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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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