J 1 



340 



T II K 



algebraical treatises of Bourdon and I.cfcbvre dc Fourey 

 Uke it more generally. 



The particular points relative to equations or the first 

 four degrees are as follows : 



1. The expression of the first decree can be reduced to 

 the form ar+b ; it vanishes when ,r= b:n, and has 

 only this one root. And r+ft is of the same sign as a or 

 not, according as T is greater or less than the root. 



i In- expression of the second degree is more import- 

 ant. It can always be reduced to the form ax* + t>j- + r, 

 and its properties are best developed by transforming the 

 preceding into 



There are three distinct cases, according as 6* u greater 



qual to, or lew Unit: 



\Vhcn A' > 4<ir, the expression tu '-\-lu-+r lias two real 

 and Jittering roots, contained in the formula* 



2a 



and has always the same sign as a, except when x }'. 

 tween those roots. Every change of signs in passing from 

 a to b and from b to c indicates a positive root, and every 

 continuation a negative root : and when one root is posi- 

 tive and one root negative, the positive or negative root 

 is numerically the greater, according as (a, A) sh- 

 change or continuation. When x= b:1/i, the expres- 

 sion is at its numerical maximum between the two roots, 

 its then value being (4ae 6 f ) : 4a. 



When r = 4>/r, the expression ax?+bx+c is a pcrfeei 

 square with respect to a; and absolutely so if a be a square. 

 The two roots become equal, and each equal to p:2fl. 

 Tlii> expression now never differs in sign from a. 



When 6'<4o'-, the two roots become imaginary, the 

 expression always has the sign of a, and is numerically 

 least when j-= "-6 : '2i, beim: then (4<if 6*) :4/i. 



:j. Thi> equation of the third degree (or cubic) has been 

 separately considered in the article IRKEDVCIUI.K CASE. 



J. Nothing belongs particularly to the equation of the 

 fourth degree (or biquadratic) except the recital of the 

 various modes in which the solution is reduced to 1 

 a cubic. The various modes are distinguished by the 

 names of their inventors. 



l-'-'rniri. Let ,r 4 +.<* + bjc+c = Q. This can be trans- 

 formed into 



(a? + V')' = (2r - a) T* - bi- + r- - r : 

 make the second side a perfect square : t i from 



b' - 4 (r 1 - c) (2;; - a), 

 or Sv* 4ac* See + 4<ir b 1 



tlir extraction of the square root then reduces the biqua- 

 dratic to a couple of quadr 



7>.v ( .;;/<*. Let .T' + cur 1 + bx + C = (a* + >Jp. <*+/) 

 (& - t/p.-r + i' , which gives 



g+f-P = a,(g-f) >Jl> = '>'.fe = <. 

 or ;/ + -Ziji 1 + ;'(i - 1 ,:/ - b- = 0: 



find a positive root of this equation (it certainly has one). 

 and from it find g and/; then the roots of X s + +fji. .1 -f / 

 = 0, and x* >Jp ,x -|-/=0, are those of the given equa- 

 tion. 



Thomas Simpson gave a modification of Ferrari's 

 method, and Kulei one of that of Des Cartes. (Murphy's 

 Theory of Equations i /.. U. A. . pp. 51. 



Tlie theory of equations of all degrees is to be divided 



into two distinct parts; the numerical solution, and the 



general properties of the roots and the expressions thcni- 



i. The iiumi'rirnl solution must be carefully distin- 



(1 from '' V miluliiin : the former term 



applying to any mode of approximating to a single root, 



the latter to any mode of evhiliitiiv.- 



for the roots. tVe shall begin by the general properties of 

 the roots: the expression in question being <{>.r, or 



a^c +n,.r + a,x +....+ rt-i- r + " 



1. If r be a root of <ftx, or if $r=0, then <j>f is dh i 

 hy r r, and the quotient is another such expression nf 



' - 1 th degree, every root of which is al.-o a . 

