr ansaureiohen scheinon," In the case D(or 2>) 3 my value is 10, Riemaun's is 9 : that the latter is correct was shown by a direct proof in the paper Brill, "Note beziiglich der Zahl der Moduln einer Ktossu von algebraischen Gleichungen," Math. Ann., t. i. (1860), pp. 401406 : the explanation of my error is given in the paper, Cayley, "Note on the Theory of Invariants," Math. Ann., t. in. (1871), pp. 268271. 400, The question here considered, viz,, the expression of a binary sexfcic f in the form u s - w 3 , and a cubic and a qimdric respectively, forms the basis of the very interesting investigations contained in the Memoir, Clebsch "Zar Theorie der binSren Formen seohster Ordnung und zur Dreitheiluug der hyperelliptisohoii Funotionen," Gdtt. Alii,, t. xiv, (1869), pp. 1 6&. Considering / as a given sextic it is remarked that the number of solutions, or wliat is the same thing the numb the functions u or -y, although at first sight = 45, is really = 40 ; supposing tha* is a given solution it, v, or that the sextic function is in tho first instance gi the form ^-w", then if any other solution is u', v', we have v 3 w 8 = v' 3 - it' 3 , wiiere v f , u' are functions to he determined : there are in all 39 solutions, a set of 27 and a set of 1-2 solutions; via. writing the equation in bhe form (v+if)(v~i/)=(uu!)(u-ev!)(u eV), e an imaginary cube root of unity, then either the v + v' and the v-if contain each of them as a factor one of the quadric functions uuf, u eu', w-eV (which gives the set of 27 solutions) or else the v + v r and the v-V are each of them the product of three linear factors of the quadric functions respectively (which gives the set of 12 a vi. 75 594 ' NOTES AND REFERENCES. solutions). It may be added that the 27 solutions form 9 groups of 3 each and that these 9 groups depend upon Hesse's equation of the order 9 for the determination of the inflexions of a cubic curve; and that the 12 solutions are determined by an equation of the order 12 which is the known resolvent of this order arising from Hesse's equation and is solved by means of a quarfcic equation with a quadrin variant = 0. As appears by the title of the memoir, the question 13 connected with that of the trisection of the hyperelliptic functions, 401, 403, On the subject of Pascal's theorem, see Veronese, " Nuove teoremi sail' hexagrainmum mysticum," It. Accad, dei Linoei (187u' 77), pp. 761 ; Miss Christine Ladd (Mrs Franklin), " The Pascal Hexagram," .Amen Math. Jour,, t. II. (1879), pp. 1 12, and Veronese, "Interpretations ge'ome'triques cle la thdorie des substitutions de n lettres, particulierement pour n = 3, 4, 5, en relation avec les groupes de 1'Hexagramme Mysti- que," Ann. di Matem., t. XI, 188283, pp. 93 23G. See also Richmond, "A Sym- metrical System of Equations of the Lines on a Cubic Surface which has a Conical Point," Quart. Math, Jour., t. xxn. (1889), pp. 170179, where the author discusses a perfectly symmetrical system of the lines on the cubic surface and deduces from them equations of the lines relating to a Pascal's hexagon : there are of course through the conical point 6 lines lying on a quadric cone and these by their intersections with the plane give the sis points of the hexagon : the interest of the paper consists as well in the connexion established "between the two theories as in the perfectly symmetrical form given to the equations, 406, 407. A correction was made by Halphen to the fundamental theorem of Chasles that the number of the conies (X, kZ} is = a/A + j3v, he finds that a diminution is in some casea required, and thus that the general . form is, Number of conies (X t 4Z) = a^t + /3y - F : see Halphen's two Notes, Gomptea -Rendus, 4 Sep. and 13 Nov., 1876, t. LXXXIII. pp. 537 and .886, and his papers "Sur la the'orio des caractdristiques pour les coniques," Proo. Lond. Math. 8oo. t t. ix, (18771878), pp. 149170, and " Sur les nombres des coniques qui dans un plan satisfont . i\ cinq conditions pvojectives et inde'pendantes entre elles," Proo. Lond. Math. Sac., t. x. (1878 79), pp, 76 87 : also Zeuthen's paper "Sur la revision, de la" theVie des caracte'riatiques de M. Study," Math. Ann., t, xxxvu. (1890), pp. 461464, where the point is brought out very clearly and tersely. . , The correction rests upon a more complete development of the notion of the line-pair-point, viz, this degenerate form of conic seems at first sight to depend upon three parameters only, the two parameters which determine the position of the coincident lines, and a third parameter which determines the position therein of the coincident points: but there is really a fourth parameter. (Compare herewith the point-pair, or indefinitely thin conic, which working with point-coordinates presents itself in the first instance as a coincident line-pair depending on two parameters only, but which really depends also on the two; parameters which determine the position therein of the vertices.} As to the fourth parameter of the line-pair-point the most simple definition is a metrical one; taking the semiaxes of the degenerate conic to be a and b (a = 0, & = 0) then we have two positive integers p and q prime to each other such that the ratio NOTES AND REFERENCES. 595 a? : bv is finite ; and this being so the fractional or it may be integer number p : q is the fourth parameter iu question. But it is preferable to adopt Halphen's purely descriptive definition, viz. we consider a conic 1 in reference to three given points y t z, t on a given line, and take is, ot for the intersections of the conic with the lino : we take a = (y, z, t, K) (y, 2, t, #') for the difference of the corresponding anhar- monic ratios of the three points with the points #, #' respectively; and 2 we consider the conic in reference to three given lines Y, Z, T- through a given point and take X ) X' for the tangents from the given point to the conic; we take l(Y, Z, T, X) (Y, Z. T, X') for the difference of the corresponding anharmonic ratios of the three lines with the . / Q> jy' 11 t lines X, X' respectively | observe that these values are a = - -. ~, , , and 1 J \ z-m.e-tf z y,tt t X ~X' Y ~-T \ b v v~v v> *" ~r7 \7-~n m } Here when the conic is a line- pair-point, to ~ &f and /j A- . Zi A A X , & 1 / X = X', where a = and & = 0, but we have as before the integers p and q such that a v : 6'-' is finite, and we have thus the fourth parameter p : q. Halphon's correction is now as follows, starting from the formula number of conies (X, 4)Z) ttfj,+ @v, we may have among the a/i-t-/3y conies Ime-pair-points any one of which if we disregard altogether the fourth parameter is a conic satisfying the five conditions, but which unless the fourth parameter thereof has its proper value is an improper solution of the problem and as such it has to be rejected: if the number of such solutions is = P, then there is this number to be subtracted, and the formula becomes, Number of conies (X, 4#) = /* + /9? P. It may be asked in what way the fourth parameter comes into the question at all: as an illustration suppose that a, l> denoting the semiaxes of a conic, or else the above mentioned descriptively denned quantities, then p, q, Is denoting given quantities (p and q positive integers prime to each other) the condition X may be that the conic shall be such that a 1 ' -h fr* = & ; this implies a" : finite, and hence clearly if the system of conies (X, 4<#)< contains line-pair-points, no such line-pair-point can be a proper solution unless this relation OP -*& k is satisfied. . NOTES AND REFERENCES. and assuming the correctness of Zeuthen's values it would seem to follow that tho four forms of surface have 12, 6, 12, 1 actual double planes respectively. 413. In the equation No. 36, tl = AP + BQ + CUZ + .. = 0, it is implicitly assumed that the number of terms P, Q,R,.. is finite, viz. the implied theorem is that any given fc-fold relation whatever (k of course .a. finite number) there is always a finite number of functions P, Q, X,'... such that every onefold relation included in tho fc-fold relation is of the form in question fl, = AP + BQ+ GE + ..., -0; this seems self- evident enough, but I never succeeded in finding a proof: a proof of the theorem has however been obtained by Hilbert, see his papers "Zur Theorie dor algobraisoheii Gebilden (Erste Note)," Oott, Nadir. No, 16, (1888), pp. 450457. 411, 415, 416. The first and second of these papers precede in date Zoufchon's Memoir of 1871 referred to in 416, but I ought in that paper to have referred also to his later Memoir, "Revision et extension des formulos nume'riques do la blidorio des surfaces reciproques," Math. Ann, t, x. (1870), pp. 446546. I compare tho notations as follows, viz. for the unaccented letters we have Cayley, n, a, , K, p, a- &. q> k *. 