THE ANALYST. Vol. I. Nov. and Dec., 1874. Nos. II and 12. THE USE OF IMAGINARY QUANTITIES IN ANALYTICAL GEOMETRY. BY WERNER STILLE, MARINE, ILL. In works on Analytical Geometry we often find the statement that for certain values of the argument x the function y beeomes imaginary and hence cannot be constructed. Yet at the same time we find that the function, although imaginary, still varies, still remains a function of For example take the central equation of the ellipse: Here y becomes imaginary for x>a, and yet y continues to vary as x continues to increase. The question therefore arises whether our notation is not deficient. If our notation were well adapted to represent the relations between the argument and its function, then we should be able to construct that function for all values of x. By Gauss' interpretation of the imaginary numbers we know that when the "real" numbers are conceived as situate upon a straight line, then the "imaginary" numbers must be taken as situate upon a line right-angular to the former and having the point zero in common with it. Hence for the purposes of the Analytical Geometry of the Plane, it may be well to introduce imaginary numbers. Let us see what notation would result from the introduction of these numbers and what advantages may be derived from it. An equation between x and y, such as equation (1), states what shall be the relative lengths of the abscissa and the ordinate of any point of the curve; but nothing is expressed as to the relative directions in which x and y shall I be counted. Therefore if, according to usage, we tacitly add the condition that y shall be perpendicular upon x, we need not be surprised if occasionally we find a construction impossible. If we wish to have the ordinates perpendicular to the abscissas, we should express this in some way. This can be conveniently done by the well known formula z = x + i.y. ..(2), * denoting the position of any point of the curve, x its abscissa, y its ordinate and i=VI, which symbol expresses that y is perpendicular upon x. Now eliminating y from (1) and (2) we have Here z is tho position of any point of the curve, x the abscissa and b a Va22 the ordinate standing perpendicular upon the x axis. The curve so generated is an ellipse, as is well known. Now let us see what becomes of the movable point z when x becomes greater than a. x = a + t, then Put Now 24 which was before of the form, has become a "real" number, that is to say, z has entered the x- axis and is still a function of x as shown by equation (4), siuce x = a + t. The point z moves on the x axis. More strictly there are two points z, as in fact there were before became greater than a. The ambiguity of 1/2at + t shows that two such points z exist; and their rate of motion as depending upon t is given by equation (4). But since all numbers, real and imaginary, occupy only two dimensions, there is in space one dimension at our disposal and nothing prevents us from constructing equation (4) in a plane at right angles to the * y plane. This equation (4), as we readily see, represents an hyperbola when a+t, as before, is the abscissa and b a V2at the ordinate. Now we can construct equation (1) or its equivalent by our present notation, namely equation (3) which the el lipse is situa ted. Let xoy be a perpen dicular plane, xov a horizontal plane, then the ellipse lies in the vertical plane and the corresponding hyperbola in the horizontal plane. The equation of the hyperbola gives an ellipse for the values of x lying between + a and -a; and an hyperbola for values of x absolutely greater than a. For, again, which is the well known vertex-equation of the ellipse, as is evident when recollecting that a-t is the abscissa and V2at ta the ordinate. b a The same figure therefore serves to show the geometrical meaning of x2 b the two equations y = a b Va2x2, and y V x2 - a2. = a In the same manner we find that the equation of the circle y= ax2 represents an hyperbola for x > a; and that the equation of the parabola y vax represents another parabola situated in a plane perpen dicular upon the xy plane. From what has been said it is evident that our notation will enable us at all times to construct the function when the ordinates are imaginary. In fact, the geometrical sense of an imaginary number being that it is to be counted in a direction perpendicular upon the line of real numbers we see that the symbol i may be taken as nothing but a coefficient of direction. By the ordinary method the equation x2 + y2 + a2 = 0 cannot be constructed; for y = √ — (a2 + x2), which shows that y is imaginary for all positive and negative values of x. But it is evident that y is a continuous function of x, and that therefore we ought to be able to construct corresponding values of x and y into some curve. Again employing our notation, z = x ± i √ − (a2 + x2) = x = √ a2 + 202, which gives an hyperbola. After these introductory remarks I proceed to some more general considerations. Any curve in the xy plane may be represe..ted by the symbol 2 = ζ + in. (5), another function of x. This method is z = x + iy, being a function of x; and preferable to putting where x and y are the coordinates, since by the former method functions can often be written in a more tractable form, than by the latter. Since is the abscissa and 7 the length of the ordinate of any point of the curve, therefore the tangent at any point of the curve will form an angle with the -axis whose trigonometrical tangent will be drie. dr We also see that the well known expressions for the lengths of the normal N, the subnormal S, the tangent T, and the subtangent S, will remain unchanged in form, hence In short, all the relations between the differential coefficients and the form of the curve are immediately applicable to our present notation. We will now discuss the geometrical meaning of some analytical relations known to exist between real and imaginary functions. A function of the form f (x + iv) we will call (as has become customary) a complex function; a number of the form x + iy a complexe number. I. From the theory of complex functions we know that Or, introducing for cos(¿y) and sin(ży) their exponential equivalents, Now x and y are independent of each other unless we establish some functional relation between these two variables. Hence this equation expresses a very general relation and will admit of the construction of a very great number of curves. Let us then assign some special value to y. At first let y be a constant number y = a, then (6) becomes which for the sake of brevity, we may write thus sin(x + a)= A.sinx + z.B.cos, so that, comparing this with the general formula z = + in, we see that |