Note on Projective Coordinates

VOL. 14, 1928

MA THEMA TICS: H. P. ROBERTSON

1;83

4 See LefsChetz (a), No. 71. 6 Lefschetz (a), formula 71.1; (b) formulas (10.5), (36.2). The reason for the fact that these formulas differ from our formula (2) by the factor (-1)" is. an inessential difference in the definition of the "index." J. W. Alexander, Trans. Amer. Math. Soc., 28 (1926), pp. 305-306. 7 See, for instance, Alexander, 1. c., p. 316, formula 10.6. 8 Alexander, 1. c., pp. 306-307.

NOTE ON PROJECTIVE COORDINATES By H. P. ROBZRTSON* PRIcNToN UNrViaRSrrY Communicated January 7, 1928

In a recent paper in these PROCUZDINGS 0. Veblen' has developed a theory of projective tensors which is based on the fact that associated with a general analytic transformation and a given point there is a unique linear fractional transformation. It is the purpose of this note to derive the associated transformation in what would seem a more direct and significant way. Let the given transformation be, in the neighborhood of the point xo, in question, i(-XO = UX(X - XO) = i

i

o

~~~1

-Xj

F 7k i - xO) + 2 (Uk)o (x xi-)(xkXO) +. (1)

The associated linear fractional transformation y=

=a

1+ bkyk is determined by the two requirements:

-a,bkyiy

(2)

+ .

(a) It shall agree with (1) in terms of 1st order. This condition alone associates a unique linear transformation with (1). (b) Its Jacobian shall agree with that of (1) in terms of 1st order; i.e., the ratios of volume magnification shall differ by at most terms of 2nd order. From (a) it follows that

(u4)o

(3)

a, = The Jacobian of the original transformation is

(U,i)o + (u.'Ik)o(xk and that of (2) is U

+

uo

+ (Vsr4k)o(Xk

ok)+

}

MA THEMA TICS: 0. VEBLEN

154

|as$

-

PROC. N. A. S.

MDo. { (U(Xt)[( obk + (k)obi]I

(a%ft + dkj)+* * *

-

o

=uo 1-(n +1)bky +*} where vi is the normalized cofactor of uj in u. The condition (b) then yields (4) (!,.') n+ 1 It is to be noted that the numerator and denominator of (2) are then, within 2nd order,

bk = - 1

-

i U[(y) )I 1u(Y)[u

1 ~~~~~~1 I 1 + + "'andd

-n

U(Y)]-"

respectively, a result which agrees with that obtained by Veblen. * NATIONAL RSsuRCH FEiLow mi MATHEMATICS.

10. Veblen, "Projective Tensors

and Connections," these PROCOEDINGS, 14, 154 (1928). Prof. Veblen has kindly shown me this paper in manuscript.

PROJECTIVE TENSORS AND CONNECTIONS By OSWALD VUBLN PRINczToN UNIvrRsrTY Communicated December 29, 1927

1. The following pages contain an introductory account of a system of differential invariants the algebraic theory of which is closely analogous to that of ordinary affine tensors. The analytic theory is quite distinct from classical tensor analysis, though in our exposition we have emphasized the points of similarity as much as possible. The projective tensors and connections seem to be exactly the tools required for a symmetrical theory of the projective geometry of paths. This will be evident on comparing our formulas with those of H. Weyl,I T. Y. Thomas and J. M. Thomas in the articles cited below. They would also seem to be suitable for the analytic development of the closely related theory of manifolds with projective connection of E. Cartan;2 and they figure in a generalization of the quadratic differential form, which has some physical applications. In this note we limit ourselves to the point of view of invariant theory and leave all applications for future discussion. The definitions which we assume known and the general invariant-theoretic background will be found in a recent Cambridge Tract.3 It is clear that analogous systems of