7 G, r, h, j8, 0, a G,B 23 letters in all, Zeuthen. n t a, S, K, p, a- b, q, k, t, 7 ; a G, r, h, /3 ; in J> B, U,0 f, i, d, y, e 27 letters in all Here for Zeuthen's k, h, I have written k, k, viz, these numbers represent the Plitckerian equivalents of the number of double points for the nodal and cuspidal curves respectively. Zeuthen considers also the general node, say ( (^, v, y-\-v),0 + & u, ti), see 416, this includes the cnicnode and off-point w, and accordingly lie includes under it and takes no special notice of these singularities, but it does not properly include, and he takes special notice of, the binode B\ it does not extend to tho case where the tangent cone breaks up into cones each or any of them more than once repeated, and accordingly not to the case of a unode U where tho tangent cone is a pair of coincident planes, He introduces this singularity, and also the singularity of the osculating point which is understood rather more easily by means of the reciprocal singularity of the osculating plane 0', this is a tangent plane meeting the surface in a curve having the point of contact for a triple point ; and lie disregards my unexplained singularity $. The letters s, m do not denote singularities ; s is the class of the envelope of the osculating planes of the nodal curve, m the NOTES AND REFERENCES. . 597 class of the envelope of the osculating planes of the cuspidal curve. Finally d denotes the number of stationary points (cusps) of the nodal curve, exclusive of the points 7 which lie on the cuspidal curve; and g and e denote, g the number of ordinary actual double points of the cuspidal curve, e the number of stationary points (cusps) of the same curve, exclusive of the points ft which lie on the nodal curve. Moreover with Zeuthen, the nodal curve has 30'+2' double points /+30' + 2', if k denotes, as with me, the number of apparent double points of the curve), and it has 7 + d + 2' stationary points. The cuspidal curve has g + QX' + l ^' + U' + 40' + 2 + 2' double points ', if A. denotes, as with me, the number of apparent double points of the curve), and it has ft 4- e + 20' stationary points and the nodal and cuspidal curves intersect in 2 + :' points; where I have written 2 and 2' to denote sums (different in the different equations) determined by Zoutheu, and depending on the singularities ( and (' respectively, For comparison of my formula? with Zeuthen's it is thus proper in my formula to write = 0, a> = 0, Q = (but in the first instance I retain 0) and in his formula to write *7=0, = 0, d = 0, # = 0, e = 0, 2 = 0, S'^0. Doing this the last mentioned formulro give as with me 3+/ double points and 7 stationary points for the nodal curve, but they give for the cuspidal curve 6^' + 12y9' (instead of 0) double points and ft stationary points; and the two curves intersect (as with me) in 3/3 + 2-y + i points. There is a real discrepancy in the number 6^' + 12y9' of double points on the cuspidal curve. 5 gg NOTES AND RKFKHKNCKS. I compare his (6 + 26 + 1=) 33 relations: (1) a = a', d = d'. k = h'. (6) (-l)a + 26 + 3c, (7) ft (((-!) = + 28'* 3/c'. (8) cW = 3(-4 (9) b(b-l) (10) [3 (b - q) ~ 7 + d - s + 2', dotonninofi sj. (11) c (c - 1) = r + 2A + 3/3 + GO' -!- 3d, (12) [3(o-r)-H-m + 20' + S', dotoi-minos + 3i H- 00' + (15) o(-2) (16) a(- (IV) K- (18) c(fl- with the liko i-eoiprooal oqutitioiw (0) to (IK) ; (19) <r+m-r-/9- 4/ - 3^ - 14(7' + ^ NOTES AND KEFEllENCES. 599 and my (3 +22 + 1 =) 26 relations as follows (1) rt = it'.. (2) /-/'. , ,. (3) i=f. (']<) a = (n-l)-2&~3c. (5) tf' = U?i(?i~ 2) (J6 8c. (0) ' in (- 2) (' -!))-&& (A.) (!!!) (11) (14) (0) (7) a (n - 2) - k - Ji + /> + 2a + (B) (8) fc(;t-2)p + 2 + 87 + 3f. (IS) (!)) o(n-2)2ff (F) (10) (tOi-2)(ji-8)2(8-0-3w) + 8(ao-3ff- x -3o (0) (11) &(<>i2)(-3)= ' 4& + (wfr-2/3-j ) + 3(6c (H) (12) o(n-2)(-8)= flA + (ao-8o %-3)4 2(6o with the liko rooiprocol equations (4) to (I) (20) 2(?i -l)(?i -2)(n -3)-120i-3)(& = 2(n' - !)(' - 2)(?i' - 3) - 12 (' - 3)(&' +c') + Gg' + 6)'' + 24t' + 42/9' + 30 7 ' A + <y + 6 + 1 2;r + D 7 + 40'- + S + S'. 600 NOTES AND BEPERBNCES. Substituting for k t h their values we have instead of (A), (5), (C), (D) the equations (A 1 ) tt-b (JS'J c 2 -c Writing as before 0=0, &> = 0; tf = 0, = 0, rf = 0, r/ = 0, e = 0, and neglecting the terms in 2, 2', the two equations (E) become Zeuthen c ( - 2) = 2a + 4/3 + 7 + 8^' + 167:2', Cayley c (, - 2) = So- + 4/3 + 7 + 0, which can be made to agree by writing = 8^' + 165'. But we have Zeuthen (5') c 9 - c = r + 2fc + 3/3 + 12^' + Cayley (5) C 2 -e = values which differ by the terms 12tf + 24ff, or if 5 has the value just written down, the term f 0. I refrain from a comparison of the two equations (I.), and of the expressions for the deficiency given by these two equations respectively but I notice hove tho expression for the deficiency obtained by Zeuthen in the last section (XIV.) of his Memoir, viz. this is 85+ 245' + 18tf + 6J7' + GO' The problem is a very difficult one, and it cannot be held that as yet a complete solutum has been obtained, Take in plane geometry the question of reciprocal curves - here, using throughout point-coordinates, we start with a curve represented by the general equation ( ffl) y, *)>- , such a curve has only isolated singularities, viz. the line-singularities of the inflexion and the double tangent, we know the expression in point-coordinates of any such singularity (inflexion or double tangent as the case may be), viz, we can at once write down the equation of a curve of the order n having a given stationary tangent and point of contact therewith, or a given double tangent and two points of on act herew^h Returning to the general curve fc y, ,)._<>, we know that the recip ocal curve has other isolated singularities, viz, the point-singularities which corre- spond to these, the double point (or node) and the stationary point (or cusp), and we know the expression of any such singularity (node or cusp as the case may be) vi, we can at on ce write down the equation of a curve of the order n having at a given Zdi^Sr &?**&"* OT S P with given tangent. And then starting afiesh with a curve of the order n having a node or a cusp we obtain the effect NOTES AND REFERENCES. 601 thereof as regards the line-singularities of the inflexion and the double tangent. We are thus led to consider as ordinary singularities in the theory the above-mentioned four singularities of the inflexion, the double tangent, the node and the cusp; and we know farther that any other singularity whatever of a plane curve is compounded iu a definite 'manner of a certain number of some or all of these singularities, But in the theory of surfaces, starting in like manner with the general equation (OB, y, z> w) n == 0, such a surface has torse -singularities, the node-couple torse, and the spin ode- torse ; each of these is in general an indecomposable torse of a certain kind (hut there is the new cause of complication that it may break into two or more separate torses), but we do not know the analytical expression of these singularities, nor consequently the analytical expression of the curve-singularities which correspond to them, the nodal curve and the cuspidal curve. Thus if we attempt to start with a surface (ft 1 , y, e t w) n ~Q having a nodal curve, we can indeed write down the equation in its most general form, viz. if the nodal curve has for its complete expression the / equations P = 0, Q = 0, 11 = 0, &c. (viz. if the curve is such that every surface whatever through the curve is of tlie form fl, =AP + J3Q+GR + ..., = 0) then the most general equation of the surface having this curve for a nodal curve is (A, B, G, ...]P, Q, R, ...) a = 0, but this form is far too complicated to be worked with ; and if for simplicity we take the nodal curve to be a complete intersection P = 0, Q = 0, and consequently the equation of the surface to be (A, B, CJP, Q) a = 0, then it is by no means clear that wo do not in this way introduce limitations extraneous to the general theory. The same difficulty applies of course, and with yet greater force, to the cuspidal curve; and oven if we could deal separately with the cases of a surface having a given nodal curve, and a given cuspidal curve, this would in no wise solve the problem ^ for the more general case of a surface having a given nodal curve and a given cuspidal curve. It is to be added that the general surface of the order n has no plane- or point-singularities, and thus that such singularities (which correspond most nearly to the singularities considered in the theory of reciprocal_ curves) present themselves in the theory of reciprocal surfaces as extraordinary singularities. END OF VOL. VI, CAMBRIDGE : PRINTED BY C. J. CLAY, M.A. AND SONS, AT THE UNIVERSITY PRESS. SKLKCJT SCIENTIFIC PUBLICATIONS OF THE OAMBlfflXlE UNIVEESITY PRESS, The I I Collected Mathematical Papers of Arthur Cayley, Sc.B., F.R.S., Sadlorian iT iT\J iw r ft ' ml 'i' 1H S " th(1 ' T '>ivi'Hity nf Oan.bri.lgo. ])cmy dto. 10 vote. Vols. 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VOL. VI. CAMBRIDGE : AT THE UNIVERSITY PRESS. 1893 [All Mights reserved.] ADVEETISEMENT. THE present volume contains 33 papers numbered 384 to 41G published for the most part in the years "1865 to 1872; the last paper 416, of the year 1882, is inserted in the present volume on account of its immediate connexion with the papers 411 and 415 on Reciprocal Sur- faces. The Table for the six volumes is Vol. I. Numbers 1 to 100. II. 101 158. III. 159 222. IV. 223 299. V. 300 383. VI 384 416. CONTENTS. 384. On the Transformation of Plane Curves ..... 1 Proo. London Math. Society, t, i, (186566), No. in. pp. 1 11 385. On the Correspondence of Two Points on a Curve ... 