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 JlIN 14 1917 
 
 DIFFERENTIAL INVARIANTS 
 UNDER THE INVERSION GROUP 
 
 BY 
 
 GEORGE WALKER MULLINS 
 
 Submitted in Partial Fulfilment of the Requirements 
 
 FOR the Degree of Doctor of Philosophy in 
 
 the Faculty op Pure Science, 
 
 Columbia University 
 
 PRESS OF 
 
 THE NEW ERA PRINTING COMPANY 
 
 LANCASTER. PA. 
 
 1917 
 
Digitized by the Internet Archive 
 
 in 2008 with funding from 
 
 IVIicrosoft Corporation 
 
 http://www.archive.org/details/differentialinvaOOmullrich 
 
DIFFERENTIAL INVARIANTS 
 UNDER THE INVERSION GROUP 
 
 BY 
 
 GEORGE WALKER MULLINS 
 
 Submitted in Partial Fulfilment of the Requirements 
 
 FOR the Degree of Doctor of Philosophy in 
 
 THE Faculty of Pure Science, 
 
 Columbia University 
 
 PRESS OF 
 
 THE NEW ERA PRfNTINQ COMPANY 
 
 LANCASTER. PA. 
 
 1917 
 
-MS 
 
 7 ■!l?<' • " 
 
DIFFERENTIAL INVARIANTS UNDER THE INVER- 
 SION GROUP. 
 
 Introduction. 
 
 The object of this paper is to study the differential invariants 
 that arise under the continuous transformation group of six 
 parameters, known as the inversion group of the plane, and 
 given by the equations of transformation 
 
 (Q!iYi+Q!272)(a:^+y^) + («i+/3m+j32T2)a;-(ata+ffi72-/327i)y+/9i 
 
 X = 
 F = 
 
 (7i''+72^)(x2+2/2)+2TiX-272y+l 
 (a27i -aiYz) (x^+y^) + (a2+/327i -/3i72)x + (ai— /3i7i -/3272)y+i8s 
 
 (7i^+72')(x2+2/2)+27ia;-272j/+l 
 
 Inversion geometry for the most part has been studied syn- 
 thetically, in connection with the theory of functions of a complex 
 variable satisfying the general linear fractional transformation 
 group 
 
 72 + 
 
 Thus it is known that the families of minimal lines, circles, and 
 logarithmic equiangular double spirals remain invariant under 
 
 Ge.t 
 
 The justification for the study of the group from the stand- 
 point of differential geometry, aside from the classification or 
 codification of the invariants under the group, is the bringing to 
 light of some new properties of the logarithmic double spirals. 
 
 In part one, the Lie theory is employed in finding the differ- 
 ential invariant (I5) of lowest order, while in part two the inter- 
 pretation of this invariant is given by making use of a certain 
 family of bicircular quartic curves. Furthermore the differential 
 equation of the logarithmic equiangular double spirals is found 
 
 *See Holzmiiller: Theorie der Isogonal Verwandenschaften; also Cole, 
 " Linear Functions of a Complex Variable," Annals of Mathematics, Vol. V: 
 No. 4. 
 
 t Throughout this paper we shall refer to the inversion group as Gt. 
 
 3 
 
 360599 
 
4 . '-. ": :/: V'. Difmrential In\^ariants under 
 
 and expressed in terms of 1 5, and from this differential equation, 
 certain properties of the family are deduced. 
 
 Part three is concerned with differential invariants of order six. 
 Here again other properties of the double spiral are found. 
 Finally, part four deals with differential invariants, intrinsically 
 expressed, and with the classification of invariants, of order higher 
 than the sixth. 
 
 I. The Absolute Differential Invariant of Lowest Order. 
 
 1. The six independent infinitesimal transformations gener- 
 ating the inversion group, Ge, may be taken as, 
 
 A solution of the complete system [2] of differential equations 
 formed by equating to zero, each of the above five-fold extended 
 independent infinitessimal transformations of G^ is a differential 
 invariant of lowest order under Ge* Extending the infinites- 
 imal transformations Uif [i = 1, 2, 3, 4, 5, 6], fivefold and 
 equating each to zero we have for 
 
 E 
 
 Bx "' dy ■ 
 * See: Lie-Scheffers, Continuierliche Gruppen. 
 
