Page:Encyclopædia Britannica, Ninth Edition, v. 15.djvu/698

666 the locus of P. The solution is obvious, and hence we learn that & conic which has double contact with the fundamental conic is a circle in the generalized sense, and the centre of that circle is the pole of the chord of contact. A system of conies which have double contact with the fundamen tal conic in the same two points form a system of concentric circles, and the centre of the system is the pole of the chord of contact. We are accustomed in ordinary geometry to admit that every circle passes through the two imaginary circular points at infinity. This is the specialized form of the general theorem which asserts that every circle has double contact with the fundamental conic. The two theorems indeed coincide if the fundamental conic degrades to the infinity of ordinary measurement. A critical case is presented when the chord through coincides with either of the two tangents which may be drawn from to the fundamental conic. The two fundamental points then coincide, and hence the distance between any two points on a tangent to the fundamental conic is equal to zero. We have thus the curious result that in every system of concentric circles, including even the fundamental conic itself, the two points common to the system of circles are at the distance zero from the centre of the system. In fact the pair of tangents from the centre may be regarded as a conic having double contact with the fundamental conic, and therefore forming one of the circles of the concentric system of which the radius is zero. The reader will at once perceive the analogy to a well-known phenomenon in ordinary geometry. The equation in rectangular coordinates a; 2 + 7/ 2 =0 denotes either a circle of which the radius is zero or the pair of lines in the latter case we are obliged to admit that the distance of any point on either of these lines from their intersection is equal to zero. We have now to consider the displacement of a rigid figure, and we shall for the present speak only of a plane movement. We shall first show that it is possible for a plane figure to receive such a displacement that the distance between every two points in the figure after the displacement is equal to what it was before. Let x, y, z be the trilinear coordinates of a point in a plane, and suppose that x, y , z are the coordinates of the position to which this point is transferred in accordance with the linear transfor mation x =ax +by +cz y = a x + b y + c z z = a&quot;x + b&quot;y + c&quot;z. There are in general three points in the plane which are not altered by this transformation ; for, if we assume x = px, y = P y, z =oz, we have for p the cubic equation a p b b -p b&quot; c c&quot;- P = 0. The three values of p which satisfy this equation determine the coordinates of the three points. It is natural to take the sides of the triangle formed by these three points as the three lines of refer ence, in which case, if o, & 7 be constants, the system of equations assume the simple form x = ax, y = frj, z = yz. It is easily shown that four collinear points before the transforma tion are collinear after the transformation, and that their anhar- monic ratio is unaltered. This general form of linear transformation must be specialized in order to represent the movement. As no finite movement can either bring a point to infinity or from infinity, it is obvious that the displacement must be such as to leave the fundamental conic unaltered. It is easily seen that the specification of the transforma tion in its general form requires eight constants ; viz., the ratios of the nine quantities a, b, c, a, b , c , a&quot;, b&quot;, c&quot;. We may imagine five_of these constants to be disposed of by the provision that the conic shall remain unaltered. There will still remain three dis posable constants to give variety to the possible displacements. Although the fundamental conic will coincide with itself after the transformation, yet it generally happens that each point thereon will slide along the conic during the transformation. It is, how ever, very important to observe that there are two exceptions to this statement. Let 0, A, B, C be four points upon the fundamental conic which by transformation move into the positions, A , B , C. If OX be one of the double rays of the systems OA, OB, 00 and OA, OB , OC, and if we use the ordinary notation for anharrnonic pencils, then we have 0(A, B, C, X) = 0(A, B , C , X). But the anharrnonic ratio subtended by four points on a conic at any fifth point is constant, whence 0(A, B , C , X) = (A , B .C , X), and therefore 0(A, B, C, X) = (A, B , C , X). Suppose the transformation moved X to X, Jlien since the an- hannonic ratio of a pencil is unaltered by transformation we have 0(A, B, C, X) = (A, B , C , X ); whence (A, B , C , X) = (A B C X ); but this can only be true if the rays O X and O X are coincident, in which case X and X are coincident, whence it follows that the point X has remained unaltered notwithstanding the transforma tion. Similar reasoning applies to the point Y defined by the other double ray, and hence we have the following theorem : In that linear transformation of the points in a plane which con stitutes a generalized movement, there are two points upon the fundamental conic which remain, unchanged. It also follows that the tangents to the fundamental conic at the points X and Y, as well as the chord of contact, must remain unaltered. These two tangents and their chord of contact must therefore form the triangle of reference to which we were previously conducted by the general theory of this transformation. It will now easily appear how a transformation of this kind is really a displacement of a rigid plane. The distance between each pair of points is expressed by an anharmonic ratio; such ratios are unchanged by the transformation, and the two points which lay on the absolute originally are also there after the transformation. It therefore appears that the distance in the generalized sense between every pair of points is unchanged by the transformation. In other words, a rigid system will admit of a displacement of the kind now under consideration. If we denote the two tangents at the unaltered points on the conic by =0, y=0, and the chord of contact by z = 0, then the equation to the absolute is xy-k 2 z 2 =*Q. Transforming this equation by the substitution x = ax, y = Py, s = 7 2. we see that the condition &amp;lt;xj8-=y 2 must be fulfilled. It is very remarkable that the fundamental conic is only one of a family of conies, each of which remains unaltered by the transforma tion. In fact every generalized circle of which the intersection of the two tangents is the centre has for its equation xy - h z 2 ; and, whatever h may be, this circle remains unaltered by the transfor mation. Hence we have the following remarkable theorem: When a plane rigid system is displaced upon itself there is one point of the system which remains unaltered, and all the circles which have as their centre remain unaltered also. It is quite natural to speak of this motion as a &quot;rotation,&quot; and thus we may assert the truth in generalized measurement of the well-known theorem in ordinary geometry that Every displacement of a plane upon itself could have been prod need by a rotation of the plane around a certain point in the plane. Notwithstanding the rotation of the plane round 0, the two tangents from to the fundamental conic and also their chord of intersection, or the polar of 0, remain unaltered; each point on the polar of is displaced along the polar, and we would in ordinary geometry call this motion a translation parallel to the polar. It thus appears that, in the sense now attributed to the words rota tion and translation, a rotation round a point is identical with a translation along the polar of the point. Another point on which the present theory throws light on tho ordinary geometry must be here alluded to. We have seen that the two tangents from to the fundamental conic remain unchanged during the rotation of the plane round 0. It certainly does seem paradoxical to assert that a plane, and all it contains are rotated around a point, and that nevertheless this operation does not alter the position of a certain pair of lines in the plane which pass through the point. But have we not precisely the same difficulty in ordinary geometry ? Let us suppose that a plane pencil of rays is rotated through an angle 9 about the origin. Then a line through the origin whose equation before the rotation was becomes after the rotation xcos The lines thus represented are of course in general different, but the&quot; will be the same if It follows that even in ordinary geometry the two lines xHy = Q remain unaltered notwithstanding the rotation of the plane which 