9 Proc. London Math. Society, t. i. (1865 66), No, vn, pp. 1 7 386. On the Logarithms of Imaginary Quantities .... 14 Proc. London Math. Society, fc. n, (186669), pp. 6054 387. Notices of Communications to the London Mathematical Society . 19 Proo. London Math. Society, t. u. (186669), pp. 67, 2020, 29, 6163, 10310']., 123125 388. Note on the Composition of Infinitesimal Rotations . . 24 Quart. Math. Jour. t. vm. (1867), pp. 710 389. On a Locus derived from T^VQ Conies ...... 27 Quart Math. Jour. t. vm. (1867), pp, 7784 390. Theorem relating to the four Conies tuhich touch the, same two lines and pass through the same four points Quart. Matli. Jour. t. vm. (1867), pp. 162167 391. Solution of a Problem of Elimination .... Quart. Math. Jour. t. vm. (1867), pp. 183185 392. On the Conies which -pass through two given Points an tivo given Lines ...... Quart. Math. Jour, t. vm. (1867), pp. 211 219 v l CONTENTS. 393. 0/i the Conies which touch three given Lines and pass through a given Point ....... , . Quart, Math. Jour. t. vm. (1867), pp. 220222 304. On a Locus in relation to the Triangle . . . L/ * * " Quart Math. Jour. t. vm. (1867), pp. 264277 305. Investigations in connexion with Casey's Equation . , Quart. Math. Jour, t, vui. (1867), pp. 33d 841 3DC. On a certain Envelope depending on a Triamjk imcribcd in a Circle ........ . Quart. Math. Join-. t, ix. (1868), pp. 31-41 imd 17fi 170 397. Specimen Table M , <M (Mod. N) for any prime or composite. Modulus . ......... Quart, Math, Jour. t. ix. (1868), pp. 96-90 ami pl,vte 398. On a Certain Sextic Developable and Sextic Surface conned t/ieremth . ' ' ' * Quail Math. Jour. t. ix. (1868), pp. I20-.ua and 373-370 890. On the Cubical Divergent Parabolas ..... Q^rt. Math. Jour. t. ix. (1868), pp. 18G-189 W. On tlu, Cubic Ct. m inscribed in a given Pencil of flfe .^,, Quart. Math. Jour. t. ix. (1868), pp. 210-221 401. A Notation tf tjle foints md ^ ^ pasmts Qa.t Math. J 011 , t Ix . (1868)| , )M68 __ 274 2. On a Singularity of Surfaces Quart. Mail, Jom . t . , x . (1868)| pp jM!uas8 ' ' ' ' ia.1 3. On Pascal's Tlieorem Quart. Maft . Jou , , I18 . . . , lg|l 404. Reproduction of Euhr's Memfr of ,,,. rt Solid Sody. . J 758 on the Rotation of a th. Jou , t . IX . - ias otvi , fe 1 p> HJ ^ . 147 CONTENTS. IX FAQE 406. On the Curves which satisfy given Conditions .... 191 Phil. Trans, t, OLVIII. (for 1863), pp. 75143 407. Second Memoir on the, Curves which satisfy given Conditions ; the Principle of Correspondence . . . . . 263 Phil. Trans, t. CLVIII. (for 1868), pp. 145172 408. Addition to Memoir on the Resultant of a System of two Equations .......... 292 Phil, Trend, t. OLVIII. (for 1868), pp. 173180 409. On the Conditions for the existence of three equal Moots or of two pairs of equal Roots of a Binary Quartic or Quintic . . . . . . . . . . 300 Phil, Trans, t. OLVIII. (for 1868), pp. 577588 410. A Third Memoir on Skew Surfaces, otherwise Scrolls . . 312 Phil. Trans, t. CLIX. (for I860), pp. 111126 411. A Memoir on the Theory of Reciprocal Surfaces . . , 329 Phil. Trans, t. CLIX. (for 1869), pp. 201229 412. A Memoir on Cubic Surfaces ....... 359 Phil. Trans, t. OLIX. (for 1869), pp. 231326' 413. A Memoir on Abstract Gfeometry ...... 456 Phil, Trails, t. CLX. (for 1870), pp. 5163 414. On Polyzomal Curves, otherwise the Curves -77+Vl 7 -t-&c. = . 470 Trans, R. Soo, Edinburgh, t. xxv, (for 1868), pp. 1110 415. Corrections and Additions to the Memoir on the Theory of Reci- procal Surfaces 577 Phil. Trans, t. OLXII. (for 1872), pp. 8387 416. On the Theory of Reciprocal Surfaces ..... 582 Addition to Salmon's Analytic Geometry of Throe Dimensions, 4th ed. (1882), pp. 593604 Notes and References ......... 593 Plates to face pp. 52, 122, 190 Portrait ......... to face Title. CLASSIFICATION. GEOMETRY : Abstract Geometry, 413 Curves, 114 BeciprocHl Sm-faces, 411, 416, 416 Cubic Surfaces, 412 Skew Surfaces, 410; Developable, 398 Singularity of Surfaces, 402 = Curve., ! pencil of aix K MS , ANALYSIS : Hotatio, ls , 388 M cmoh . f of eight,, M enioi ' ' > 8, 40D 384. ON THE TRANSFORMATION OF PLANE CURVES. [JTi'om the Proceedwcjn of the London Mathematical Society, vol. I. (1865 1866), No. in.. pp. 111. Eead Oct. 16, 1865.] 1. THE expression a "double point," or, as I shall for shortness call It, a "dp," is to be throughout understood to include a cusp : thus, if a curve has S nodes (or cloTuble points in the restricted sense of the expression) and K cusps, it is here regarded as having 8 4- K dps. 2, It was remarked by Cramer, hi his "Theorie des Lignes Courbes" (1750), that; ft curve of the order n has at moat (?&-l)(ji 2), =s -J- (n 9 3n) + 1, dps. 3, For several years past it has further been known that a curve such that the coordinates (fa : y \ z) of any point thereof are as rational and integral functions of tlie order n of a variable parameter 6, is a curve of the order n having this maximum mi rnboi 1 |(?i !)( 2) of dps. 4 The converse theorem is also true, viz. : in a curve of the order n, with (ra l)(?i 2) dps, the coordinates (so : y : z} of any point are as rational and integral functions of the order n of a variable parameter Q or, somewhat lesa preciselv. him coordinates are expressible rationally in terms of a parameter Q. OS THE TJUHSFOKlttTION OF PLANE CUBVBS. fgg. series of curves of the order n-1, given by an equation V+9V = (\ eontaininrr ,,, rbitrary parameter 6; any such curve intersects the given curvo in tl.o ,1,,, ,,| >.mt,, lg as two points in the S.-S points, and in o e other point; hone,, w ' tl,,,, ' only one vamble pomt of intersection, the coordinates of HUH point, vii! ., ,,,', ,,, :.tes of an arbitrary pomt on the given curve, are expressible mtio mllv in t, r "*" * = the foregoing theorem is tha o expressible rationally in terms of a parameter 9 S fet po ints ^succeed each other in ^^ AR ^" ^ T" V " ',"" tamed by g lv i llg to the parameter it s different real Un fron , " ,'"' rve may be termed a , curve. "^ '"' ' lllui inciation it , s necessary to refer to nae-e l<f! , Tr PP ' , " <mi ' lll!to th " ; Abelschen Functioned Or*, t J "( 8 7, 1 'TT-- '""' ^P' " Thll '' rio I lei L1V - ^ a7 >: PP' Ho-],,,,, Vlai> th(J ( ,, mil(!i(l |. ilm of any order form (1, f)(i, , ) = . P aiamete " (6 ';), connected by nu (!(I , llUi , m >0 - rationally in terms of the p a mn rn IK \ a certain form, viz.: ' te ' S (fi ') O ' mn *<l by an o,,,,atioii of fl = l, the equation is (1, t\t n ~f-n , , j^'-^erootof;^;,^: oT "' tho """" "-4 the equation is (1, py> a x,_ n v i. the square root of i 6 'exU 'fltbn O f P" to " 10 * mo " > 2, viz. ; ' 5 odd, = 2/1 -3, the equation is a tw, , that treating (f ^ , r,^' f r(1> '" =. and is besides such represented Lyi 2 )f d J rteSmn CM '*^s, the curvo thoT' that treating (f^' 1?' P" (1 ' '' ) " = . a "d is besides sue -" ^ & of this, f , . , f coordmates (. s , ; , ^ ,- f. place of (f, ,), Possible rationaUy and hooge neously i n teni 384] ON THE TBANSFOKSfATION OF PLANE CURVES. 3 of (f. % D connected by an equation of the form ( )* ( ^ = 0. Such an aquation, treating therein (, 1;, f) as coordinates, belongs to a curve of the order 2/t, with a /i-tuplc point at (f = 0, =0), a /*-tupIe point at (77 = 0, = 0), and which has besides (M-2) 3 or (/*- !)(/*- 8) dps, according as D = 2/i-3, or 2/*-2. The coordinates (x-.y.s) of a point of the given curve are expressible rationally in terms of the coordinates ($''>}' ) of a point on the now curve ; and we may say that the original curve is by means of the equations which give (to : y \ z} in terms of ( ; 7; : ) transformed into the new curve. 11. A curve of the order 2/a may have ^(2/4-1) (2/t- 2), 2^ s -3/t+"l dps; hence in tlie new curve, observing that tho /t- tuple points each count for Kffft we have In the caso jD = 2/i ~ 8, Deficiency = 2/t fl In the case D = 2/j, ~ 2, Deficiency = 2/i 3 8/t + 1 -/*"+ 9,i 9 n *""* fffM ' ' >Wj " J^/ Moreover for 7J = 0, tho transformed curve is a conic, with dps, and therefore wibh deficiency =0; in' the case J)~l, it is a quartic with 2 dps, and therefore deficiency = 2; in the caso D = 2 it is a quintic with a triple point =3, and a double point = 1, together 4 dps, and therefore deficiency = 2. Hence in every case the new curve has the wamo deficiency as the original curve. 12. The theorem thus is that tho given curvo of the order n, with deficiency D, may bo rationally transformed into a curvo of an order depending only on the deficiency, and having the same deficiency witli the given curve, viz.: D = 0, the new curve is of tho order 2(=D + 2); = 1, it is of the order 4 (=. + 8); JD2, ifc is of the order 5 (= + 3); and D>2, it is for D odd, of the order D + 3; even, of the order .0 + 2. It will presently appear that these are not the lo which it is possible to give to tho order of the now curve, Riemanu' r ON THE TRANSFORMATION OJT PLANE CUIWES. - of ( " i * ): H is con to iew - and [384 1-1 the s toV* with deflcie 7 A which is by the '^z^^^Htf eacl i i / -r, 1 w> 7 ?