THE Inversion Group. 5 
 
 "" dx^y dy^^ dy' y dy" ^^ dy'" "^^ di/- ^^ dy^ "' 
 ^2^' " ^'^ ^ " ^"^2/ fy - %(! + 2/") ^ + [2^2/" - 6y2/'2/" 
 -2y'{\+y'')^, + {W''-^yy'''-SyyY'-'^'^y'Y]^, 
 
 + [80:2/- + lOi/^^ - I2yy'y- - SOyy'Y^ - SOy'Y^ 
 
 - myyy" - 20yy"" - dOy"^^ = 0, 
 
 2^2/^ + if - ^') ^ - 2x(l + y'') f-, - [2yy" + Qxy'y" 
 
 n] ^ - [42/2/-+ 6a:2,"^+8a^2/'2/-'+122/'^"] ^ 
 
 /..///21 ^/ 
 
 - [62/2/- + lOxyY- + 182/"2 + 20a:2/"2/'" + 24yy"'] ^^ 
 
 - [182/2/^ + I2xy'y^ + 40z/'2/'^ + mxy"y'^ + SOy"y"' 
 
 jn2, ^f 
 
 -^20xy'"\^^ = 0. 
 
 In order to obtain a solution of the complete system of differential 
 equations 2, in the simplest manner, it will be necessary to find 
 first the differential invariants under the Hauptgruppe.* 
 
 2. Let us make use of Lies' differential parameter theory and 
 assume that ^ is a differential invariant under the Hauptgruppe. 
 By extending each of the infinitesimal transformations Ui{f) 
 [i = 1, 2, 3, 4] three-fold and then with respect to <p and equating 
 each to zero, we have a complete system of linear partial differen- 
 
 * lOein has called the group generated by df/dx, df/dy, — y{df/dx) +x{df/dy), 
 xidf/dx) + yidf/dy), the Hauptgruppe. 
 
6 Differential Invariants under 
 
 tial equations 
 
 1=0. 
 
 + [4yy" + 32,"1 ^, . . . + 2/ V ^ = 0, 
 
 having two independent solutions. Since 
 
 dx dy * 
 the system S reduces to 
 
 5': 
 
 y dy" ^y dy'" 'f d<p'~^' 
 
 and the two independent solutions of S' are 
 
 3yy^' - y"'il + y") 
 
 ym - c, 
 
 and 
 
 (1 + y'^<p' 
 
 77 = ^2. 
 
 y 
 
 These two invariants* expressed in terms of the radius of cur- 
 vature, p, the arc length s, and the derivative of p with respect 
 to 8 are 
 
 dp d<p 
 
 d^ = ''' Pdl = '" 
 
 * It is interesting to note that the Hauptgruppe may be expressed as a one 
 parameter intrinsic transformation group whose infinitesimal transformation 
 is p{df/dp) + 8{dfjds) and the invariants above are readily obtained from this. 
 
THE Inversion Group. 7 
 
 Since the differential invariant of lowest order is known, i. e., 
 dp/ds, it follows at once, by making use of the differential para- 
 meter p{d<pfds), that those of higher successive orders are 
 
 dp ^ 2^ n-3^li 
 
 ds' ^ds" ^ ds^' '"' ^ ds^-^' 
 
 From the preceding discussion we may conclude: 
 
 Theorem I. The most general differential invariant of order n 
 under the Hauptgruppe is of the form 
 
 f 
 
 [dp ^^ ... ,n-3^Pl 
 
 Ids'^ds^' '^ ds^-^]' 
 
 Moreover since the Hauptgruppe is a subgroup of Ge, we have: 
 Corollary I. All differential invariants under Gt must he of 
 the form 
 
 
 3. Let us now return to the system S of section one. Since 
 
 dx dy * 
 2 becomes 
 
 (1 + y'') |i + Sy'y" ^ + Wy'" + Sy"'] j^, 
 
 + [52/Vv+102/'y'1^ 
 
 + [62/y + ir>y"t- + \^y"J'\ ^ = 0, 
 
 y dy" ^ dy'" y dy'^ ^ dy- ^' 
 - 22/(1 + y") ^ + \2xy" - ^yy'y" - 2y'{\ + y")] ^ 
 + [Axy'" - Gyy'" - SyyY" - l2y'Y] ^^ 
 
 dy'" 
 2J': + [Qxt- - lOyy'r- - 20y'Y' + 4y"'] ^ 
 
Differential Invariants under 
 
 + [Sxy^ + lOy'^ - I2yy'y^ - dOyy'Y'' — SOy'Y^ 
 
 - UOy'y'Y' - ^Oyy"" - 30i/"^]^ = 0, 
 
 - 2a:(l + y'') ^ - [2yy" + Qxy'y- + 2(1 + y") ^ 
 - [W + ^xy'" + ^xy'y"' + l^y'y"] ^, 
 
 
 - [18yr + l2xyY + 40y'^^^ + SOa;?/"?/- + SOy'Y 
 
 and S' is in turn reducible to.* 
 
 + 20xy'-']^^=0, 
 
 
 -2/ 
 
 . ^/ 
 
 S'^ 
 
 [4y'"(l + 2/'^ - 122/'2/"»] 
 
 - 2y'"^ 
 
 dy' 
 
 dy'^ 
 
 df df 
 
 + [i02/-(i + y") - Wy"y"' - W"]^. = 0, 
 
 * It is known that the complete system of linear partial differential equa- 
 tions, Ai(f) = 0, Aiif) =0, • • •, Ar(f) = 0, has the same solutions as the 
 system 
 
 Bkf = ftkiAJ + pLkzAsf + • • • fXkrArf [fc = 1, 2, 3 • • • r], 
 
 where the fi'a are functions of the variables provided 
 
 + 0. 
 
