an angle with the Z-axis whose trigonometrical tangent will be 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 N T ་་ 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 a + iy a complexe number. I. From the theory of complex functions we know that sin(x+y)=sinx.cos(iy) +cosx.sin(ży). Or, introducing for cos(iy) and sin(y) their exponential equivalents, sin(x + iy)=sinx. ( đen + icosx ( zen).... 2 2 (6). 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 + ia) = A.sinx + z.B.cosx, = (7 so that, comparing this with the general formula zin, we see that Now in order to find what curve is represented by (7) let ur express No. This we recognize at once as the central equation of the ellipse. (7) therefore is the equation of an ellipse whose major axis A = A, and the curve becomes a circle. For ao the imaginary part vanishes and therefore the ellipse degenerates into a straight line. These results may be expressed in the following The construction will be this: Since the argument is xia, the imaginary part of it is constant and x describes a straight line parallel to the x-axis and at a distance from it equal to unity, namely the line in the figure marked 1 i, i + x. When x is 1, 2, 3, &c., then the argument is + i, 2 + i, 3 + i, &c. At the same time that the argument describes this straight line, the movable point z (the function) describes the ellipse. The values of the function belonging to any given value of the argument are calculated after reducing the function to the formin. In the present case II. In No. (6) let x be constant, x = a, while y varies, then This is the well known central equation of the hyperbola. For a ➡o we obtain a straight line parallel to the axis of ordinates whose distance from it is (e-). For a 2 the curve becomes a similar straight line parallel to the x-axis. Construction: The argument moves on a line which is parallel to the 7-axis and at a distance from it equal to unity. This straight line passes through the intersection of the axes of the hyperbola, which point is in the x-axis at the distance = I from the origin of the coordinates. The abscissas of the hyperbola are real, the ordinates imaginary. We now have this theorem: The formula z = sin (a + iy) is the equation of an hyperbola whose axes are sina and cosa. III. In No. (6) let y=x, then icosx| sin(x + ix) = sinx (+) + cos("=") 2 Now the argument x + ix describes a straight line forming an angle of 45° with the x-axis, while the function describes a curve in the plane of these two lines. For x = л, 27, зπ, &c., the abscissa vanishes, because ex + ex then sinx = 0, while the factor increases very rapidly. For 2 x= }π, }π, §π, &c., the ordinate vanishes while the fator et ex in 2 creases very rapidly. The curve therefore is a kind of spiral. Its origine is in the origine of coordinates. = 1, When sinx = o, then the abscissa = 0; but for sinx = o, dn÷d: therefore the curve at every intersection with the y-axis forms an angle of 45° with that axis. Again, for cosxo the ordinate = 0; but for COST = o, dn ÷ d I; therefore the curve at every intersection with the -axis forms with that axis an angle of 45°. In order that did may be = o, we must have hence = Hence The expression on the right hand side of this equation shows that as x increases, the value of tanx approaches rapidly to unity. dr de will become = 0 for tanx = 1, that is to say for x = 杌,杌, the approximation begins, but for ,, &c., t The construc tion is given in the figure. But the curve soor assumes such enormous proportions that i has been tracec only a little beyond = 135° The argument describes the straight line o x + ix. On the axis of abscissas there are marked the points corresponding to x = 20. 30, 50, &c., degress. We have thus far considered only one complex function, namely sin(x+y), and we see that it will produce a very great number of curves, according to the relation which we choose to establish between x and y. The same is true of all other complex functions. To recapitulate: The introduction of the imaginary numbers into Analytical Geometry enables us to construct all functions of the abscissa, whether real or imaginary. For example the equations without the introduction of imaginary numbers cannot be constructed; but when expressing the condition that y is to be perpendicular upon x, it is easy to construct them. And indeed they must each represent some curve since y varies when x does. THE PLANE TRIANGLE AND ITS SIX CIRCLES. BY ASHER B. EVANS, A. M., LOCKPORT, N. Y. The six circles whose properties are discussed in this article are the circumscribed, the inscribed, the nine-point, and the three escribed circles. The first two of these circles are familiar to every student of elementary geometry. The nine-point circle in a triangle is that circle whose circumference passes through the feet of the three perpendiculars from the angles upon the opposite sides, the three middle points of the sides, and the three middle points of the segments of the perpendiculars between the angles and their common point of meeting. The escribed circles are three circles situated wholly without the triangle, each of which is tangent to one side of the triangle and to the other two sides produced. Three points being in general sufficient to determine a circumference, it is necessary to show that the nine points enumerated in the definition of the nine-point circle are always on the same circumference. To this end let ABC (Fig. 1) be a triangle, O' the centre of its circumscribed circle, O'a, O'ẞ, O'r the perpendiculars from O' to the sides BC, AC, AB, respectively. Produce O'a to A', O'ẞ to B', O'r to C', making aA' = O'a, ẞB' O'ẞ, TC' O'r; complete the triangle A'B'C', let L be the centre of its circumscribed circle, and let a', B', r' be the intersections of AL, BL, CL with B'C', A'C', A'B'. 2 = = |