> 6; ~u. ine transformed <leficiency D ^ ^ ^^ ^^ equations f f-P - n . /? f i ^ ' J * -, t0 each of fch i ^ / ' " 5 ' * : ^ theie C01 'responds a ainde set of of tlie nf /fc . M ' IT. If, however, the curves Po = ff-n - curve is equal to this number. I assume that we have k < n : P though t , le same ii(*8)-l pia i"t," ' ' f "* " ler sections common, the equations bein/f fee onms have all their inter the tan P + ^ + Jlo To mX t ***, "T" ' ^ " Manti<lal relati pebble, TO ^ tike the c , v f P -* (P^'o' f T*"? m ' Ve " ' W a as many poiuta ns possiblej and it ""'3^ from T]^ m89 ' .r "" on > ff = groateat possible number i s =*/,% + 31^2 i,f? , , ""J"* , J " St " lade ' that the number of points m the eSvt ^-0 rflTff'' T ^T'' ""^ 8 -- 3, K'^- 3^) -2, respectively. " """" ^ to 7 e ; 00 f de S ^e curve ff = with deficiency D, or with i(tf -lh,)- i Ai = ?t 1, we mav asmmn flinf I-K i. r ^^ ' of the order -!, ^ pas f *** ^ '"-fa* Curves P-0 ( -0, J2-0 through the * (m - 3n) - .D + 1 dps, and through 2 + JD - 4 other poiiitB, together j^j^ T^ points of thfl 384] ON THE TRANSFORMATION OF PLANE CUEVES. This being so, each of the three curves will meet the curve U=0 in the dps, counting as ?i a 3?i 2.D + 2 points, in the 2)i + D~4> points, and in D + 2 other points, together ?t a )i points ; whence the order of the transformed curve is D -I- 2, 20. In precisely the same manner, secondly, if k = n~ 2, then we may assi that the transforming curves P = 0, Q = 0, R - 0, of the order n - 2, each pass through the 4( a -3w)-D + l dps, and through n + D-4 other points, together ^(?i a ~?i) 3 points of the curve 7=0; and this being so, each of the three curves will meet the curve U = in the dps counting as n* - 3n W + 2 points, in the n+ -D 4 points, and in 7) + 2 other points, together a - 2??, points ; whence the order of the transformed curve is also in this case = D + 2. 21. I was under the impression that the order of the transformed curve a not bo reduced below D + 2, but it was remarked to me by Dr Clebsch, that in case JD > 2, the order might bo reduced to D + 1. In fact, considering, thirdly, ease & = w-3, we see that the transforming curves P~0, (2=0, .# = of the o , 8 may bo made to pass through the (w 9 - 3) - + 1 dps, and through JD - 8 other points, together (n a - 8w) - 2 points of the curve U = ; and this being so, each of the three curves meets the curve U= 0, in the dps counting as (?t a - 3n) -2.0 + 2 points, in the -8 points, and in > + ! other points, together n* - 3 points ; whence the order of the transformed curve is in this case =/)+!, 22. The general theorem thus is that a curve of the order n with deficiency can be, by a transformation of the order n-I or n-2, transformed into a curv 6 ON THE TRANSFORMATION OF PLANE CUEVttS. [8g the order + 2; and if D>2, then the given curve can bo by a transformation of the order 3 transformed to a curve of the order .Z) + l: tho trans formwl curvo having iu each case the same deficiency D as the original curve. 23. In particular, if jD-1, a curve of the order n with deficiency 1, or with ^ (n s - 3) dps, can be transformed into a cubic curve with tho aamo doluuoiioy, thud is with Odps; or the given curve can be transformed into a cubic, Thin WIBO is discussed by Olebsch in the Memoir "Ueber diqjenigou Curvcn doron G'ooi'diiuiton elliptisehe Functioiien eines Parameters sind," Grelle, t. r.xiv., pp. 210271. And lu< has there given in relation to it a theorem which I establish as follows: 2-1). Using the transformation of the order n 1, if besides tho 2H + .H 4(=: g w ;j) points on the given curve U~0, we consider another point on tins curve, thon wn may, through the (w 3 -3) dps, the 2w-3 points and tho point 0, draw a wrios of curves of the order n-l, viz., if P 0t Q 0t R , are what tho function* P, Q, M, become on substituting therein for (ts, y, z\ the coordinates fa, y 0i *) of tin; givuu point then the equation of any such curve will be aP + bQ + olt^Q, with tho relation aP + bQo + cll between the parameters a, b, c; or (what ia tho same thing) olimimitiii"' o, the equation will be a^R*- 2\R) + b(QR,- Q ^) = 0, which contain* tho Hind?, arbitrary parameter a : b. In the cubic which is the transformation of tho givuu ourvi- we have a point O' corresponding to and if (f a , %] ,) bo the Gi.ordinatoH of this point, then coi-re spoil cling to the series of curves of the order n-l, wu havu a HOHCB of lines through the point 0' of the cubic, viz,, the linos f + * + < with thu relation a D + 6^ + cft = between the parameters; or, wluit is tho samo tliinif, w, have the series o lines <&- ffi) + 6- ^o) = 0, containing tho Hmnn inglo parameter - I By determining this parameter, tho CUVVOB of tho ordor n~l, will vlicMoiich .h * rder ^7 h t dp3 ' th 2?1 - 8 P 01 ^ ^ * 1*** ^ hich touch the pven curve ^=0; and the lines will bo tho tangent* to tho onbu nm 1 ' tane " tS *" ' theorem, viz.: p0m; mt 18 ' ATO have the following * cency V 1 wo curve, four curves of the order Ul - vifTl ^ draw S M to touch tho where the ratio ft : 6 is determined bv .' -T-^ 9 ^ by ftn equation ~, ttmi ULUVBS oi uiue order n i' viV tl vuu\,n unu the ratio a : 6 is determined by a certai^ 9 ^7" 7 ^ eqUati n ^ + ^'"0, ( wn the absolute invariant J*-.J O f tho '1,^ f 8 4 ( ! ua1a ? 1 (*!* & ) 4l=0 i then 01 the quartic funchon, is independent of the 384] ON THE TRANSFORMATION OF PLANE CURVES. 7 positions of the "2 2 points on the curve U 0, and it is consequently a function of only the coefficients of the curve U = 0, being, as is obvious, an absolute invariant of the curve U= 0. 26. And, moreover, if the curve U= is by a transformation of the order n l, by means of 2 3 . points on the curve as above, transformed into a cubic, then the absolute invariant I s -s- J- of the quartic equation which determines the tangents to the cubic from any point 0' on the cubic (or, what is the same thing, the absolute invariant S 3 -=- 2 lJ of the cubic, taken with a proper numerical multiplier) is independent of the positions of the 2 3 points on the curve U=(), being in fact equal to the above-mentioned absolute invariant of the curve U=Q. The like results apply to the transformation of the order 11 ~ 2. 27. Suppose now that we have D>2, and consider a curve of the order n with the deficiency D, that is with (n a 3n) D + 1 dps, transformed by a transformation of the order n 3 into a curve of the order D-|-l with deficiency D', then, assuming the truth of the subsidiary theorem to be presently mentioned, it may be shown by very similar reasoning to that above employed, that the absolute invariants of the transformed curvo of the order JD + 1 (the number of which is =47) G), will be independent of the positions of the D 3 points used in the transformation, and will be equal to absolute invariants^) of the given curve 7 = 0, 28. The .subsidiary theorem is as follows: consider a curve of the order D + l, with deficiency J), that is, with ^J) (D - 1)- D = $(&- 3-D) dps; the number of tangents to the curve from any point ()' on the curve is -(JD + l)D-(7> -3D) - 2, =3 4kb -2, (this assumes however, that the dps are proper dps, not cusps,) the pencil of tangents has 4>D - 5 absolute invariants, and of these all but one, that is, 4>D 6, absolute invariants of the pencil are independent of the position of the point 0' on the curve, and are respectively equal to absolute invariants of the curve. 29. To establish it, I observe that a curve of the order D + 1 with deficiency D, or with |(J9 3 -3Z)) dps, contains ^(D + l)(D + ^)^^(D^-W) 1 ~4D+2 arbitrary constants, and it may therefore bo made to satisfy 4D+ 2 conditions. Now imagine a given pencil of 4-Z* - 2 lines, and let a curve of the form in question be determined so as to pass through the centre of the pencil, and touch each of the 4.D-2 linos; the curve thus satisfies ID - 1 conditions, and its equation will contain 4D + 2 - (4-D - 1), - arbitrary constants. But if we have any particular curve satisfying the 4.D - 1 oonditu then by transforming the whole figure homologously, taking the centre of the pencil as pole and any arbitrary line as axis of homology, ao as to leave the pencil of linos unaltered (analytically if at the centre of the pencil 0, y = Q, then by writing <uu + (3y -\- yz in place of z) the transformed curve still satisfies the 4>D-l conditions, and wo have by the homologous transformation introduced into its equation 3 arbitrary constants, that is, we have obtained the most general curve which satisfies the conditions in question. The absolute invariants of the general curve are independent of the 1 It is right to notice that the absolute invariants spoken of hew, and in what follows, are not in gaucral rational ones. OE P LANE . 3 arbitrary constants introducer] 1 *], i consequently functions of on y the cn.ffi 'T 8 '7 te sfo >*tion ; and they are "'-being eo.it is obvious that Lr^'t 8 ^ f" ^ f * D ~* "- of the pencil of _ 2 ,. at ^ " be respectively equal to absolute invariant, curve of the order D + l i, =J(/+ nTfl ' f, he j* 50 '"' 6 ln * of the general each of the dps, honee i,, the prLt oL *n, , ~ i^n '^ ^ " MdaoHon =1 ' for -40-6i and there are th, 4^-7 a h !,7 ' * ( ^ + !) ( ^ + ^-H*-SI))-8, oql to absolute invariant the pet l^TT* * %' ' ^ f ">"> of the pencil, there a re 4D-6, each o he ', t ^~ 3 abs Ulte """* cm-ve, and consequently indepe, <l e f O f H ?" '" " absolllte "variant of the d,ffe out planes, thail joilli ^ ^ e ^ d ransformed curve as situate in point on the transformed curve we hnvf g " Wl m ' Ve with the oorrespondi lw *): if the t,vo curves : %r;.; ol , s S 7 S , of "-. Arming a scroll" see ,on by the plane of the oriri lm l "! vn ; ' " res P eo "ly, then the eon pleto d of ' Derating En.. J'ti , y l^! ,"? f ^ of the order transformed curve i s made of ^^ &l* sect.on by the plane of the inc.. Conversely, given a sc ^,, of ' *e order ', and of genei ^ tic od . , c o be.ng ln ,,. Olu , ves rf the tic od _ + sMtona ^ e _ of the other; but for t !lo ^ scroll of ,1 ' , "^ ta 8f otion e the one . breaking up as above th <ndar + ' ; ' i> not possible to find UN TIIK ('uKUKHl'nNUKXi'K OK T\Vi) I'nlNTrt ON A (IIMtVK. A.ii .l/Mf/mdM/t't-if ,S'.,,!V/y. tul, 1. {iMli V, II* i<l A|'iil Itl, iNiiii } VII. fiiul lllf '! 