Ill 



TIFKt'UY <>K EQUATIONS. 



THKOUY OF EQUATIONS. 



212 



S. A pair of ooupln to the aria of x (v, f) (T, /), the total moment 

 i'f which i ry t:. 



> pair of couples to the axil of y (x,o) (t, m), the total moment 

 Mcli ii xj w. 



-\j.|.ly this to every point in the system, and l.'t 2\ itand for 

 x, + x, -f , *c., and ao on : hence it appears that the whole of the force* 

 are equivalent to force* 2x, Jr, 2z, applied at the origin in the 

 directions of x, y, and t, together with couples in the planes of xy, y:, 

 vc, of which the moment* are 



a (M-ly), 3 (ly-Ti), 3 (xt- ix). 



Let A= 

 B = 

 c-Sz, 



vc) 



sy) 



(articular points relative to equations of the first four degrees 

 are as follows : 



1. The expression of the first degree can be reduced to the form 

 + t> ; it vanishes when z= -6:0, and has only this one root. 



And ax + b is of the same sign as a or not, according as x is greater or 

 than the root. 



2. The expression of the second degree is more important. It can 

 always be reduced to the form ax* + bx + c, and its properties are best 



leveloped by transforming the preceding into 



4ac-V 



Then it appears that all the force* can be reduced to one force, v, 

 acting at toe origin, making angles with the axes whose cosine* are 

 A : T, B : v, c : v ; and one couple having a moment w, and whose axis 

 makes with the axes of co-ordinates angles whose cosines are L : w, 

 ll : w, H : w. But when there is equilibrium, both the force and the 

 moment of the couple must vanish, for the single force cannot equili- 

 brate a couple. Consequently the conditions of equilibrium are v = 0, 

 w = 0, which give A = 0, B - 0, o = 0, L = 0, X = 0, S = 0, the six well-known 

 conditions of equilibrium. 



Th<> forces will have a single resultant when v falls in the plane of 

 the couple whose moment is w ; that is, when the direction of v is at 

 right angles to the axis of the couple. This takes place when 

 .\r. + DM + CN = o, a well known condition. 



(For further information, we may refer to Poinspt's EUmrnt de 

 Statique ; or, in English, to Pratt's Mathematical Principles of Natural 

 PhilotnpTiy ; or Pritchard's Theory of Couplet.) 



THEORY OF EQUATIONS, t'nder this term is expressed all 

 that part of algebra which treat* of the properties of rational and inte- 

 gral functions of a single variable, such as ax + b, atf+03+t, 

 ax t +bx s +ex + e,aDd so on; o, 6, c, Ac., being any algebraical quanti- 

 ties, positive or negative, whole or fractional, real or imaginary. 

 Unless however the contrary be specified, it is usual to suppose these 

 co-efficients real, not imaginary. 



The great question of the earlier algebraists was the finding of a 

 value for the variable which should make the expression equal to a 

 given number or fraction : as what must .r be BO that Sor + 2x may be 

 .t*a? + 6x may be 40, and so on. In modern form it would be 

 asked what value of x will make 8x=+2x 11 = 0, or x>-xi+te-40 

 = 0, and ao on. To find values of a variable which should make an 

 expression vanish, or become equal to nothing, was then the first desi- 

 deratum ; and these values are now called routt of the expressions. 

 Later algebraists made the finding of these roots subservient to the 

 discovery of other properties of the expressions. 



The Hindu algebraists communicated to the Arabs, and through 

 them to the Italians, the complete solution of equations of the first 

 and second degrees. The Italians added the solution of equations of 

 the third degree, and of the fourth imperfectly. These last two 

 degrees have been completed in more recent times, so that it may be 

 now said that the equations of the first four degrees have been com- 

 plrtely conquered: that is to say, having given the equation 

 ox* +* +r +M +/ 0, an algebraical expression can be found, hav- 

 ing four values, and four values only, and being a function of a, b, c 

 t,f, which being substituted for x on the first side of the equation 

 shall make that first side vanish. But the student would look in vain 

 through the books of algebra to see this expression : it is both pom 

 plicated and useless, and it is more desirable to indicate how it is to 

 be found, than to find it, 



The equation of the fifth degree was attempted in all quarters, with 

 out success : means were found of approximating to the arithmetic* 

 value of one or another root in any one given equation ; but never a 

 definite function of the co-efficients which would apply in all cases 

 A proof was given by Abel, in Crelle's Journal (reprinted in his works) 

