524 



CURVE LINES. 



i'urve 

 Lints. 



Pl*t« 



In the Figure, Ave have supposed the new origin A' to 

 be in the angle XAY, in which a and b are both to be 

 accounted positive ; if it had been in one of the other 

 three angles, we must, in like manner, have given to a, 

 h, the signs which belonged to that angle. Thus, in 

 the angle XAy, a would have been positive and b ne- 

 gative ; so that we should have had x—x' -f- a, y—y' — b. 

 19- Having given the relation of AP=x (Fig. Q.) 

 j?BL and PM=y, the co-ordinates of a point M, referred to 

 the axes AX, AY ; let it be required to find the rela- 

 tion of AQ=a:', and QMzzy', the co-ordinates of the 

 same point referred to other two axes AX', AY', having 

 a given position in respect of the former, the origin of the 

 co-ordinates A being supposed in both cases the same. 



Put /S=XAY, the angle which the axes make in 

 their original position, also «=X'AX, and *'=Y'AX, 

 the angles which the new axes make with AX, the ori- 

 ginal axis of the abscissae. Draw QKI parallel to AX, 

 meeting PM in K, and QL parallel to AY ; then MKI 

 =/3, AQL=/3 — *, and QMK=/3 — «'. In the two tri- 

 angles ALQ, QKM, we have, by trigonometry, 



Sin. /3 : Sin. (/3 — «) : : x' : AL, 



Sin. /3 : Sin. « : : x' : LQ=PK, 



Sin. fi : Sin. (fi—»') : : y' : QK=LP, 



Sin. /3: Sin. a! ::y' : KM. 

 Hence, ALxSin. p—x' Sin. (p — «),. 



LP x Sin. /8=y Sin. (£—«') 



PK x Sin. fi—x' Sin. », 



KM x Sin. /3—y' Sin. «' ; 

 and since by the position of the lines, AP=AL + LP, 

 PM = PK+KM, Ave have 



_ a'Sin. (/3— -a)+y' Sin. (/3— «') , 

 Sin. /J 

 x' Sin. a, +y Sin. «' 



•(!•) 



#= 



Sin. ,6 



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

 tion which expresses the relation of a: toy, the result will 

 be a new equation, expressing the relation of x' to y'. 



In applying these formula to any particular case, re- 

 gard must be had to the position of the new axes AX', 

 AY', in respect of the former. Thus, if the new axes 

 of the abscissa? were to have the position AX" on the 

 other side of AX', then, instead of £ — «, we would have 

 #+« ; and, in this case also, Sin. «. must be reckoned 

 as negative. 



20. If the original axes contain a right angle, then 

 Sin. /3=1, Sin. (/J— «)=Cos. <*, Sin. (& — «')=Co8. «! ; 

 and, in this case, 



xzzx' Cos. «+#' Cos. «'? , * 



y=x' Sin. »+y' Sin. »' J ' ' ' ' ^ 

 Again, supposing the original axes to contain a right 

 angle, if the new axes also contain a right angle, in this 

 case, a! — «=90°, or <*'=r90°-f.«, and Sin. «'=Cos. <*, 

 Cos. a'z= — Sin. a. See Arithmetic of Sines. 

 Hence these last formulas give us 



£=#' Cos. a. — y' Sin. «\ .„■. 



y=x' Sin. *+«' Cos.* J ^-/ 



Examples of the application of these formulas to the 

 transformation of equations, may be found in the con- 

 cluding section of Conic Sections. 



21. Let us now consider the other mode mentioned 

 in Art. 6. by which the position of a point, or the na- 

 ture of a line described on a plane, may be indicated. 



Fig. 10. Let HK (Fig. 10.) be any line whatever. Assume 



4X a straight line, having a determinate position in 

 the plane of HK, and take A, a given point in AX. If 

 now, from M, any point in HK, a straight line MA be 

 drawn to A, and we put r for AM, and <p for the angle 

 MAX, it is evident that the position of M will be de- 



termined, if the line r and the angle q> are both known, 

 and therefore that the nature of the line HK will be 

 indicated by an equation expressing the relation be- 

 tween r and <p. 



By supposing the line AM to revolve about A as a 

 pole, such a relation between r and <p may be assigned 

 as shall determine its extremity M to describe any pro- 

 posed curve. In this mode of generating curves, the 

 angle <p, or the arc of a circle described on A as a cen- 

 tre with a radius =: 1, which serves to measure that an- 

 gle, may be regarded as an abscissa, and the corre- 

 sponding line r as the ordinate. The two are com- 

 monly called polar co-ordinates of the line HK ; and 

 the equation expressing the relation of <p to r, its polar 

 equation. 



22. It is easy to pass from the equation of a curve 

 referred to two rectilineal axes, to its polar equation. 

 For let AX, AY be rectangular axes passing through 

 the pole A ; and let x=AP, yzzYM. ; then, by trigono- 

 metry, we have 



x=r Cos. <p, y—r Sin. p. 

 These values being substituted in the equation of 

 the rectangular co-ordinates of a curve, it will imme- 

 diately be transformed to its polar equation. 

 Thus the equation of a circle, viz. 



x z +y* — 2 ax — 2by=c* — a 2 — 6 2 

 gives us, for its polar equation, 



r l — 2r (a Cos. <p + b Sin. <p)=c 2 — a 2 — b\ 

 The most simple polar equations of the conic sections, 

 are given towards the conclusion of the article Conic 

 Sections. 



The polar equation of a curve being given, we may, 

 on the contrary, find the equation of its rectangular co- 

 ordinates. To do this, it is only necessary to put — 



V r 



for Cos. <p, and -£- for Sin. tp, and afterwards to put 



-v/(x*+y) for r. 



23. As there may be an endless variety of lines, in 

 considering their relations one to another, it has been 

 found necessary to class them. Accordingly, they have 

 been divided, in the first place, into two kinds. 



1. Such as may have their nature indicated by an 

 equation of a finite number of terms, composed of inte- 

 gral powers of the indeterminate quantities x, y, (co- 

 ordinates to two rectilineal axes,) and given quantities. 

 Lines of this kind are called algebraic, also geometrical. 

 Any straight line, a circle, and the conic sections, are 

 particular examples of this kind of line. 



2. Such lines as do not admit of their equations be- 

 ing expressed by a finite number of terms composed of 

 integral powers of x, y, and known quantities. These 

 are called transcendental curves, and sometimes, though 

 improperly, mechanical curves. The cycloid is a curve of 

 this kind ; its equation may be deduced from these two, 



x=a (1 — Cos. <p), y—a (ip-f-Sin. <p), 

 by eliminating <p, and its functions Sin. <p, Cos. <p; but 

 it will then necessarily consist of an infinite number 

 of terms. Such curves also as have their equations 



x x 



of these forms, y— a", y — x",tkc. are transcendental. 

 They are also sometimes called exponential curves. 



24. As algebraic curves still admit of an infinite va- 

 riety, they have been divided into classes; and all lines 

 whose equations are of the same degree in respect of 

 the indeterminate co-ordinates x and y, constitute a 

 class, or order, of the degree of the equation. The 

 foundation of this mode of classification is the ana- 

 lytical fact, that the degree of the equation of a curve can- 

 not he changed by any change in the position of its axes. 