111 Mtiriii M ti J*| I h .'I.' t** 6?.*'"!i r hit'.*' im l'|6Utl"l| (I s ', I) 4 '* II ul lll' 'lrl)M' I * 1 at I ', I- thcit t-f 10 OX THE CORRESPONDENCE OF TWO POINTS ON A OUHVK [{J8. r J HirvL' (-) (the equation of the curve @ will of course contain the coordinntGH of J* us |m-aim:ters, for otherwise the position of P would not depend upon tlmt of ./>), I iiml that if the curve has with the given curve k intersections ut the point /*, hon in the system of (P, P) t tlie number of united points is '.vhenc. in particular, if the curve does not pass through tho point 1\ thon "ibvr oi united points is = + ', as in a unicursal curve. * The foregoing theorem is easily proved in tho particular liasu whom th X- - y, -') m,d of the order in u . 'Tf fu oto " t tho ordur iu ' rf the,,, vanish, id.nBc.ny. o, b^* 1P '7 T BU h "^ "'"^ <lu '" y.) = fc ,. 4 ' "* " = , of ft- .8 o the n ooo,,, . have ff=o, an el similarly if' rv / J" 1 *' ' } ' m tho !"'. J- wo have ff' = . The eo Z -n ( !' ^ *? a '' " 10 "'''i^ of tho wnluiatcs of the given point P ^7 ' COH8ld( g therein (,, ,, s ) , t ,, n /, as L ters " = have ft,, a the case above SUHM8 .j"^^ a havmg a /{4l , 1)lo iu|i 1 "f h , but i T ? 0+ii the total "umber of 7,,, 7 ' W ' UOh m ' of un e ;"v;d,tha t upon writing (- , are ~ iv ' " ni vJii- Is, *'. 'i. -i !< *,( < ( s oj I * /. , f MtlVr. / " It, Julcl ()( tmtf I "1 .Mn]si):i'i -. S'i tie- [!. tfi"it "I n rune i-l ill- ini,i m" lli- mid i i n.'H M! llii" IMIV- v, ifh <h !!l" nmd'l J"mf i S'i !hl',' (D (.,i I t/,, /' (Sti .-. d in * w i'i|"i, - 12 ON THE CORRESPONDENCE OF TWO POINTS ON A CURVE. [385 tangents from F to the curve, that is, ct - n - 2. Hence the number of inflexions is (-2)+4J} I =m + n-4 + 2(-2ro+2), =3(w-m), which is right. 8. For the purpose of the next example it is necessary to present the fundamental equation a = a + a'+2/ i ;D under a more general form, The curve may intersect the given curve in a system of points F each p times, a system of points Q 1 each 7 times, &c, in such manner that the points (P, P'), the points (P, Q'), & c ., are pairs ot points corresponding to each other according to distinct laws; and wo shall then have the numbers (a, a, a'), (b, ft 0'), &c, belonging to these pairs respectively j via. (1,1) are points having an (a, a') correspondence, and the number of united points is =a; similarly (P, Q') are points having a (ft ff) correspondence, and the number ot united points is = b ; and so on. The theorem then is 9. Investigation of the number ef double tangents :-Take P', an intersection with the curve of a tangent drawn from P to the curve (or what is the same thing P P' "* "" Cm ' Ve); th Unikd P^ ta hero the "o nts "of c ero the po nts of ontact of the several double tangents of the curve; or if T be the number of double " r r - s^wKvTsv 1 ^ , v;> ana ^T, /y, ) to the points (P, P'), 2(a~a-') + 2T-/3- / 3'= Moreover, n,m the last example the value of .-._, is ,, D , ftnd fchfl 2r-^-^2(n-0)JDj but from above it appears that we have fi-ff- (n - 2)(m _ 3)) whence ON Til 10 milWWI'ONI>l'JN(!K Oh 1 TWO PO1NTM ON A (HWVIO. henre, il' (/i, ;>) he tlm iOmm<!lerinl,i<!M n|' Mm nyHtein of (lOiiinn (4#), Mm number of tlm nonien through ./' in '/ A l l!IU ' n ' these I'"-" 1 with Mm tfiveu <mrvn I iiitnrriiutlJnn ut /', mid (KinmtipmnMy k.--fi, Mnrnover, eaeh ni' 1,1m (umii-H bi^idus nmntH Mm e.urve in (2m -1) pointN, and (uiimnmmnMy a t .; a.' /L (2w, - I), jlnncii Mm rnnniila ^iviiH l'.ho ininilici 1 of nnili'il or, HM tliis may bn ': IUI -I- l'l -I- ill (tyi I'}. llulp Mm HyHlinu !' rnnicM (-l-^) rniidiiim (iljt - 1/) |i(inlr|iiurH (t'Ofw/Hfls wfnmnuiit 1'iinli M|' whir.h, vi'Kimlril us a ptiir nl 1 r.iiiii(!)(lonf> lini'M, nmnln tlm givnn r.nrvn in nuiuniilt'iil, pninUM; thai, SH, UIM ]Hmil,-]tuir in l.n bit niiimidiiniil IIH IL i- hlit! tfivi'ii c.iirvn in HI puiulH; mid Mmm in on thin lumouiit n t'nliu^ () in tlm number nl 1 Mm unil.i-d pniuln ; whounti, Ilimlly, Mm niinibm 1 f Mm H (-I./) (!) in M/WC -i-nm. It- in liardly nrnwary l,u remark Mini. il> IH iwmnimtl Mm nmdiMiniH {$%} \\\\\ ntntUl>mim having im Hjmdiul ri'lal-imi do M II. An H linal example, HiipptiHi- that Mm puinl, 7' uu a ^ivnu curvo of Mm inilnr nt. and t.lm point. (,? nu a K' v "" nill ' Vl ' "' )lln nnll<r m '' ' iavn au t. ') nim'iiHpinHlounn, ami ln|, il, In- ivipiinid hi linil t-lit- fla^t ul' Mn> mirvi' cnvolnpcd by Mm linn i'Q, Take un iirbitravy pitinl O, jnin fUJ, and Ittl. Miw meet llm uni've m in /", then (/', /") iti-t* puintN nu Mm eurvn m, Imvin^ a (Vet, ma') nurn i Hpniitlitin!o; in limt., l.n a ^iven pciHitiinu ul' /' I here (tuiTi'Mpuml ' piwilimiM "!' Q, and t,o eiL^h id' Lhese. ? puKiMuiiH til 1 /*', lhat in, I,M 1'iieh pusitinn ul' /* there mnvNpund waf pnMiMntin nl' /"; and Himiltivly In each piiHiMnn nl' /'' there mm'spttiiit m' pnsiiiunH "I" I". The dill've H in the- nyHleni ul' Mm lines drawn IVmu i-twh of llm ' puMitioiiH nl' Q to Mm pnint U, henen Mm IMIIVC H ilne-i imt JIUHH Mtniit^b 7 1 , mid we have /,:-'(). Uniirm the uiLinht'i' nl 1 tlm united puinls (/', ]"). Mind in, Mitt mnnher !' the liimw ./'^ whicih thrungh tht pninl- f>, in i 7itV -I- ?it', r thin ! the rliiHM til' tlm niii've envnlnped by 1JJ, 1 1. may bit remarked, I hut ii' Urn two enrve.s are tiurvi-H in Mpatie (plann ni- nl' dimhln rnrvatiirc). tlmn the liki? retmoning H||(IWM that Mm numhor if tlr '" ..... '"' whinli nittub a given lino in ja'H-Ht'o, tliitt. in, Mm tirdnr ol' Mm HCIM by Mm Un /'(,| in MHI' + ?'. ON THE LOGARITHMS OK IMA<;|,\.\ | !V 10* O.f tlm ./.< ," [From the PIWQBC 1'P- ^0 H Knit | y ol' (ihu low but wo may ah) fl p ca i f Writing tluis and similarly wo have of course We have _ _ P KJ j' fl rt moreover bo ^ a,. HO that wo Jmvd a^t^^y i( > ON THE LOGARITHMS 01? IMAGINARY QUANTITIES. mid similarly log P = log r' + iff ) and logp=:logp + f Hence log P - log P = log + i (0 -ff- ,/,), so that, by what precedes, log P- log P', if the chord 1"P, coiMuloral H drawn I,, P' to P, cuts the negative part of the axis of . upward.,, fc = l (lg .,. 2lV . if ^ chord out, the negative part of the axis of , downwa,*, iO is = log- 2i7r , a,,<l in every other case it is = IOET * Consider the integral path of the variable , ; in order fco ? , Iimi J *~ P > <' <V it -nay dopcmd on bho * 1 p rt vAvi\j4. uu yivti IjiiO llrtrnrii\ r. & ^ therefore fix the path of the variable ,- and I -r T^ H ^ mlicaliion ' must the right line P'P w -f ' j 7 y tlUt " lff tho l )ath fco bo gw nne j P. Wnte Mw ^ p/> ^ du PI " ilUIVU , ! fflrtss / J (Vllr,M, ... . 'I . -1, ^ w ******>> ** * is a, on g t , le right lino , , P (fclwt is| from uio coordinates whereof are 1 *-o M H *-> y-u, to the pomt We have thus '*"* ^! U the path in each case beW a ririlt ,- fl n , p g UG ^ ab Ve ' The indefinite integral ^^lo,^ and as u pa8ses f rom j t P . ^ / w Io tt ' P" there 18 no diseontinuity in the value of lo gw; tho 386] ON THE LOGARITHMS OF III AGINARY QUANTITIES, 15 It is to be observed that 6 has always a determinate unique value, except in the single case y 0, us negative, where we have indeterminately Q ir, It is further to be remarked that, taking A for the origin of coordinates, we have 6 angle asAP, considered as positive or as negative according as P lies above or below the axis of is. Starting from the equation. P = re ie , we have similarly P'-rV ', and P ' u ~ ft 1 ? P' r ' p where </> is derived from - in the same way as Q from P t or & from P'. Consequently e i<0-0'-<) l f and therefore 6' <f> a multiple of 2??-, say 0-6'-$ = 2flwr, and in this equation the value of in is determined by the limiting conditions above imposed on the values of 6, &, <$. To see how this is, suppose in the first instance that the finite lino or chord P'P, considered as drawn from P' to P, cuts the negative part of the axis of as upwards ; P is then above, P' below, the axis of as ; that is, 6, 0' are each positive ; and drawing the figure, it at once appears that the sum & + ( Q'}> that is Q 6', is a positive quantity greater than IT. And in this case the angle </> will be equal to 2?r (Q &') taken negatively, that is, $ ~ - {2?r (0 0')}, or ~ & iJ3 = 2vr. Bub, in like manner, if P'P out the negative part of the axis of so downwards, P will be below, P' above, the axis of CD ; 6 and &' are here each positive, and the figure shows that the sum + 6' is greater than ?r; and in this case the angle $ is = 2?r ( + 6'} ; that is, we have & & <f> = - 27r. In m other case, (that is, if the ' chord P'P cither does not meet the axis of .