PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 327 
For odd spaces, if 
n — iu + 1 = m or m = ^ -j- 1), 
the flat and its reciprocal, [a\, and are of the same order. This is the case 
for a line in three dimensions, and vve recover from the general formuhe 
[ah'] = — (a'b') ; (ah) = [(I'b'] , 
relations which I have elsewhere given connecting the symbols of recporocal lines. 
We are now prepared with all the necessary machinery for the geometry of flats 
and of their reciprocals. 
Table of Contents. 
Introduction. 
Section I. Fundamental geometrical properties of a linear quaternion function 
,, II. The classification of linear quaternion functions. 
,, III. Scalar invariants. 
,, IV. The relations of a pair of quadrics, S(/Fi 2 = 0, S(/Fo(/ = 0, which 
depend on the nature of the function Fo“ ^Fi. 
,, V. The square root of a linear quaternion function. 
,, VI. The square root of a function in relation to the geometry of 
quadrics. 
,, VII. The family of curves ^ = (/ +and their developables . . . 
,, VIII. The dissection of a linear function. 
,, IX. The determination of linear transformations which satisfy certain 
conditions. 
,, X. Covariance of functions. 
,, XL The numerical characteristics of certain curves and assemblages 
of points.. . 
XII. On the geometrical relations depending on two functions and oi 
the four functions /, /', /o and /. 
XIII. The system of c{uadrics '1 = ^> some questions relatin 
to poles and polars. 
XIV. Proqierties of the general surface. 
XV. The analogue of Hamilton’s operator V. 
XVI. The bilinear quaternion function. 
XVII. The four-system of linear functions. 
XVIII. The quadratic transformation of points in space. 
XIX. Homography of points in space. 
XX. The method of arrays. 
XXL The extension of the method to hyper-space. 
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