CURVE LINES. 



525 



We have seen, (Art. 18.) that the origin of the co-or- 

 dinates is changed by making 



izd-f 1 ') y = ° + y'y 



tlie new axes being supposed parallel to the former, and 

 next that the direction of the new axes may be changed 

 (Art. 19.) by taking 



x'=zm x" + ny", y'=m' x" + n'y", 

 a, b, m, m', n, n' being given quantities; therefore, to 

 change both the origin and the position of the axes at 

 once, we have only to assume 



x=a-\-m x" -\-ny", y=b+m' x" -\-n' y". 

 But these values of x and y, when substituted in any 

 equation in which x and y are the variable quantities, will 

 always produce another equation of the very same de- 

 gree, and having x" and y" for its variable quantities ; 

 and hence it happens, that by no transformation can 

 the degree of the equation of a curve be changed. 



25. A line of the first order, has its equation of the form 



a -\-bx-\-cy = ; 

 this class consists of the straight line only. Lines of 

 the second order, or curves of the first order, have their 

 equations of the form 



a -f- b x + c y -f- d x 2 -f- e x y -\-fy t zz 0. 

 This order comprehends four species, viz. the circle, 

 the ellipse, the parabola, and hyperbola ; or the two 

 first may be considered as one species. 



Lines of the third order, or curves of the second or- 

 der, have for their equation 



a + bx+cy+dx' + exy+fy*\ -O 

 _ +gx* + kx i y + ixy* + kyi J 

 This order may consist of more or fewer species, ac- 

 cording to the principle of classification that is assumed. 

 Newton, adopting one principle, subdivided them into 

 72 species ; but to these, six have been added by Mr 

 Stirling and Mr Stone. Euler, again, following another 

 principle, has comprehended them in 16 general spe- 

 cies ; these, however, admit of being divided into many 

 varieties : and Cramer, taking a different view of the 

 subject, makes 14 classes. 



Lines of the fourth order, or curves of the third or- 

 der, have for their general equation 



a + bx+cy + dx l + exy+fy* ~i 



+gx* + hx i y+ixy 1 + kyi [■ =0. 

 -f- Ix* + mx i y -f- n x 1 y i +pxy* -f- qy< J 

 The lines expressed by this equation have been di- 

 vided by Euler into 146 classes; and by Warring they 

 have been comprehended in 12 cases of equations. 

 The various species of curves, however, into which 

 this order may be divided, amount to many thousands, 

 and have never been distinguished individually. As to 

 the fifth and higher orders, their number has precluded 

 any attempt to arrange them in classes. 



26. When the terms of an equation of any degree 

 are put =0, if it represent a curve of that degree, it 

 ought not to admit of being resolved into factors, which 

 are rational in respect of x and y. If it does admit of 

 such resolution, then each factor put =0 is the equa- 

 tion of a curve of any inferior degree, the co-ordinates 

 of which satisfy the general equation. 



The equation 



ay — ax-f-x* — 2 xy-\-y i zz0, 

 which is of the second degree, is the product of x—y 

 and x — a — y, so that it may be expressed thus, 



(x—y) (x—a—y) =0. 

 This equation is satisfied either by making yzzx, or 

 y—x—a, and y can have no other values ; now, these 

 equations belong to two straight lines, therefore any co- 

 ordinates of either of the two lines will satisfy the equa- 

 tion ay—axJ r x*—2xy+y~-—0, and consequently it 



does not represent a line of the second order, but two 

 lines of the first order. 



In like manner the equation 



yl — y l x-\-yx 2 — x 5 1 _A 



-fay— 2axy + 3ax*— Zatx] ~ U ' 

 has the appearance of belonging to a line of the third or- 

 der : but as it is the product of" the two equations, 



y — x+a=0, yi — 2ax+x*=z0, 

 the former of which belongs to a straight line, and the 

 latter to a circle. The above equation of the third de- 

 gree belongs at once to a straight line and a circle, and 

 it cannot represent any other line. 



27. From the connection which subsists between 

 curve lines and equations, they may be reciprocally ap- 

 plied to the illustration of one another. As the nature 

 of every curve generated according to some determinate 

 law, may be expressed by an equation peculiar to that 

 curve ; so, on the other hand, corresponding to every 

 equation involving two indeterminate quantities, there is 

 a plane curve, the co-ordinates of which are the geome- 

 trical representatives of the variable quantities of the 

 equation ; so that all the circumstances regarding the 

 latter are, as it were, graphically exhibited to the eye 

 by the former. 



28. The line, the co-ordinates of which represent the 

 indeterminate quantities of any equation, is called the 

 locus of the equation. The locus of an indeterminate 

 equation of the first degree, is therefore a straight line ; 

 and that of an equation of the second degree is a conic 

 section. 



29. The position, the figure, and course of a curve, 

 are known, when we can determine the points through 

 which it passes. This may be done, by supposing one 

 of the co-ordinates, as x, to have all possible successive 

 known values, and by determining from these the cor- 

 responding values of y. Let the former be a, a', a", &c. 

 and the latter b, b', b", &c. then the points whose equa- 

 tions are x=a, y=b; xzza', yzzb' ; x=a", y=b", &c. 

 will all be in the curve which is the locus of the equa- 

 tion, and may be readily found. 



The determination of the points of a curve in this 

 manner requires the resolution of its equation, which 

 cannot in every case be effected ; as, however, we can 

 always find approximate values of the roots, the finding 

 of the points of a curve is subject to no other difficulty 

 than the labour of calculation. 



30. As the co-ordinates of a curve admit of being the 

 representatives of the roots of an equation, the pro- 

 perties of the latter will also belong to the former. It 

 is upon this principle that the modern analysis has been 

 applied with such success to the investigation of the 

 various affections of geometrical figures ; and these, in 

 their turn, have been employed in illustrating some of 

 the more intricate theories of pure analysis. 



31. It is a fundamental proposition in analysis, that 

 an equation of any degree may have as many real roots 

 as there are units in the exponent of the highest power 

 of the unknown quantity : hence it follows, that if the 

 equation of any curve be resolved, so as to express the 

 value of one of the co-ordinates in terms of the other ; 

 corresponding to any given value of the latter, the 

 former may have as many values as there are units in 

 its highest power contained in the equation ; and this 

 will be true, whatever angle the co-ordinates make 

 with one another. This is an important proposition in 

 the theory of curve lines ; and from it we learn, that 

 a straight line cannot cut a line of any order in more 

 points than there are units in the number expressing that 

 order. That this is true of the straight line and conic 



Curve 



Lines. 