-u <w meets the positive part of the axis of to,) 6 Q' and </> are each in absolut' less than IT, and we have #' (= 0, So that w '' ' - 1? - 16 ON THE LOGARITHMS OP IMAGINARY QUANTITIES. and similarly , lg P' = log r' -f- i&\ and ll> 15J llf->-M<f. J^ o A' 7^ Hence .0 that, by what p.eede, logP-^P', if the choi , d * to P, cuts the negative pa,t of the axis of Hpwards , is .^ if chord cut. the ^ w part of the ax , rf m m every other case it is = loo- L & p ' or from telow to above ti o f PM P C m & m ab Ve * Wow the logarithm chan g e S from +1 to ' "1 o f ". rf ^ " le imf P art '"' Consider the integral path of the variable ,; iu order to fL , ' ^ ^ '* may de P end on tlu! ' ta Wrfh now ^ p ,, ( _ we lmye _ -p; a,d it is easy to see tllat , the path of the righfc Une x fo P coorfinatea whereof are -l, v . 0> to the point |) . We have thus -'J-* Ji ' the path in each case being a right line as abov e. The and aa * passes from 1 to ~ 386] ON THE LOGARITHMS OF IMAGINARY QUANTITIES. W value of the right-hand side is thus =log~. As regards the left-hand side, the indefinite integral is in like manner = \ogz; hut here, if the chord P'P cuts the negative part of the axis of to, there is a discontinuity in the value of log 2, viz., if the chord P'P, considered as drawn from P' to P, cuts the negative part of the axis of as upwards, there is an abrupt change in the value of log z from - iir to + iV ; and similarly, if the chord cut the negative part of the axis of at downwards, there is an abrupt change from + iir to ITT; in the former case, by taking the definite integral to be log P- log P', we take its value too large by 2zV, in the latter case we take ib too small by 2wr; that is, the true value of the definite integral is in the former ctiso = logP - logP'-2wr, in the latter case it is = log P - log P' + 2m, But if the chord PP' does not cut the negative part of the axis of ?, then there is not any discontinuity, and the true value of the definite integral is log P - log P', We. have tints in the three oases respectively P jp which agrees wilih the previous results. It may bo remarked, that it is merely in consequence of the particular definition adopted that there is in the value of log P a discontinuity at the passage over the negative part of the axis of to ; with a different definition, of the logarithm, there would bo a discontinuity at the passage over some other line from the origin ; but a discontinuity somewhere there must be. For if, as above, the chord P'P meet ^the negative part of the axis of IB, then forming a closed quadrilateral by joining by right p p MUCH the points 1 to P, P to P, P' to p r and -p to 1 ; the only side meeting the fdz negative part of the axis of a> is the side P'P ; the integral J , taken through the closed circuit in question, or say the integral +v\ dz h has, by what precedes, a value in consequence of the discontinuity in passing from P' 'to P; viz., this is =-2t7T or == 2wr, according as the chord P'P, considered as drawn from P' to P, cuts the negative part of the axis of at upwards or downwards ; but) this value - 2wr or + 2wr must be altogether independent of the definition of the logarithm ; whereas if, by any alteration in the definition, the discontinuity could bo avoided, the value of the integral, instead of being as above, would be = 0. The foregoing value - 2wr or +2tV is in fact that of the integral taken along (in the one 0. VI. 3 18 ON THE LOGARITHMS OF IMAGINARY QUANTITIES. [386 or the other direction) any closed curve surrounding the point 2 = for which the function - under the integral sign becomes infinite : but in obtaining the value as z above, no use is made of the principles relating to the integration of functions which thus become infinite. The equation log P = log r + id gives PIU ~ gm log P J.TO gi'iiiO or say (ea 4- iy} = r m e !me , where, m being any real quantity whatever, r m denotes the positive real value of r m , We have thus a definition of the value of (0 + fy)* and the value so defined may be called the selected value. And similarly, for an imaginary exponent m^p+qi, we have r p which is the selected value of ( It may be remarked, in illustration of the advantage (or rather the necessity) of having a selected value, that in an integral fa, taken between given limits along a given path it is necessary that we know, for the real or imaginary value of z correapondmg to each point of the .path, the value of the function Z, and consequently! Z is a function involving log, or.*, the indeterminate,! ess which present itsol m these symbols (considered as belonging to a single value of ,) i S) P SO to indefinitely multiplied, and fa is really an unmeaning combination of symbols, by selecting as above or otherwise, a unique value of log* or & we rondo,, function to be integrated a determinate function of the variable. ' 387. NOTICES OF COMMUNICATIONS TO THE LONDON MATHE- MATICAL SOCIETY. [From the Proceedings of the London Mathematical Society, vol. n. (1866 1869), pp. 67, 2526, 29, 6163, 103104, 123125.] December 13, 1S66. pp. 67. PROF, CAYLEY exhibited arid explained some geometrical drawings, Thinking that the information might be . convenient for persons wishing to make similar drawings, he noticed that the paper used was a tinted drawing paper, made in continuous lengths up to 24 yards, and of the breadth of about 56 inches 0); the half-breadth being therefore sufficient for ordinary figures, and the paper being of a good quality and taking colour very readily. Among the drawings was one of the conies through four points forming a convex quadrangle. The plane is here divided into regions by the lines joining each of the -six pairs of points, and by the two parabolas through the four points; and the regions being distinguished by different colours, the general form of the conies of the system is very clearly seen. (Prof. Cayley remarked that it would be interesting to make the figures of other systems of conies satisfying four conditions; and iii particular for the remaining elementary systems of conies, where the conies pass through a number 3, 2, 1 or of points and tor ' n "* "" A " f K """- -0 XOTICES OF COMMUNICATIONS TO THE [387 algebraical sum of the distances of a point thereof from three given foci is (this V,;H ^lecU-fl for facility of construction, by the intersections of circles and confocal <'uni<-s). The ijiuirtic consists of two equal and symmetrically situated pear-shaped curves, exterior to each other, and including the one of them two of the three given f'R:i, the other of them the third given focus, and a fourth focus lying in a circle with the given foci: by inversion in regard to a circle having its centre at a focus the two pear-shaped curves became respectively the exterior and the interior ovals of :i Cartesian. There was also a figure of the two circular cubics, having for foci four iriveu points on a circle; and a figure (coloured in regions) in preparation for the '"iistniction of the analogous sextic curve derived from four given points not in a circle, March 28, 1867. pp. 2526, _ Professor Cayley mentioned a theorem included in Prof. Sylvester's theory of Imvation of the points of a cubic curve. Writing down the series of numbers 1 2, 4, o 7, 8, 10 11, 13, 14, 16, 17, &c,, viz., all the numbers not divisible by 3 then (repetitions of the same number being permissible) taking any two numbers of he H*IC.,, we have m the series a third number, which is the sum or else the ifiom.ce of the two number (for example, 2, 2 give their sum 4, but 2, 7 give their ftore.ce o) and we have thus a series of triads, in each of wnich one number elLruct I ,e!t tf loirT ***** *' **"* * ****** m & CuWc rhCH ^ ^l?^ 1 ?^^ ^^l^" ' . H >. the third point g of f anf 7 th tc'ond nf ?? **!, *""' ^ fil ' 8t f them ' Wo have here L theo rem J't! t W o'l ^ *"", " " "f ****** of 4. "f i Similarly, 10, 11, 13 ,, each J Ten /I L m t - 7 ,", a ' S It. 10, 17, 19, each of them by t l u , e e nslSnl } i ". tamaaA ^ two tractions increasing by nnity L , ZB of ' 1 n; &e numhor of that the.0 constactions, 2, I ^^ SLJ?* ""^ ^ ^ theorem is - o point. Prof. Cayley mentioneTthat on , 7' b ' 5 al " OM md *e p. 29. O to on a straight line. ' ^ f * heae P mte >=eing flxed , whi|e 387] LONDON MATHEMATICAL SOCIETY. 