 that such an expression was impossible, but this proof was not gene 

 rally received : it was admitted by Sir W. Hamilton, who illustrat 

 the argument at great length in the 'Transactions' of tin- l.oy.i 

 Irish Academy, vol. xviii, part ii. ; but the singular complexity of th< 

 reasoning will probably prevent most persons from attending to tb 

 subject. We do not mean in this article to enter into the history o 

 tin- theory of equations, but only to place its general state befor 

 the reader by an exhibition of the principal theorems, withou 

 proof. For works on the subject we may refer as follows : Button 

 Tract*,' vol. ii., tract 33, which contains a full account of the earlie 

 algebraists ; Peacock, " Report on certain Parts of Analynix," in th 

 ' Report of the Third Meeting of the British Association ;' or th 

 recent works of Murphy, Young, or Hyiners; all of wb 

 and written on such different plans Uiat any one who makes 

 particular study of the subject will find it advantageous to consul 

 them all. In French the standard works are those of Btidan 

 Lagrange, and Fourier, which however all treat of particular topics 

 the algebraical treatises of Bourdon and Lefebvre de Fourcy treat i 

 more generally. 



4o 



There are three distinct cases, according as 6* U greater than, equal 

 to, or less than, 4ac. 



When l>" > 4c. the expression ax- + Ix + c has two real and differing 

 roots, contained in the formula* 



- b V (6* - 4ae) 

 2o 



and has always the same sign as a, except when x lies between those 

 oots. Every change of signs in passing from a to ft and from A to 

 c indicates a positive root, and every continuation a ne. 

 and when one root U positive and one root negative, the posit i 

 negative root is numerically the greater, according as (o, ft) shows a 

 change or continuation. When x~ ft : 2o, the expression is at 

 its numerical maximum between the two roots, its then valu< 

 (4acli*) : 4a. 



When 6 J = 4a, the expression ax? + ox + c is a perfect square with 

 respect to x, and absolutely so if a be a square. The two roots 

 become equal, and each equal to ft : 2o. The expression now 

 differs in sign from a. 



When fr*<4<K:, the two roots become imaginary, the expression 

 always has the sign of a, and is numerically least when x= - ' 

 teing then (4ac - ft 5 ) : 4o. 



3. The equation of the third degree (or cubic) has been separately 

 considered in the article IRUKDVCIBLE CASB. 



4. 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 that of a cubic. The various modes are dis- 

 tinguished by the names of their inventors. 



Ferrari. Let x 4 + ax* + b + e 0. This can be transformed int" 



make the second side a perfect square ; that is, find the va' 

 t> from ft* = 4 (r' c) (2i> o), or 



8* 4aiP 8ev + 4ae I* = ; 



the extraction of the square root then reduces the biquadratic to a 

 couple of quadratics. 



Det Cartet. Let x 4 + ax* + bx4-c=( L X* + \/p.x+f) (x* ^p-x+g), 

 which gives 



g + f-p = "1(3 f) Vp = jfy = e, or 

 p* + 2ap* + (a 1 4c) p - 6 = : 



find a positive root of this equation (it certainly has one), and fr 

 find g and / ; then the roots of x" + \Sf.x+f*Q, and 3? /;>. z + 1/ = 0, 

 are those of the given equation. 



Thomas Simpson gave a modification of Ferrari's method, and 

 Euler one of that of Des Cartes. (Murphy's Theory of Equations 

 (' Library of Useful Knowledge'), pp. 54, 65.) 



The theory of equations of all degrees is to be divided into t\\ 

 tinct parts; the numerical solution, and the general i ( the 



roots and the expressions themselves. The numrriVa' nilulion niii.-t U' 

 carefully distinguished from the general tolntion ; .the former term 

 applying to any mode of approximating to a single root, the latter to 

 any mode of exhibiting a general expression for the roots. We shall 

 y the general properties of the roots : the expression in ques- 

 tion 1 icing <px, or 



1. If r be a root of tfw, or if <t>r- 0, then <px is dirisiSle by .r r, nn.l 



'ticnt is another such expression of the (n l)th degree. 

 root of which is also a root of ^r> ami ovi-ry immbrr wb'. 

 root (r cxcepted) is not a root of <. Hence ^r cannot liavi 

 roots than it has dimensions, or cannot b > <>i roots. 



2. When the expression ^c is divisible by (.< rr , ii i- K.ii.1 to have 



m roots each equal to r; and when this is the case, the sulti>u 



tion of r + y for x would give an expression in which y" is the lowest 



i if y. 



very expression has as many roots as it lias dimcnnionn. This 



Thin formula ihonld be committed to immory, and quadratic equation* 

 tlwnjrn tolrcd hy It. No'.hinn l mnrt imimlng than the vitality of thr eld 

 method of romplctlnir the nqiiare and rxtncting the root in c-vcry particular 

 case. No doubt a itudrnt ihould hare Mine training in thin laiUmentioned 

 proceu ; but hit ultimate method thonld be tbat of remembering, once for all, 

 the formula In the text. 