 4>r, and every number which is not a root (r except 



* Thfo frrmnU houM be mmmlUM to memory, And quadratic equation) 

 !) Ml*d bv H. Nnihuut U mom iraiutng than U>r vitality o! 



*^ ei ^ *rf eplttn< UM iqun and cxtr*cliii the root in 



. So doubi a Xwlrat ihouU have torn training hi thu lart-ni. 



*MM 



! f 



: b.1 h ali4B.. 



fWMft OI M ^ 



aboald br thai of riDro.bctin|. one* for all, 



not a root of </ i/,j canivit have more roots 



than it h:w dime! ft than n roots. 



J. \Vhenthe < - r)", it is 



said to have m roots each equal to r ; and when tlii- 

 case, the substitution of r-f y for x would give an expres- 

 sion in which y is the lo\\. of y. 



it. I ' H ii lias dimen- 



sions. Th; im is one which has only lai 



been demonstrated in elementary works, and we *ha!l 



ittion with tin' view of extending the 

 knowledge of a remarkable theorem of M. Caucliy. which 

 is just Mii-li a :': victory over tile (liflieur 



finding how many \\ leral he Intv. 



limits, a Stum'* theorem is relatively to real roots. \\ V 

 shall assume the extended algebra explained in NKGATIVK, 

 &c. 



Take any rectangular axes, and let r and y be the co- 

 ordinates of a point, and consider the expression tf> (r 

 +yV- 1) which can be reduced to the form P + Q ,/- 1, 

 where P and 1) is each a real turn ( 

 the point move round the contour ABCD in 1! 

 direction of revolution, and let the fraction P : Q be formed 

 for all the points in the contour (or a sufficient nun 

 in succession. Examine every case in which P : Q passes 

 through Oand changes sign : let it change sign from -f- to 

 . A times and from to +, / times. .\e\t. whcin , 

 and / have such values that .r + y V 1 is a root of the 

 expression, or <j> (r-f-y / 1) = 0. let the point whu 

 ordinates are .r and y be called a radical point of the 

 expression. The theorem to be proved is as follows: the 

 number of radical points which lie irithin the contour 

 AI(( 'D is ^ (A -/i. neither more nor fewer. It mi 

 understood that the contour is so taken that no radical 

 point lies upon it. 



Take nny point P within tne contour, and round it draw 

 an infinitely small contour, round which a point is to be 

 first carried. Four cases arise : neither P nor Q \;n 

 within nor on this contour ; P vanishes, but not Q ; Q 

 vanishes, but not P; or both vanish. 



If neither P nor Q vanish, there is never change of 

 sign in either(for being integral functions, they cannot be- 

 come infinite for any finite values of x and y). and the 

 theorem is true for the infinitely small contour; for ft and 

 / are both = 0, and there is no radical point. 



If P alone vanish, the curve P = ''remember that P is 

 a function of T and y) passes through* the contour two or 

 somu other even number of times. The fraction P : Q 

 may vanish and change sign as often as the curve passes 

 through the infinitely small contour: but there must 

 be as ina ny changes from -t- to as from to +. For 

 suppose P to be positive at the commencement of the 

 revolution; it is therefore positive at the end. \V:itc 

 down the sign + twice, ana between it write any signs 

 whatever, as 



+ + + - + -+; 



it will always be found that -I and + occur equal 

 numbers of times. Hence the theorem is true in this 

 case; for A = /, and there is no radical point. 



If Q alone vanish, the curve Q = pa> - tin- 



point, and everything is as in the last, except that P : Q 

 always becomes infinite when it changes sign. Hence 

 the theorem is true ; for k and / are each = 0, and tin 

 cal point. 



Lastly, let there be a radical point within, hut not on, 

 the infinitely small contour: which may be supposed to 

 contain not more than one distinct radical point. Let Z be 

 the radius vector drawn from the origin to the point of the 

 contour whose co-ordinates arc x and y ; so that, using the 



Prarcntlhc curve P=0 from touching the contour liy (mlarptng the 

 a little U neCMftrjr. 



