2i c' cos '"f' : c c s where, again, c' is an arbitrarily chosen con stant, the same for all pencils of the plane.
The significance of the generalized concept in hand may be exemplified by a problem. Let it be required to find the locus of a point xi that so moves as to be always a given distance from a fixed point x'i. The distance depends only on the cosine; we may. therefore, obtain the required equation by setting it equal to a constant k. On clearing of fractions and the radical, the equation is =le'S zyr 8 which, being of second degree in the xi, repre sents a conic section. This conic contains the four points common to the double line S'xx'=0 and the absolute Now the line S xx' =0 is the polar of the point x'i as to the abso lute. Hence the conic =PS xx' Sxx is tangent to the absolute at the points where it is cut by the line =0. We have thus the theorem: The locus of all points xi equidistant from a fixed point x'i is a conic tangent to the absolute at the points where the absolute is cut by the Polar of the fixed point.
Pouring new wine into old bottles, we may call the locus in question a (generalized) circle. Its centre is the point x'i and its radius is 2 i c k. Regarding k as a parameter, the equation xxl=k' S represents the family of concentric circles about x'i as centre. They are all tangent to the absolute at the same contact points. The circle corresponding to is degenerate, being the polar (counted twice) of the centre x'i. Its radius —2c i cos—' =ff ci. If k=1, the radius is 2i c cos" 1=0, and the circle, x' xx' Sx x= 0, is the pair of tangents from the circle centre to the absolute. (Compare the isotropic lines of ordinary Euclidean geometry). The distance between any two points on either of these lines is zero, for the point of contact is the double fixed element (the absolute points) for metric determination in the range in question. To render such distance non-zero, we should have to take c= ao , and then the distance of any two points not on a tangent to the absolute would he 00 Finally suppose k= ao ; the radius =2 i c cos' co=rn, and the circle is Sxx=0, the absolute itself. Hence: The absolute is the
locus of all points infinitely and equally distant from any point whatever not on the absolute, lust as the infinite line of the ordinary pro jective plane is the locus of points infinitely and equally distant from all finite points of the plane. The reader may readily solve and dis cuss the solution of the reciprocal of the fore going problem. The reciprocal problem is: To find the envelope of a line ei that moves so as to make always the same angle with a given line f'i.
Displacements of the There be ing 8 collineations of the plane, and a conic depending on five essential constants, it is seen that a conic can be transformed into itself by collineations of the plane. By such a trans formation any point P of the conic is converted into a point P' of the conic. P and P' are in general different points; but any transformation converting a conic into itself converts at most two of its points each into itself. Sup pose a transformation t converts the absolute into itself and that P and P' are the two points of C that are unchanged in position. Then t con verts into themselves the tangent at P, the tangent at P and the point Q common to the two tangents. That is, t not only leaves C un changed but also the triangle, P, P', Q. Query: How many of the oc ' collineations leaving C unchanged leave also unchanged the particular triangle in question? Taking this triangle for triangle of reference, the equation of C is mix:— 0. The most general collineation that leaves fixed the sides of the triangle is that defined by the equations: xi= d, x'i, xi= a2 x'2, xs= as x's. This transformation converts C into a, a2 x' x'2— as' 0. This is identi cal with C when and only when a, ch—ae=0, i.e., when the ratios, ch: as, a, : as, are subject to one condition. Hence the answer to the query is a simple infinity of collineations. All these transformations convert into itself the quantity xix2/ xs'. Hence they convert into it self each conic of the pencil xi x2— k xe=0 of conics. This pencil consists of all the conics tangent to C at the points where it is cut by the polar line of the point common to and x2-0.