 Serret-Scheffers: Lehrbuch der Diff. u. Int. Rech., Vol. III. If aiss = m 
 = ms = ftw = 1, /XM = M«3 = 2z, fiu = 3y', mss = M3« = ws = /^ = M« = p« 
 = 0, MM = — 2y', H63 = 2y, uti = — 2y, S' reduces to S". 
 
 Mil 
 
 M12 • 
 
 • Mir 
 
 M21 
 
 M22 • 
 
 • M2r 
 
 Mrl 
 
 Mr2 • 
 
 • Mrr 
 
THE Inversion Group. 9 
 
 - 2(1 + y") ^, - Uy'y" ^,+ [AyV - 2py'y"' 
 - \2yy" - W']^y- [302/'y- - SOy'Y' 
 
 -402/yy"-302/y'^]^=o. 
 
 The following change in variables in S" is suggested by Corollary 
 
 I, section two. 
 
 Let 
 
 zi = y', 22 = y", 
 
 dp Sy'y'" - y"'{l + y") 
 ^'~ds~ y'" 
 
 dp (Sy"' - 2y'y"') d + y") (v'Y'' - ^y'"') (1 + 1/")' 
 
 ^'-Pds- y'" y 
 
 (Pp Wy"'y''' - Qy"" - y'Y)a + y")' 
 
 //4 
 
 
 ds^ y'" 
 
 {y"Y' + ^y'y"" - ^yy'r'')a + y")' 
 
 + y^n 
 
 {2y'y"' - ^y'y"^{^ + y") 
 
 -t- ym 
 
 In consequence of this change 2" becomes 
 
 2z3^^+(5z4 + 223^)^^=0, 
 
 2z3 ^^ + (3z4 + 223^) ^^ + (4z5 + 6Z3Z4 - 223) —^ = 0, 
 and reducing 2'" to its equivalent involution system, we have 
 
 423'^ + (8Z3Z5 - 1524' - 423^24 - 423* - 523^)^ = 0. 
 
 023 dZr, 
 
 The solution of S»^ is 
 
 42325 — 524^ — 423^24 + 423* — 423^ 
 
 Z3 
 
 C. 
 
10 Differential Invariants under 
 
 If we pass now to the original variables, we have for the solution* 
 of the system S 
 
 As'"''' 
 where 
 
 A3 = ^y'y" - y"'{l + y") 
 and 
 
 A5 = Wr - 52/-=')(l + y"Y 
 
 + iWy'VY^ - Wy"Y-\-2>0y"Y--4Qy'y""){l+yy 
 
 +^Qy"Y'Ky"- 2) {l+yy+ ISOyYY'a - y") (1+2/'') 
 
 -452/"«(l + Qy"-Sy% 
 
 The following results may now be stated: 
 
 Theorem II. The first absolute differential invariant (the one 
 of lowest order) under Ge is of order five and, symbolically expressed 
 is As/As^ = h- 
 
 For symmetry of notation let us write the differential equation 
 of the minimal lines 1 -\- y'^ = as Ai = 0. The differential 
 equation of the circles of the plane is A3 = 0. We may think, 
 of Ai and A3 as the first and second relative differential invariants 
 under Gq. For I5 = 0, A5 must be zero. Hence, 
 
 Corollary I. The third relative differential invariant under Gt 
 w A5 = 0. 
 
 In part two the integral curves of A5 = will be found. 
 
 II. The Interpretation of h and the Logarithmic Equi- 
 angular Double Spirals. 
 
 1. For an interpretation of 1 5 we turn to the family of bicircular 
 quartic curves, given by the equation 
 
 Q: {x'-^fy-{-{ax + by){x'+y') + cx' + 2djcy + ef 
 
 + 2fx + 2gy+h = 0. 
 
 * Of course any function of As/As* is a solution of S. We shall however 
 speak of the simplest form of a solution, as ' the ' solution. Likewise we shall 
 call the simplest form of an invariant of order n ' the fundamental ' invariant 
 of order n. 
 
