CURVE LINES. 



523 



Curve 

 Lines. 



Plate 

 CCXXV 

 Fift i. 



the one angle of the axes. On the other hand, this 

 conventional hypothesis being assumed, all the points in 

 w a plane, whatever be their position, may be compre- 

 hended in the same formula. Accordingly (Fig. 2.) 

 in the angle YAX, x is positive and y positive ; 

 in the angle \ r Ax, x is negative y positive ; 

 in the angle XA_y, x is positive y negative ; 



in the angle x Ay, x is negative y negative ; 

 Consequently the equations x=a, y—b, which deter- 

 mine the position of a point in the angle YAX, become 

 x= — a y=-{-b 

 xz=-\-a y= — b 

 x= — a y=. — b 

 according as the point shall pass into one of the angles 

 YAj, XAy, xAy. By supposing a and b to be any 

 quantities, positive or negative, the two first may repre- 

 sent all the others. 



13. Every line that can be traced by a point moving 

 according to some determinate law on a plane, may al- 

 ways be referred to two axes. The nature of such a 

 line being expressed by some common property which 

 belongs to all its points, that property will, in every 

 Case, furnish an equation expressing a common relation 

 between the co-ordinates at any point whatever in 



■ the line, and as this equation will be characteristic of 

 the line, it may be called the equation of the line. 



14. As a particular example, let us consider the 

 straight line BA'M, (Fig. .5.) which meets the axes AY, 



"• Ax in A' and B. Then MP being drawn from any 

 point in the line parallel to AY, we have, agreeably to 

 the notation/ AP=.r, PMrry. Let us denote generally 

 the segments which the line MA'B cuts off from the 

 axes A x, AY by a and b, giving to a and b the signs 

 which belong to them from their position in respect of 

 the origin A. In the Figure under consideration, we 

 must, agreeably to what has been said, make AB= — a 

 and AA'= + b. 



By the nature of a straight line, PM has to PB a given 

 ratio. This property gives immediately j=Hi-)-K 

 for the general equation of the line, H and K being put 

 for invariable quantities. To determine H and K, it is to 

 be observed, that when xz=0, then y=.b, and that when 

 y=0, then x= — a, therefore in these two particular 

 cases the general equation becomes bzz0-\- K, 0= — aH 



4-K, hence we find K=b and H= — = — , and thus the 



a a 



• n b , 



equation of tne line is t/= • — x-\-b. 



If a, be put for the angle B which the straight line 



makes with AX the axis of the abscissae, and /3 for the 



angle A contained by the axes AX, AY, then in the 



triangle AA'B, we have A'=z/3 — x, and since, by Trigo- 



b Sin. a, , . 



nometry, - = ^-. — ; -, the general equation of a 



•" a bin. (/3 — a) ° ' 



straight line may also be expressed thus, 



x Sin. ee 



Curve 



Lines. 



y=\ 



,+*• 



Fig. 6. 



"Sin. (/3— «)' 



If we suppose the axes at right angles to one ano- 

 ther, then Sin. (/3 — «) = Cos. «, and in this case y= 

 x Tan. u-\-b. Each of these equations gives a positive 

 value ofy for every positive value whatever of x, and also 

 for all negative values between x=0, and ,r=r — a ; but 

 for every negative value of x beyond — a, we find that 

 y is also negative ; by which it appears, that the nature 

 of a straight line, as well as the circumstance of its pas- 

 sing through the three regions YAX, YAz, xAy, are 

 correctly indicated by its general equation. 



15. To find a general equation for a circle in respect 

 of two rectangular axes, let AB and BO, the co-ordinates 



of its centre, Fig. 6. (which we shall suppose situated in 

 the angle YAX,) be denoted by a and b, and put c for its 

 radius. From M any point in the circumference draw "'" 

 MF perpendicular to BO; then supposing .r and^ to 

 be co-ordinates of M, we have PB=MF=a — x, and 

 FO=b — y, and since MO ! =MF ! -f-FO% we have for 

 the equation of the circle, 



or, X'+y* — 2 ax — 2by=c i — a 2 — b\ 

 If the origin of the co-ordinates be at the centre, thea 

 «—0, bzzO, and the equation is simply .r 2 -f-7/ z :=c e . 



16. To find the equation of any conic section HMK re- 

 ferred to the rectangular axes AX, AY; let F be a focus, 

 (Fig. 7.) OD the directrix, and AO its distance from A. Plate 

 Put AO=d, and let AC and CF, the co-ordinates of the CCXXVH. 

 focus, be denoted by a and b ; and let a denote the angle ^'S- '• 

 which the axis of the conic section makes with AX the 



axis of the abscissa?. The lines AP=x and PM==y being 

 the co-ordinates of any point M of the curve, draw PBI 

 parallel, and MDB perpendicular to the directrix ; also 

 ME perpendicular to FC. The angles IAP, BPM, are 

 manifestly each = a ; therefore 



AI =r AP x Cos. <* = x Cos. «., 



MB=MP x Sin. a=y Sin. «, 

 and hence MD=^ Sin. »-^-x Cos.* — d. Moreover, in 

 the right angled triangle MEF, we have 



MF 2 =ME J + EF 2 =(rt— xy+(b— yy. 

 Let the determining ratio of the conic section be that 

 of 1 to n ; then because 1 : n 1 : : FM 2 : MD' (See 

 Conic Sections,) we have MD a =n ! x FM' ; in this 

 expression substitute the values of FM 2 and MD 2 , and 

 we get 



(y Sin. * + x Cos. <e — d) l =B* | (a— x) l + (b— y) 1 ] 



for the equation of the conic section, and which, by 

 putting 



A=w 2 — Sin. 2 a, ; ~D=2d Sin. «. — 2n z b ; 



B= — 2 Sin. ce. Cos. «; E=2d Cos. u—2n i a; 



C=n s — Cos. 2 *; F=a 2 +6 2 — d 1 ; 



may also be expressed thus, 



A^-f Ba-,y-f Cr 2 -f.Dz/-f Ex + F=0. 



17. From the values of A, B, C, we find that 



4AC— B ! =4« 2 (w 2 — 1). 

 This property of the coefficients deserves attention, be- 

 cause it affords a criterion by which the kind of section 

 to which the equation belongs may be immediately de- 

 termined. Thus, if 4 AC — B 2 be a positive quantity, 

 n must be greater than 1, and the section must be an 

 ellipse. (See Conic Sections.) If, again, 4 AC — B z =:0, 

 then n=zl, and the section is a parabola; and lastly, if 

 4 AC — B 2 be negative, n must be less than 1, and the 

 section is a hyperbola. 



18. In the analysis of curve lines, it is often necessary 

 to change the direction of the axes, as well as the origin 

 of the co-ordinates, that is, to refer the curve to two new 

 axes. The object of this change, in general, is to render 

 the equation 'of a curve more simple in its form. Let 

 us first suppose that the origin is to be changed, so 



that AX, AY (Fig. 8.) being the original position of the p- g# 

 axes, they may have a new position A'X', A'Y', paral- 

 lel to their former position. Let AB, BA', the co-ordi- 

 nates of A', the new origin in respect of the original 

 axes, be a and b ; and supposing AP, PM, the co-ordi- 

 nates of a point M, referred to the original axes to be 

 x and y ; let their new values A'P', P'M be x' and y'. 

 Then, because from the position of the lines, AP=AB 

 -f BP, PMrzBA' + P'M, wehavex=y + a, y=zy' + b. 

 These values of x and y being substituted in the equa- 

 tion of any curve, it will be transformed into another, 

 expressing the relation of x' to y', the new co-ordinates. 



