On Differential Inversive GeometryAuthor(s): Frank MorleySource: American Journal of Mathematics, Vol. 48, No. 2 (Apr., 1926), pp. 144-146Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2370744 .

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On Differential Inversive Geometry. By FRANK MORLEY.

?1. The Integral Invariant.

Through the Columbia dissertation of G. W. Mullins, " Differential In- variants under the Inversion Groups," and the memoirs of Liebmann (Munich ]3erichte, 1923) and Kubota (Tohleuk University Reports, 1923 ?), the foundations of differential inversive geometry for a plane curve are established. But the following self-contained mode of approach seems desirable.

Denote the Schwarzian derivative of x as to y by {x, y}. Then we have for any number n of related numbers x, y, , 1. Cayley's * cyclic formula

C.: {x, y} (dy)2 + {y, z} (dz)2 + {l, x} (dx) 2 O.

Let the curve in question be given by a self-conjugate equation in x and. its conjugate T. Then by 02

{x, } (dX)2 + {, x} (dx)2 0,

so that {X, y} (dX)2 is an imaginary, say

1) {X, X} (d) 2 = + 2t (dX) 2 where X is real.

Apply a homography x aY+ b and use Cayley's rule for the cycle CY ? d

x, T, y, y. Then since {x, y} 0 and {y, x} _ 0,

{X, 5} (d)X2 = _ {9, y}dy2

{y, y}dy2.

Hence x is an invariant under homographies and is the proper real parameter. Using C3 for the cycle 7, X, x

{x, d} (dX) 2 + {A, x} (dx) 2 + {X, A;}d32 0,

so that {Y,A} + 2L.

Hence {x,X} = ?-- + I, when I is real.

I is then the fundamental differential invariant.

* Cayley, Camb. Phil. Trans., Vol. 13, 1880.

144

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MORLEY: On Differential Inversive Geometry. 145

If the curve is given in the form x f f(/) where ,u is any real para- meter, then from C.

(dA/dfL) 2{X., A} = {x, } - {A, z}, so that + 2t(dA/dk)2 I {x,,u} {t,t}

and 2I (dA/dj)2_ {x,p + {A,}42{A,}.

?2. The Curve whose Intrinsic Equation is Linear.

The intrinsic homographic equation of a curve is the relation of I to A. It remains the same for all the homographic transformations of the curve. It is known that the curve

1= const.

is the loxodrome, or isogonal trajectory of aTes of circles from a point to another. We seek the curve given by

I = aX + b

or taking a proper initial point on the curve from which to measure A, namely a point such that at it I = 0 (or the closest loxodrome has the angle 7r/4).

I = aX.

The curve is given by the equation

{x, }-a(A ? t/a).

Whence x is a ratio of two solutions of

(d 2v/dA2) + 'va(X t la). (Forsyth, Treatise on Dif. Equations, ? 61). Writing A ? 'a =/ K/I, where K3a = 2, this equation becames

d2v/d/2 + V1% 0

and has the solutions in power-series

v1=M-2/L4/4! + 2.5./7/7!-*

V2= 1- -3/3! + 4,u6/6 ! -4.7. 7u9/9! +-- The curve is then x:= v1/v2, where u moves parallel to the axis of reals. To see what happens when tt is large, we note that our series are expressions for Bessel functions; apart from a constant factor, v1 is J (/,U3/2) and

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146 MORLEY: On Differential Inversive Geometry.

v9 is J/ (23%U3/2). (Forsyth, loc. cit., ? 101 and 111). Now the develop- ment of a Bessel function in negative powers of the argument was given by Lommel. (Forsyth, loc. cit., ? 105).

Applying Lommel's formula we have for z large to a first approximation

J%, ( z)-( 2/rz)/2 cos (z-7r/4 - 7r/6 ) +*

Jy, (z) (2/rz) '/2 cos (z-r/4 + r/6) +

Thus x is to a first approximation homographic with exp 2tz, where z 2/303/2. The general question of passing from the intrinsic equation of a curve,

I = f(A) to the map equation x = f(A) is that of integrating

{x,A}= f(A)+?.

Thus it comes under the method sketched by Klein, " Ueber Gewisse Differ- entialgleichungen dritter OrdnLng," Math. Ann., Vol. 23 or Works, Vol. 3, p. 721.

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