THE Inversion Group. 11 
 
 It is known that the family Q remains invariant under G^* Let 
 us suppose that under Ge the curve 
 
 c: y = cix + c^x^ + Czx^ + • • • 
 
 is transformed into the curve 
 
 C: Y= CiX + C2Z2 + CzX' H , 
 
 where both c and C are assumed to be analytic in the neighborhood 
 of the origin. The relation between the coefficients of c and the 
 successive derivatives at the point in question is 
 
 1 
 1 .2 .3 ..-n^""^" 
 
 and similarly for C 
 
 1 
 
 1 . 2 . 3 ••• n 
 
 r« = C„. 
 
 lf,mQ,f=g=h = 0,Q has a double point at the origin and 
 its equation becomes 
 
 Q': (a;2 + y^y + (ax + by){x'' + y^) + ca^ + 2dxy + ey^ = 0. 
 
 Since there are five independent constants in Q', one member of 
 the family may have contact of order five with c at the origin 
 P, and this curve will serve as the geometric image for the fifth 
 derivative. Expressing the fact that a member of the family 
 Q' may osculate c at P we have Q' subject to the five conditions 
 
 1. c+2y'd + y"e^0, 
 
 2. (1 + y")a + 2/'(l + y")b + y"d + yY'e = 0, 
 
 3. 12(1 + yy+ I2y'y"a + W + I82/V - ]h 
 
 -\-4y'''d+[Sy'"+4yy"]e = 0, 
 
 4. 2iy'y"a + y") + Wy"' + ^y'> 
 
 + W" + Qy'Y' + ^y'y"']h + y'^d + [2y'Y' + y'y'^]e = 0, 
 
 5. 240i/'2/'"(l + 2/'^) + l^Qy"\l + Bi/'^) + {2,0yY- + my"y"']a 
 
 + [152/- + 45i/'V^" + l^Oyyy + 452/"']6 
 
 + Qy-d + [15^''2/>v + IO2/"" + Qyy]e = 0. 
 If 6 is the angle between the two branches of Q' that pass through 
 
 ♦Casey: On Bicircular Quartics, Royal Irish Academy, Transactions, Vol. 
 24. 
 
12 Differential Invariants under 
 
 P we may write 
 
 (6) Cot "^ "^ ^ 
 
 Vd2 - ce * 
 
 Since angles are preserved under G^ it is evident that ^ is a 
 function of As/As^. On solving equations 1, 2, 3, 4, 5 for c, d 
 and e and substituting in (6) we have 
 
 ® Cot0=-2^. 
 
 From Theorem II and equation (6') we have 
 Theorem III. 7/, under the inversion group G^, the curve 
 
 c: y = cix + 023^ + c^pc^ + • • • 
 
 he transformed into the curve 
 
 C: Y= CiX + C2Z2 + CzX^ • • • 
 
 {where c and C are analytic in the neighborhood of the origin P) 
 the angle 6 between the branches of the osculating bicircular quartic 
 having a double point at P, remains invariant and 
 
 2. It is known that the family of the 00^ logarithmic double 
 spirals of the plane distribute themselves under the inversion 
 group into 00 ^ families of co^ curves each, each one of the co^ 
 families being called a family of logarithmic equiangular double 
 spirals. In such a family, the angle between the polar axis and 
 the inflectional tangent is the same for each curve of the family. 
 Intuitively we may expect Ag/Aa^ = h to be the equation of the 
 00* logarithmic double spirals, where h is a function of the angle 
 just mentioned. The equation of the logarithmic double spirals 
 in so-called bipolar coordinates is (7) ri/r2 = Vl-e^*. If the 
 coordinates of the poles are (a, b) and (c, d) equation (7) expressed 
 in rectangular cartesian coordinates becomes 
 
 ^7,^ {x-ay-^(y-by 
 
 (x - cY -\- {y- dy 
 = ke 
 
 2A t>.n-i {d-b)x+(a-c)y + bc- ad 
 
 {x'-{-f)-ia+c)x-ib+d)y + ac + bd' 
 
THE Inversion Group. 13 
 
 If we put 
 
 — 71 ,72 
 
 a = o . » . = 
 
 7i' + 72'' 7i'+722' 
 
 ^ — (oClPl 4- <X2^2) , ^ — («li32 — CX2^l) 
 
 ais + «2^ ' «1^ + «2^ 
 
 (7') becomes 
 
 , _ 71^ + 72' 2;,tan-i «27i - «172 
 «i + «2'' aiji + a272 
 
 ^'^"^* _ 2A tan-1 («2 7i " «i 72) (a:*^ + 2/^) + («2 + 182 71 - |8i 72)3; 
 
 («i2 + a2^) (a:2 + 2/2) + 2 (ai i8i + aa (32) 
 
 4-2(«i)82-«2/3i) + /3i'+/32 « 
 
 (yi' + y2') {x' + 2/2) + 2^1 a; - 22/1 2/ + 1 
 
 ■f 2/') + («2 + i82 7i 
 
 + (q;i — )3i 7i — ^2 72) y + /32 
 
 (ai 7i + «2 72) («^ + y^) + («i + /3i 7i + i32 72)2: ' 
 
 — (a2 + )3i 72 — 02 7i) 2/ + /3i 
 
 which is a form more suitable for the purpose of using the curve 
 for contact. 
 