21 March 26, 1868. pp. 61 G3. Prof. Cayley made some remarks on a mode of generation of a sibi-reciprocal surface, that is, a surface the reciprocal of which is of the same order and has the same singularities as the original surface. If a surface "be considered as the envelope of a plane varying according to given conditions, this is a mode of generation which is essentially not sibi-reciprocal ; the reciprocal surface is given as the locus of a point varying according to the reciprocal conditions. But if a surface be considered as the envelope of a qitadrio surface varying according to given conditions, then the reciprocal surface is given as the envelope of a quadric surface varying according to the reciprocal conditions; and if the conditions be sibi-reciprocal, it follows that the surface is a sibi-reciprocal surface. For instance, considering the surface which is the envelope of a quadric surface touching each of 8 given lines; the reciprocal surface is here the envelope of a quadric surface touching each of 8 given lines; that is, the surface is sibi-reciprocal. So again, when a quadric surface is subjected to the condition that 4 given points shall be in regard thereto a conjugate system, this is equivalent to the condition that 4 given planes shall be in regard thereto a conjugate system or the condition is sibi- reciprocal; analytically the quadric surface oaf + by* + cz* + dw- = is a quadric surface subjected to a sibi-reciprocal system of six conditions. Impose on the quadric surface two more sibi-reciprocal conditions, for instance, that it shall pass through a given point and touch a given plane, the envelope of the quadric will be a sibi-reciprocal surface. It was noticed that in this case the envelope was a surface of the order (= class) 12, and having (besides other singularities) the singularities of a conical point with a tangent cone of the class 3, and of a curve of plane contact of the order 3. In the foregoing instances the number of conditions imposed upon the quadric surface is 8; but it may be 7, or even a smaller number,. An instance was given of the case of 7 . conditions, via., the quadric surface is taken to be o# a -}- 6?/ a + c# 3 _+ dw* = (G conditions) with a relation of the form Abo + Boa, + Gab + ffacl + Gbd -I- Red = between the coefficients (1 condition) ; this last condition is at once seen to be sibi- reciprocal; and the envelope is consequently a sibi-reciprocal surface viz., it is a surface of the order (= class) 4, with 16 conical points and 16 conies of plane contact. It is the surface called by Prof. Cayley the " tetrahedroid," (see his paper "Sur la surface des ondes," IAQU/V. torn. xi. (1846), pp. 291 296 [47]), being in fact a homo- graphic transformation of Fresnel's Wave Surface. [Prof. Cayley adds an observation which has since occurred to him. If the quadric surface am? -I- faf + ex* + dm* = 0, be subjected to touch a given line, this imposes on the coefficients a, I, G, d, a relation of the above form, viz., the relation is A z lo + B*ca, + &al) + Pad + G*bd + H*od = ; where A, B, G, F, G, H are the "six coordinates" of the given line, and satisfy therefore the relation AJF + BG+ <7I/= 0, It is easy to see that there are 8 lines for which the squared coordinates have the same values A*, B*, 0\ F\ G 2 , f/ a ; these 8 lines are symmetrically situate in regard to the tetrahedron of coordinates, and 22 NOTKWS Ol' 1 rOMMinVK'ATIONH TO TIM; I moreover they lio in a hyporbolmd. Tim i|uiulrin mirliuv, instnid ..!' Imin^ ilrliinul above, may, it in cloav, bo dnfinod by thn i!i|iiiviilniil. u lil.iuu>i n|' fniii'liimf ( . m ')i ,,f , i" 8 given linos: that is, wo havo Um oiivrlnpi> nf a ipmdrin Miirliuv hnu-liiiiir ,,(,[, ii( 8 given lines; thosu HHUH nob bring urliit.niry MUCH, bill liciii^ a K.Vdfrm il' u \vi'v liiM-ri'il form. By what procedoH, tho nnvnL.pn i>t n .jiiai'Uc Hiiriinv. li. 'II|I|H-IU;I, linwovcr I'l'^r in virtue of tho rohition AF + JH}* (!Jf ?>(), Uim \ i,u tnu^.r M piupn-'^uuili,' nm-fi,!!^ but that it rosolvon itsoU' into MK- abdvc-iiioiil.i d liyp.'-rbi.lni.l mKru i.win- ' TrY is, i-catoi'ing tho oi-iginal A, it, &,!., in p| llt v n f /| '/.' J . ,<T M ll ivrln|.. ' !' i')", Xic -I- Jka + (Jab + 2i\id -\-Qbtl -\- Jfal - 0, (which in in ijPi'irnil u h'tiahrdmitl), in wi,".*" ^1, A (7, /'', ff, // am tin) Hi]imnl (!oonliiml,oH uf u Im,- (nr, wltiif i^ il,r -^i,,,, u^)"." passing through tho givon lino and tlirnngh the HyimMulnVMlly''^!., 'u,' L'-v'n nllMrii!',!!' J-' 1 la, IHli.S. pp. ion, nil, Frofiw Onyloy g,tvo an uooouni, h, i| lr M-.-tin,/ ul' ,i M,,,,,,,. liv ,.. ((i , H(( ,. Comus rmnnhohor ( on,ploxo ndo. Vonill^nH-MimiMK ,|,., Ktilrr',,!,,,!, H,,.,,J v u % ^ ....... -''-" ) - number of spaces, un,l .orUiu Huppl.n.nnlnry mnu.riH,, A' " J 1 . H tho bottom bo also We,,,, away, ,,,, H i . t ' ' ^ ' , 7 ,"T"i " '' "" '' ........ " supplomonhuy (1 nni.lilii!H ,., i,, ,,, ,,,,, ,T, .',, ' .' "'" , ..... ' ..... "' " f "' The ohiof diniculty ,! i ..... ,, r ,,, ''., "'.' """ lv "' ' -*'l---(./..-*")-.. 8l Wlomo,,tavy ...Jatta ^ J ,?, ..... '' '"" '" ' ........ "" ...... ""' ...... '' ...... December 10. 1808, ini l' 1P- l OlMorratioiw by I'rofoas , - I 1. 2, 3, au,l any ,, tl lr(!u i jT '," , '"' ""* >' liui " ..... > io C T OS ' caoh ^ * Hi \ ;. T , ,; v ;r"' r-- *- *. o, 8, wo havu a tmnfo, vt io n ,, . ,, . i " y ' ,:' ''' " "' ,! hl ' "'"'I' 1 " ! system " '' ' *"-A, .- 1, ; ^, l,, u ,| illK U| u t , lll|V( , nM , Si o > points 4, s, 0, rat i n >0 8implo ^ r "' 2 , "f," ' ^'" Ol " ro 'l 1 ""' 1 ^ having 387] LONDON MATHEMATICAL SOCIETY. 23 Analytically, Cremona's transformation is obtained by assuming the reciprocals of "ai 2/a 2* fa ke proportional to linear functions of the reciprocals of as lt y lt z l (of course, this being so, the reciprocals of *' lp ft, ?i will be proportional to linear functions of tho reciprocals of flj a , 7/ 3) z.j). Solving this under the theory as above explained, write M a , /> , c \ i i_ r T a' 3 u6i i/i Z\ 1 } = . L J, /_ . = 1 : + + i if Honco n Q-JB^O, &c., arc qnavtioH, or generally aQ.12, + ^R l P l + -yP, Qt = is a quartic, having throo double points (y^O, ^ = 0), (^ = 0, ^ = 0), (^ = 0, ^ = 0), and having beaides the throo points which are tho remaining points of intersection of the comes (&-0, ^-0), ( Ji, = 0, P, = 0), (Pj = 0, Q! = 0) respectively ; viz., these last are the points "1 'I 'I o :L . . : - = ei hf : fy - id : dh (/e> &c. &o. Tho double and simple points are fixed points (that is, independent of a, /3, 7). and the fonnulm coino under Cremona 1 * theory. It is, however, necessary to show that if the point* < 5', 0' are in a line, the points 1', 2', 3' are also in a line. This may be rtono as follows : Lot thoro bo throo pianos A, B, 0, and let the points of the first two correspond hv ordinary triangular inversion in respect of the triangle , on the_plane A, and _/3, oil the piano S. Lot also the pianos S, C correspond by ordinary triangular inversion in respoot of the triangle ft on tho plane B, and 7 , on the plane lh " "^ Hpondonoo between A and is tho one considered, the points 188 to ming the r,,.^',:-;".- r ^-^ r^-j-s, -i~ A^nlnrmim nrnnP.vtioa musl) apparently belong to uiemonuB uw interesting part of the theory. i's "TJeber- t ip" Grelle, torn. but in the present Paper wwnwty proportional to 388. NOTE ON THE COMPOSITION OF INFINITESIMAL ROTATIONS. [From the Q mrtef ly J omd of Pw . e and ^ vm . pp. 710,] n the last Smith , -^ COS ion to Denote M infinitesinml *X rigidly ^ in the point 388] NOTE ON THE COMPOSITION OP INFINITESIMAL ROTATIONS. 25 where I, m, n, p, q> r are constants depending on the infinitesimal motion of the solid body. Hence, first, for a system of rotations &>! about the line (a lt b lt c,, f lt </,, h^, 0)3 ,, (tt a , y 3 , C a , J 2l (/ 2 , fly), the displacements of the point (#, y, z}, are Bo) = . ?/Sca> s2ow 4- Sy 2cw , + zZaw 4- and when the rotations are in equilibrium, the displacements (80, Sy, 8,2) of any point (a), ?/, z) whatever must each of them vanish ; that is, we must have 2wa = 0, S<u& = 0, SMC = 0, Sa>/ = 0, "Za>(f = 0, Sw/i 0, which are therefore the conditions for the equilibrium of the rotations w lf a) 2 , &c. Secondly, for a system of forces P l along the line (a,, &,, c,, / 1( (/j, /M, * 3 )i Jl V*3) ^3) C a , _/ a , (/ 2 , ft fl ), &c. the condition of equilibrium as given by the principle of virtual velocities is SP (al + Imi + on+fy + gq + kr) == ; or, what is the same thing, we must have which are therefore the conditions for the equilibrium of the forces P u P 3 , &c. Comparing the two results we see that the conditions for the equilibrium of the rotations WD w a , &c. are the same as those for the equilibrium of the forces P l} P a , &e. ; and since, for rotations and forces respectively, we pass at once from the theory of equilibrium to that of composition; the rules of composition are the same in each case. Demonstration of Lemma, 1, Assuming for a moment that the axis of rotation passes through the origin, then for the point P, coordinates (as, y, z], the square of the perpendicular distance from the axis is = ( , y cos -y + z cos /3) 9 + ( x cos 7 . z cos ) a + ( to cos /S + y cos a . ) a , C. VI. 4 2ti .VOTE OX THE C03IPOSITION OF INMNIT.ESLVIA1, IIO'I'A'J'IONH. aid tho expressions which enter into this formula donoto (is fnllmvH ; via if l.lih Hll r] ^ l-mt * at right angles to the plane through P ftnd t h axm nf mlul,i, m Mil ""f n* : Iie rt Ular diStanCG f ' < Mirmtw oi Q refonwl to P as origin are a cos 7 , _ g cos ffj - A'cos/9+7/cosa oach ' .V-*, *-) in place of ( an H coordinate., (, J, c , f . n : '' ^' allt " 4l < cto . of l emma 2. gts ~- ,"' "'" '''"' ........ ' ""'i'-' to t ], ..inilind.],,,^ ( ,,| u , n,l, ...... . l,v , I G f M . Motion of the 389. ON A LOCUS DERIVED FROM TWO CONIOS. [Prom tho Quarterly Journal of Pure and Applied Mathematics, vol. vm. (1867), pp. 778*.] REQUIRED tho locus of a point which is such that the pencil formed by the tangents through it to two given conies has a given auharmonic ratio, Suppose, for a moment, that the equation of tho tangents to the first conic is (HJ ay) (as by) = 0, and that of tho tangents to the second conic is (co cy) (at dy) 0, and write G (a- so that write also /,--* /- - **~ A* lv *~A' then tho anharmonic ratio of the pencil will have a given vali (&-/<;,)(& -70 = 0; that is, if or, what is the same thing, if that is, if - (B-0}* =0, ^ 8 ON A LOCUS DERIVED PROM TWO CONICS. [389 where are each of them symmetrical in regard to a, b, and in regard to G, d, respectively. Let the equations of the two conies be U =(a, b, c,f, (J , A>, y, 2) a = 0, V (*',&', ef i f,g',kJa ll y l z^ = Q i and let fa, ft 7 ) be the coordinates of the variable point. Putting as usual (A, B> G, F, G, tf) = (i c -/ 2 , ca-f, i-/ t , g} t -af, kf~bg t fg-ch), K = abo - af* - hf - c /i the equation of the tangents to the first conic is (A,B, 0, F, G, where X^W-fy, r^ and therefore 5l ,b sti t u ti, lg for 5 the vall>e _i (tf tangents, an equation of the form aZ + 2hZ7+h7*-n w l,i n i i ^ , to b e (Z-7)(X~ & 7) = 0; that is, we ht e ^ " efa ' ^ taken 1 : ft+& : 06= a ; -2h : b; an<l, in like , Mme , if the accented letters refer to the second code 1 : c -f d : cd = a' : - 2h' : b'. Substituting for a, h, b their values, and for a' h' b' th . ana 101 a, h, b the corresponding values, wo find ' ' a + b : ab i c 0/aa We then have ...Ja, A 7 ), 389] ON A LOCUS DERIVED PROM TWO CON1GS. 29 and similarly ( -d)' = -4W. ..., /3, 7)'- We have, moreover, (a + 6) (c + d) - 2 (06 + cd) = 4 (//7 3 - ^7 - Gf3>y - 2 (7V - 2 # and substituting the foregoing values, we find or putting for shortness = (EG 1 + '0 - 2J!f, ., ., Gil 1 + (?'// - ^J" - ^'^, ., . Ja, ^3, 7 ) a . the equation of the locus is where (a, /3, 7) arc current coordinates. The locus ia thus a quartio curve havi quadruple contact with each of the conies Z7=0, U' Q; viz. it touches them at th points of intersection with the conic = 0, which is the locus of the point such ti the. four tangents form a harmonic pencil. The equation may bo written somewhat more elegantly under the form 30 ON A LOCUS DERIVED PROM TWO (.IONICS, so that A - 2k ,.F=* mr (BO -^ ,..) = -(w)' -jiff, H/J -/-, and substituting these values the oqimtion in which, if A, B, denote a, p, 7 . /?, 7 i fl, j0, fy a > ft> 7 I , m, 11 f , m', n' i , W, ?l Xi tf'i '' P, 9. r p'. g', ?' f, wi', n' ^. 2, ' }' J , i P> 7 a a /3 > 7 a_{ i /3, 7 a, ft 7 _. i, m, ?i /' / / f , m, n ^ fli, w J?>> f/. '' P. ff ' X. ?', *' r, W', ft' ^'. '/, '' , 7 respectively, (A + B + 0=0} is, in fact, tho (2fi + l)M 9 or, what is the same thing, that i s ', m', *r' , 7 jugate axes, the we have K^abo, K'^ O. ON A LOCUS DERIVED FROM TWO CONICS. I suppose in particular that the two conies are a? 4- inif 1 = 0, fcho equation of the quurtic is 1 ma? + - 1 _ a m ? - m - 1 = ,, -N ,, . . or putting A, = jf/o/nrv5 tnis 1S njiS _L 1 To fix the ideas, suppose that m is positive and > 1, so that each of the conies in nu (illipso, tho major semi-axis being =1, and the minor semi-axis being = ,, -; . 3?or any roul valuo of Js the coefficient X. is positive, and it may accordingly be assumed thai) X in poHitivo, Wo have ..^..."t.. 1 ,-> < 1, or the radius of the circle is intermediate between m (m -h I) in liho aomi-axoH of tho ellipses, hence the points of contact on each ellipse are real points. Writing for shortness ^ - > ??i a + m tho equation in (a? + iwf - 1) (ma? + f - 1) - X (& + if- a) 9 = 0. For thu points on the axis of eo t we have (f-l)(flMJ a -l)-M< 1 -a) B = 0, that i ( m - \) & + {- (1 + m) + 2X] a; 3 + (1 - Xa fl ) == 0, mid bhoucu or, substituting for a its value, this ia Eemarking tlmt tho values are 32 ON A LOCUS DERIVED FROM TWO CONICS. [389 1 (m + l) a and considering successive values of X; first the value \ = ,, = - ~, wo have + ) m/ m 1 m or observing that this is ^ = 0, or \ w m The next critical value is X-m. The curve here is (a? + mf - 1) (met? + f-i\- m (& + ,* - a )a = Q that is ' ' m (tf + # 4 ) + (l + m s ) a,' 2 ?/ 3 - (m + 1) (a 3 + f) + 1 that is v ,/ y * f m - I) 3 .ty + ( 2)w - m - 1) (^ + /) + 1 _ , ?ia a = o, or, substituting for a its value, _ > l-mtfl--&??li a8 _( m -l)'(m rt ,. . m( a the equation is . m(m+iyt-' or, as this may also be written, 1 m H in or, what is the same thing, in 389' ON A LOCUS DERIVED FROM TWO OONICS. 33 which has a pair of imaginary asymptotes parallel to the axis of so, and a like pair parallel to the axis of y, or what is the same thing, the curve has two isolated points at infinity, ono on each axis, linos TKo next critical value in X = J(m-t-l) 3 ; the curve hero reduces itself to the four , (m ' m 0; and it in to be observed that when X exceeds this value, or say \ >(* + 1) 8 , the cui'vo has no von! point on either axis; but when X=oa, the curve reduces itself to (? -I- y 3 -)"<), i.e. bo liho eirclo ro 3 + f - a. = twice repeated, having in this special ciwo real pointw OH the two axes, Ik i iww eauy to trace the curve for the different values of X, The curve lies in ovory eano within the unshaded regions of the figure (except in the limiting coses aftor-muutiomsd); and it also touches the two ellipses and the four linos at the eight points /i), at which points it also cuts the circle ; but it does not cut or touch the four HnoH, tho two ellipses, or tho circle, except at the points k Considering X as varying by HUCCUBnive stops from to oo; A, = 0, tho curve is tho two ellipses, \, ^^ M.!?JljA Vj the curve consists of two ovals, an exterior sinuous oval lying in the \ W four vegionH a and tho four regions 6 ; and an interior oval lying in the region e. C. VI. 5 q * ON A LOCUS DERIVED FROM TWO CONICS. Tggc, 1) 3 - there ,s still a s i oua oval M above> aove> ^ .^^ \ w . dwindled to a conjugate point at the centre. ; X "" ; TO ; there is no interior oval, but only a J fc , the onive becomes the four lines. k>i(m-+l) 2 , the curve lies wholly in tho f,,. eonasting thereof of f olll . detaehed siuL ov a l s A TT - ^ ^ *" ^ !( + !). e^h oval approaches mm nea r' tnf ' ^ Y , lateS ' eSS from the and inflate l ine . porti l w hieh temd " ' "^ W * ini1 f ed ^ the eide And a X depart/ f fom the 1 tit T + , Z"" * V WhMl th OTal be approaches more nearly to the circuil^ h PP '' 0a heS * "' each si cont ^ 6eparates the 4 e crcu h contains the dnnon. oval. ^ 6eparates the 4 ^giona 4 e , w hich Finally, x = 0, the curve is t ], e ci role Wee repeated. 390. THEOREM DELATING TO THE FOUR CONICS WHICH TOUCH THE SAME TWO LINES AND PASS THROUGH THE SAME FOUll POINTS. [From the Quarterly Journal of Pure and Applied Mathematics, vol. vm. (1867), pp. 162167.] THE aides of the triangle formed by the given points moot; one of the given linos in tliroo points, nay P t Q, 11; and on this same line we have four points of contact, Bay A lt A tl A a , Ai\ auy two paii^i say A lt A t \ A B , A 4t form with a properly selected pair, sn,y Q, R, out of the above-mentioned three points, an involution j and we have thua the three involutions (A lt A; A aj Ai\ Q, .fi), (A,, A s \ A it A,- } H, P), (A lt A,; A 2> AS\ P, Q). To prove thia, lot ffi = 0, i/^O ^ G tno equations of the given lines, and bake for the equations of the sides of the triangle formed Toy tho given points b & + a y ~ab =0, If oi + ' y - a' &' = 0, the equation of any one of the four conies may bo written Lab LW . L"of'b" \j - ab b'os + a'y - a'V b"ai + a!'y - and if this touches the axis of ic, say at the point co= a, then we must have _^L JL' .. V/ = -/f^") a , tv-acK-aiv a" () )(;- a') (a; - a") ' 52 Uti THEOREM RELATING TO THE FOUft CONICS WHICH TOUCH THIS >r, aiming as we may do, #=-('-<)("-) (-'), this gives L a =( ft -op (a 1 -a"), MUW manne, if the conic touch the axis of y, 8il y at the poi.,; ifi =(b -^(fi'-fi", ' and thence p, ming - P = a & (a' - ft ") (y j/^ f =a'b' (a," - a )(b"~l)), wo have ^-V(a-0( ~V), and thenco ; * ; wo have in like mamel . - _ "' ' p , Putting equation is '" ' -^ 390] SAME TWO LINES AND PASS THROUGH THE SAMS TOUH POINTS. 37 and "by attributing the signs + and - to the radicals, we have, corresponding to the four conies, the equations (a - ,) V (X) + (a 1 - ffl ) V (A") + (ft" - a.) V (X"} = 0, - (a - ,) V (X) + (a' - a a ) V (A") + (a" - a 3 ) V (X") = 0, (a - .) V TO - (a' - a,) V (T) + ( a " - ,) V ( JT") = 0, (a - Oi) V (X) + <' - ,) V (*') - (" - <0 ^ (A) = 0, where a n 3( a 3) a., aro the values of a for the four conies respectively, Eliminating a" we obtain the system of three equations (2a - , - a,) V (X) + (a, - ,) V (A") + (a, - ,) V (A r ") 0, foi - ) V (A') + (2a' - , - ,) V (A r/ ) + (a 3 - ,) ^ (.Y") - 0, (a, + a - a, - a.,) ^ (A r ) + (a, -}- a 3 - a, - a,,} V (A r/ ) + (a, + 4 - a, - a 3 ) V (^") = 0, and then oliminating the radicals we have , a - a, + 4 ~ a - ==0, which is in fact 1, . +'j act,' - 0, as may bo verified by actual expansion; the transformation of the determinant is a peculiar one. The foregoing result was originally obtained as follows, viz, writing for a moment a V (X) + (if V (*"') + " V (X") = , tho four equations aro - a, <I> = 0, - <E> = 2 these give (a, 2 (a - B 38 THEOREM RELATING TO THE I'OITR CONICS WHICH TOUCH THE From the last equation we have (a, - ,) fc = 2 (0 - a V (X) - a' V (Z')J - 2a 4 {> - V (X) = 2(i-^)*-2(a- that is or substituting for J(X), *J(X') their values in terms of <]>, we find which way be written that is T or ft Of., that is or finally which