INFINITESIMAL CALCULUS (X, Y) those of any point on the line passing through these points, then the equation of the line is If now the point Q be supposed to approach P, and ultimately to coincide with it, the line becomes the tangent PT at the point P, and its equation becomes For example, in the curve represented by x m = ay", dy my we have = - ; dx nx and the equation of the tangent at the point x, y is in n = m - n . x y This furnishes a simple construction for the tangent at any point on a parabolic curve. (Compare Ricci s construction given in the Introduction, p. 7.) If the equation of the curve be given in the form u=f(x,y) = 0, du du, di/ we have -T-+T- -^ = 0, dx dy dx and the equation of the tangent is du . , . du , 70. Again, the normal at the point (x , y), being perpendicular to the tangent, has for its equation du . . du , v dy dx the curve being referred to rectangular axes of coordinates. 71. The line TM in fig. 6 is usually called the snbtangent and RM the subnormal. It is easily seen that as it may written. Again, if the PTM-0, dy dx the length of the normal we have tan .^ ; and Fig. 6. PR = y sec < = also that of the tangent dx" 72. In general, if the equation of a curve be given in terms of any two variable coordinates, the position of the tangent at any point can be determined by finding the ultimate ratio of the corre sponding elementary variations of the coordinates at the point. Newton gave, in his Opuscula, several applications of such systems of coordinates. In particular, it may be noticed that he considered the case of what are now called bifocal curves, i.e., where the equa tion is expressed in terms of the distances from two fixed points. Newton illustrated his method by finding the tangent to a Car tesian oval, styled by him an ellipse of the second order. The same problem, in a more general case, was studied by Leibnitz (Ac. Erud., 1693), who gave a method of drawing tangents to curves given in terms of the distances from any number of fixed points. 73. At a double point on a curve (see CuiiVE, vol. vi. p. 719), we have = and : and at such a point becomes inde- i " dy dx du dx terminate, being of the form , since dy dx dx du dy Applying the method of in this case that of du dx 58, the true value of -=- becomes ft 11 du dx 2 dxdy dx d *u d~u dy I ^_ dxdy dy 2 dx (AJ U (t~"lf/ Ct]/ dx* dxdy dx d*u d*u dy dxdy dy* dx Hence d?u d?u_ dy d?u dx 2 dxdy dx dy 2 The roots of this equation in -J- dx the tangents to the two branches of the curve at the double point. Double points are distinguished into three classes, according as the roots of this equation are (1) real and unequal, (2) real and equal, or (3) imaginary, i.e., asf-^-Y- ^ S is >, -, or <0. Of these the first are called nodes, the second cusps, and the third conjugate points. They are frequently also styled by Professor Cayley s nomenclature as crunodes, spinodcs, and acnodes. See vol. vi. p. 723. 74. In the general discussion of curves it is usually more con venient to refer them to a system of trilinear coordinates (see vol. vi. p. 719), in which the position of a point is determined by its distances from three fixed lines. The equations of curves in system are homogeneous. Again, if (a;, y, z), (x , y , z ) denote two points in such a sys tem, the coordinates of any point on the line joining these points may be represented by Hence, to determine the points in which the line joining x, y, z to a/, /, z intersects a curve of the ni degree, we substitute x + KX 1 , y + Ky , z + KZ for x, y, z in the equation of the curve, u = ; then by Taylor s theorem ( 57) the result may be written . / , du . du . du "u + " - I K ( x + y -T- + z -r- )
dx dy dz J
d V L * T" }u + & dz) T 1 . 2 d x
dy A" 2 * 2 ^- where A stands for the symbol of operation / , d , d , d { x ~r + y -^~ + z ~T )
dx " dy dzj
The roots of this equation in determine the coordinates of the / du x - =0 , or Au=Q, points of intersection of the line and the curve. If the point x, y, z lie on the curve, we have M=0 ; if in addi tion we have , du dx dy dz then a second point of intersection of the line with the curve will be consecutive to x,y,z; and Au = is the equation to the tangent at the point x, y, z. Again, if the latter expression Ait, vanish identically, the point x, y, z is a double point on the curve ; or, in other words, every line passing through it meets two branches of the curve there. The equation A 2 u = is in this case that of the pair of tangent lines at this point to these two branches. This method is evidently susceptible of much extension. Asymptotes. 75. The method of the calculus furnishes a ready mode of deter mining the asymptotes to algebraic curves. By an asymptote we understand a tangent whose point of contact is situated at an in finite distance. To find the asymptotes to a curve of the ?ith degree, we suppose its equation written in the form where u n is a homogeneous expression of the 7ith degree in x and y, &c. Again, writing u n = x n f NM , Un-r^x"-^ ( -M , &c., the equation becomes Let y = KX + v be the equation of any right line ; then, to find its V II points of intersection with the curve, we substitute K- for -^- in X *C the preceding equation, and, after expansion by Taylor s theorem, we arrange according to powers of x ; this gives
- n /oM + s"- 1 {AM +tfM}
+ x * {f,( K ) + vf( K ] + ^/"oW; + &c. = o. Now if the line y=KX + v be an asymptote, two of the roots of this equation in x must be infinite, and consequently we have / O (K) = O, and/ 1 W + "/ = 0- XIII. 4