 Since, in the equation of a logarithmic equiangular double 
 spiral (X being regarded as fixed), there are five independent con- 
 stants, the curve may be used for contact of the fourth order. 
 Let us seek the condition under which the curve may hyperos- 
 culate the curve c (section 1) at P. Taking the successive de- 
 rivatives, as far as the fifth of (7") at the point P, the condition 
 that (7") hyperosculate c at P is that the equations, 
 
 8. (2/' + X)^2 + (1 - \y')li = 0, 
 
 9. • (1 + y")h + 2X(1 + y")k + y% - X2/'7i = 0, 
 
 10. - 2(1 + y"){l + \y")k + 2(1 + y'){\ - y')k 
 
 + (32/'2/" - \y")U + (62/'2/"X + 2y")h + y"'l2 - \y"'h = 0, 
 
 11. - I2y'y"{l + \y')k+ Wy"{\ - y')h 
 
 + (32/"^ + 42/'2/'" - 2X^"0^4 
 
 + (6X2/"' + SXy'y'" + 4y"')h + r% - y^y'^^h = 0, 
 * The six constants ai, aa, /3i, /Sj, 71, 72 are now connected by the relation 
 
 Ti^ + 72* «i7i + "272 
 
14 Differential Invariants under 
 
 12. -i20yY'+d0y'"){l+\y') 
 - ^y"'{y' - X) 
 - ZQy"^ 
 
 k+(20yy''+30y'"){\^y') 
 + 4y"'a + \y') 
 - I2y"\ 
 h + (52/'2/- + lOy'Y' - 3X2/-)Z4 
 
 + (lOXi/'z/- + 20X2/''2/"' + 6^-)Z3 + 2/^/2 - X^^i = 0, 
 shall be satisfied, where 
 
 ^1 = («i^i + a2^2) - 71(^1' + ^2'), 
 h = (aii82 - «2)Si) + 72()8i' + /322), 
 h = 7i(«ii82 — Q!2iSi) + 72(71^ + 72^^), 
 ^4 = (ai^ + a^') - W + 132^) (71' + 72^), 
 
 ^5 = (71^ + 72^) (0:1182 - OLi^l) + 7l(«l^ + Ot2^), 
 
 k = (71' + 72')(aii8i + a2^2) - 7i(«i' + 0C2'), 
 and where the new constants are connected by the relation 
 
 (13) hh - hk + hh = 0. 
 
 Solving equations 8, 9, 10, 11, 12 for the ratios of the Vs and 
 susbtituting in (13) we have 
 
 (14) ^_ 4(1^25 = 
 
 If rp is the angle between the polar axis and the inflectional 
 
 tangent of the logarithmic double spiral tan ^ = 1/X and (14) 
 
 becomes 
 
 (14') A5 + 8A33 Cot 2^ = 0. 
 
 Equation (14') is condition under which a logarithmic equiangular 
 double spiral, hyperosculate curve c at P, or stated differently, 
 equation (14') is the differential equation of the 00^ logarithmic 
 equiangular double spirals (^ being fixed) expressed symbolically 
 in terms of two relative differential invariants of G^. 
 
 Theorem IV: The differential equation of the logarithmic equi- 
 angular double spirals, expressed in terms of the invariants of G^, is 
 As + 8A33 Cot 2iA = 0. 
 
THE Inveksion Geoup. 15 
 
 If ^ = 45°, A5 = and from Theorem IV and Corollary I, 
 Theorem II, we have 
 
 Corollary I. The integral curves of the third relative differ- 
 ential invariant A5 = are the curves of that family of logarithmic 
 equiangular double spirals with an angle of 45" between their polar 
 axis and their inflectional tangent; or the integral curves of A5 = 
 are the inverse transforms of those logarithmic spirals cutting their 
 radii vectors at an angle of 45°.* 
 
 As a consequence of equation (14') and equation (6') of 
 section 1, we have 
 
 Theorem V. At any point P on a logarithmic double spiral 
 with an angle xf/ between the polar axis and the inflectional tangent, 
 the osculating bicircular quartic, with a double point at P, has 
 an angle 6 between the branches, such that d = Cot""^ f Cot 2x1/. 
 
 Corollary I. At any point P on a 45° logarithmic double 
 spiral, the branches of the osculating bicircular quartic having a 
 double point at P are orthogonal. 
 
 Corollary II. At the point of inflection of a 45° logarithmic 
 double spiral the polar axis of the spiral bisects the angle between 
 the branches of the osculating bicircular quartic having a double 
 point at the point of osculation. 
 
 III. The Differential Parameter for G^ and Differential 
 Invariants of Order Six. 
 
 1. In order to determine the fundamental differential invari- 
 ant of the sixth order, it is necessary to find the differential param- 
 eter for Gq. By making use of the extension of the independent 
 infinitesimal transformations, as far as the fourth order, for S 
 of part one, section three, and then extending with respect 
 to <p, which we shall assume is a differential invariant under 
 Ge, we have the complete system of linear partial differential 
 
 * This corollary in Inversion Geometry is the analog of one of Halphen's 
 theorems in Projective Geometry, i. e., " Les courbes int^grales de I'equation 
 A = sont des transform^es homographiques quelconques de la spirale log- 
 arithmic qui coupe ses rayons sur Tangle de 30 degrees " A is the third relative 
 differential invariant under the projective group. 
 
 Halphen: — Thesis: Les Invariants Diff^rentiels. 
 
16 Differential Invariants under 
 
 equations, 
 
 df df 
 
 4- Wy"' + ^y'") -~r, + \hy'y^ + \^"y"') ^ 
 
 + ^vg.= o. 
 
 "^ax^^dy^^a/ ^ a/' ^y by'" ^y dy^- 
 
 (,^-^)|-2.,|-2,(l + ,'^| 
 
 T: + {2xy" - Qyy'y" - 2y'{\ + y'^)] ^ 
 
 + [4a:2/'" - Qyy'" - Syy'y'" - WY']^r 
 
 + [Qxy-^- - lOyyY- - 20yV - SOyY" + 42/'"] ^ 
 
 -2{yy'-x)<p'^,= 0, 
 2xyl+(f-^)f^-2xa + y'^)§ 
 
 - [2yy" + &xy'y" + 2(1 + y")]^, 
 
 - [W + ^xy"' + ^y'y'" + I2y'y"']-^, 
 
 - [62/2/-+ 10a:^'2/-+ 182/"2+20a;2/"2/'"+242/y"] ^^ 
 
 -2(2/ + ar2/')^'^=0. 
 
THE Inversion Group. 17 
 
 The same reduction process that was applied to S, brings T to 
 the form 
 
 (l + 2/'')|i+(3r-22/y")^. 
 
 2". 
 
 + iWY' - ^y'y'-) ^. - y'<p' ^ = o, 
 . -2(i + 2/-)|-,-i2,y'^ 
 
 + [42/' V" - 202/y" - Uy'Y' - 181/"^]^ = 0, 
 
 and T' is equivalent to the involution system. 
 
 df <p'[{2y'y"' + ^y"')iX + y") - l2y'Y'] df _ 
 dy' 2[1 + y'V'il + y") - Zy'y"'] d<p' "' 
 
 ^ • %" 2/'"(i + 2/") - 32/y'=' a^' "' 
 
 a/ y'"(i + 2/'=^) - 32/y'' a/_Q^ 
 
 = c 
 
 dy'" <p'il + 1/'=^) d<p 
 
 The solution of T" is readily obtained and is 
 
 cp'd + y") 
 y"(i + y") - SyY'Y" 
 
 or symbolically expressed 
 
 Since the differential invariant of lowest order, As/As^ = Is,. is 
 known (Part I, section 3), we put tp — As/As^ and by making 
 use of Ai{d<p/dx)/Az^'^ = c, the fundamental differential invariant 
 of order six is Ai[A3A5' — SAs'Asl/As^'^ = h, where the primes 
 denote differentiation with respect to the dependent variable x. 
 The following theorem may be stated. 
 
 Theorem VI. The most general differential invariant of order 
 
18 Differential Invariants under 
 
 six is of the form 
 
 -fA^ Ai[A6^A3-3A3^A5] '] 
 •^LAs^' A3«/2 J-c. 
 
 In order to give a geometric interpretation of a differential 
 invariant of order six, we may make use of a bicircular quartic 
 of deficiency zero. Let M be the discriminant of 
 
 Q: {^ + f) + {ax + by){x^ + y') + cx^ + 2dxy + ef 
 
 + 2fx + 2gy+h = 0. 
 Since Jf is a function of the coeflBcients, a,h, • - • h, we may write 
 M = m{a, b, c, d, e,f, g, h). 
 
 Let m = 0, then Q is of deficiency zero, and Q, subject to the 
 condition m = 0, may have contact of order six with c (part 
 two, section one) at P. The coeflBcients a,h, - • - g are expressible 
 in terms of y', y", • • • y^K Since it is known that angles remain 
 invariant under Ge, we have 
 
 Theorem VII. 7/ a bicircular quartic of deficiency zero, osculate 
 a curve c at a point P (c being analytic in the neighborhood of P) 
 the angle between the branches of the real node of Q is expressible as 
 a function of 1 5 and Iq. 
 
 2. It is interesting to note that the differential equation of 
 the 00^ logarithmic double spirals, arises in connection with a 
 certain family of bicircular quartics, and that this class of double 
 spirals are made up of a special kind of points, which are closely 
 analogous to Halphen's points of coincidence. This is brought 
 out as follows. Let c be a curve analytic in the neighborhood of 
 point P, and let U and V be two bicircular quartics, each having 
 contact with c at P of order six. The family of bicircular 
 quartics have fifteen points in common, i. e., seven points at P 
 and four points at each of the circular points at infinity. Besides 
 these fifteen points of intersection, the members of this family 
 have a sixteenth point in common, which we shall call P'. It 
 may happen that P' will coincide with P. If such is the case 
 we shall speak of P as a " bicircular quartic point of coincidence."* 
 
 * The " bicircular quartic point of coincidence in Inversion Geometry is 
 analogous to Halphen's point of coincidence in Projective Geometry. 
 
THE Inversion Group. 19 
 
 Among the penosculating* bicircular quartics of the family 
 U + W, there will be one member of the family, let us say W, 
 having a double point at P and one of the branches of W will 
 have contact of order five with c at P and with each member of 
 the family U -\-W. If then W should have contact of order six 
 with another one of the family, say U, the sixteenth point of 
 intersection of the members of U -{- W, will coincide with P. 
 Conversely, if the sixteenth point of intersection of U -^W 
 should coincide with P, then one branch of W will have contact 
 of order six with each member of the family U -{-W. 
 
 Theorem VIII. The necessary and sufficient condition jor the 
 existence of a " bicircular quartic point of coincidence " at a point 
 P on a curve C is that the osculating bicircular quartic, having a 
 double point at P, have contact with c of order six. 
 
 To express the content of the above theorem analytically, we 
 assume that P is the origin, and that W, having a double point 
 at P, has contact with c of order six. The equation of W is 
 
 W: (x^ + 2/2)2 _|_ (^3. + 52/)(a;2 + y^) + cx^ + 2dxy -\- ef = 
 
 subject to the conditions 
 
 15. c + 2y'd + y'^e = 0, 
 
 16. (1 + y")a + 2/'(l + y'')b + y"d + yY'e = 0, 
 
 17. 12(1 + y'y + I2y'y"a + [62/" + WY]b 
 
 + ^y"'d + W + ^y'y"']e = 0, 
 
 (18) 2Ay'y"iX + v") + Wv'" + ^y'> 
 
 + [2y"' + ^y'Y' + ^y'y"']h + y'^^d^WY'+y'r^le = 0, 
 
 19. 2402/'2/"'(l + y'^) + 180y'"(l + ^") 
 
 + myY + 60i/"t/'"]a + [152/- + ^^y'^ 
 + imy'y"y"' + ^by"^]b 
 
 + 62/vcZ + [Iby'Y + IO2/"" + 62/'2/v]e = 0, 
 
 * Prof. E. J. Wilczynski has used the term ' penosculants ' to denote a 
 class of curves for which the order of contact falls short of the maximum by a 
 single unit. 
 
20 Differential Invariants under 
 
 20. 60y'2/-(i + y'^ + noy'Y'a + ^y") 
 
 + \Wy"' + Wr + 152/''^- + 102/"'']a 
 
 + [3r + 92/'y + ^hy'y"r^ + 452/"^'" + 302/'2/""]6 
 
 + y'd + [32/'V + 52/'"2/J^ + 2/y ']e = 0, 
 
 and this set of linear equations is satisfied if and only if the de- 
 terminant D vanishes, the columns of D being 
 
 (1) 
 
 
 
 12(1 + yy 
 
 2^y'y"{\ + y"^ 
 
 240^y"(l + 2/") + 1802/"^(1 + y") 
 
 602/'2/-(l + y'^ + l20y"y"'iX + 32/") + ISOz/Y'' 
 
 (2) 
 
 (3) 
 
 (4, 5, 6) 
 
 
 
 
 
 i + y" 
 
 
 Wy" 
 
 
 ^y'y'" + 32/"^ 
 
 
 ZOyY- + 602/''^"' 
 
 
 62/y + iW'y'^ + 102/"" 
 
 
 
 2/'(l + y") 
 
 
 
 62/" + 182/'y' 
 
 
 2y"' + 62/'y " + 92/'2/"' 
 
 15^'^ 
 
 + 452/'V'^ + 1802/'2/'y " + 452/'" 
 
 3r + 92/'y + 452/'^"^^- + 452/"V" + 2>0y'y"" 
 
 \ 1 
 
 2^' 2/", 
 
 
 
 2/" 2/y', 
 
 
 
 42/'" 32/"' H- 42/'2/"', 
 
 
 
 2/iv 2y"y"' + 2/'2/*", 
 
 
 
 62/- 152/"2/- 4- IO2/"" + 62/'2/^ 
 
 
 
 2/vi 3y'y _|. 5y'"2/'" + 2/'2/"' 
 
 Geometrically it is evident that D is of the form J{If>, h)- Upon 
 evaluation and substitution we have 
 
 D = SAiAslAaAg' - 3A3'A5] 
 
THE Inversion Group.'-, ; ', l^ ''';].' \'\\it\ 
 
 and the necessary and sufficient condition for a " bicircular 
 quartic point of coincidence/' analytically expressed, is 
 
 (21) A3A5' - SAs'As* = 0. 
 
 Again consider the bicircular quartic 
 
 Q: {x' + y') + (ax + hy){x'' + y') + cx^ + 2dxy 
 
 + ey^ + 2fx -\-2gy+h = 0. 
 
 If at the point P, (0, 0), Q be given contact of order seven with 
 the curve c, i, e., if Q osculate c at P, the coefficients of Q can be 
 expressed in terms of y', y", • • • y""". On expressing the con- 
 dition for osculation and solving for g, and h, it is found that 
 ^ = gr = 0, if AsAs' — 3A3'A5 = 0, and we have as a corollary to 
 theorem VIII. 
 
 Corollary I. At a " bicircular quartic point of coincidence " 
 the osculating bicircular quartic has a double point at the point of 
 osculation. 
 
 If rp be eliminated from equation (14') by differentiation, 
 equation (21) is obtained and we have 
 
 Corollary II. The logarithmic double spirals are curves, all 
 of whose points are " bicircular quartic points of coincidence." 
 
 Corollary III. At any point on a logarithmic double spiral, 
 the osculating bicircular quartic has a double point at the point of 
 osculation. 
 
 In the last two corollaries we have new properties of the logar- 
 ithmic double spiral. 
 
 IV. Differential Invariants Intrinsically Expressed. 
 Differential Invariants of Order Higher than the 
 
 Sixth. 
 
 1. According to the change of variables in part one, section 
 three, the first absolute differential invariant may be expressed as 
 
 ^^^^ {dp/dsy ~^'' 
 
 * Ai = and A3 = appear as extraneous solutions in finding the condition 
 for such points. 
 
"22: ■ Differential Invariants 
 
 where p is the radius of curvature and 5 is the arc length. Simi- 
 larly the differential parameter for G^ may be written 
 
 p{d(p/ds)/{dp/ds). 
 
 The intrinsic differential equation of the oo^ logarithmic equi- 
 angular double spirals is 
 
 Equation (23) expressed in terms of the first four successive radii 
 of curvature p, pi, p2 and pi, becomes 
 
 (24) 4p3Pip2 - 10p2Pi2p - 5p2V + 15pi^ - 4piV 
 
 + 8piV Cot 2x1/ = 0, 
 and we have 
 
 Theorem IX. At any point on a logarithmic equiangular 
 double spiral, the first four successive radii of curvature satisfy the 
 relation expressed by equation (24). 
 
 2. The general expression for the fundamental differential 
 invariant of order n is readily found. If we put A3A5'— SAa'As 
 = 06, we have for the fundamental invariant of order six 
 AiQ^3~^^l^\ Again if 67 represents the numerator of the fraction 
 found by taking the derivative of h with respect to x, we have 
 for the fundamental invariant of order seven 
 
 AiOzAs"*. 
 
 It is easy to generalize this process. If then 0„ be the numerator 
 of the fraction formed by taking the derivative, of the funda- 
 mental differential invariant of the (n — l)st order, with respect 
 
 to X, we have 
 
 AiGnAa-^^^-^^^ = j^^ 
 
 and we have 
 
 Theorem X. The most general differential invariant of the 
 rUh order, under G^, is of the form 
 
 and every invariant family of <x>^ curves is the solution of a differ- 
 ential equation of the nth order, expressible in the above form. 
 
VITA. 
 
 George Walker Mullins. Born Fayetteville, Ark., Feb. 11, 
 1881. Graduate University of Arkansas, A.B., 1904; Columbia 
 University, M.A., 1913. Professor of Mathematics, Simmons 
 College, Abilene, Texas, 1905-1912. Student University of 
 Chicago, Summer Sessions, 1907, 1909, 1910. Graduate Student 
 and Instructor in Extension Teaching, Columbia University, 
 1912-1913. Instructor in Mathematics, Barnard College, Colum- 
 bia University, 1913-. Member of the American Mathematical 
 Society. 
 
 Grateful acknowledgment is made to Prof. Edward Kasner 
 for his helpful suggestions and encouragement in the prepara- 
 tion of this paper. 
 
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