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 B II A. 104 A L G E where there mud be always one change of the figns, “ another that d^all have its roots greater or lefs than fmce the fad term is pofitive and the lad negative. And “ the roots of the propofed equation by fome given difthere can be but one change of the figns, fince if the ad “ ference.” 3 1 term is negative, or a--b lefs than ct the third mud be Let the equation propofed be the cubic x —px --qx negative alfo, fo that there will be but one change of the —r=o. And let it be required to transform it into anofigns. Or, if the fecond term is affirmative, whatever ther equation whofe roots ffikll be lefs than the roots of the third term is, there will be but one change of the this equation by fome given difference (?), that is, fupfigns. It appears therefore, in general, that in cubic e- pofey=zx—e, and confequently x—j+e • then, indead of quations, there are as many affirmative roots as there are x and its powers, fubditute j+e and its powers, and changes of the figns of the terms of the equation. there will arife this, new? equation. There are feveral confeftaries of what has been al(•^ll3+3^ >4- Pe<?5'> ' fyl +3<2pe} ready demondrated, that are of ufe in difcovering the " ~ Z >=o, roots of equations. But before we proceed to that, it + V+ie C will be convenient to explain fome transformations of equations, by which they may often be rendered more whofe roots are lefs than the roots of the preceding efimple, and the invedigation of their roots more eafy. quation by the difference (e). If it had been required to find an equation whofe roots be greater than thofe of the propofed equation by Chap. XVI. Of the Transformation of E- ffiould the quantity (e), then we mud have fuppofed /=x4-<%. quatiom ; and exterminating their interme- and confequently x==y—e, and then the other equation diate Terms. would have had this form.% {B)y'—yy'+y y— e' We now proceed to explain the transformation of e— Pf+lpey—pe'C —q quations that are mod ufeful: and, fird, “ The affirma+ qy—qe r “ tive roots'of an equation are changed into negative “ roots of the fame value, and the negative roots into “ affirmative, by only changing the figns of the terms If the propofed equation be in this form x'+px'+qx “ alternately, beginning with the fecond.” Thus, the 4-'—o, then, by fiippofing x+e=y, there will arife an eroots of the equation x4—x3——30=0 are quation -agreeing in all refpetts with the equation {A), +1, -f-2, -H3> —5 5 whereas the roots of the fame e- but that the fecond and fourth terms will have contrary quation having only the 4figns3 of thel fecond and fourth figns. terms changed, viz. x 4-x —i9x 4-49**-—30=0 are And by fuppofing x—e—y, there will arife an equation agreeing with (i?)'in all refpects, but that the fecond —i»To —2, —3, +5underdand_ the reafon of this rule, let us afiume and fourth terms will have co'ntrary figns to what they (Z?). an equation, as x—aXx—l>Xx—tXx—dXx—e, See. =0, haveThein fird of thefe fuppofitions3 gives this equation, whofe roots are +a, +b, -fc, +d, See. and another, having its roots of the fame value, but affedted with contrary figns, as x+tiXx-ftXx+cXx+dXx+s, See. =0. 4- py1—2pey +pe1 (Xf0' It is plain, that the terms taken alternately, beginning 4qy -qe from the fird, aref the fame in both equations, and have +r J the fame fign, ‘ being products of an even number The fecond fuppofition gives the equation, “ of the roots;” the produd of any two roots having the fame fign as their produd when both their figns are (Z>)j34-3<9-i4-3<?yF <r3^ changed; as-faX——aX+t. + py'+irpy+pt'C-n But the fecond terms, and all taken alternately from + qy+q? them, becaufe their coefficients involve always the pro+ r )C duds of an odd number of the roots, will have contrary The fird ufe of this transformation of equations is to figns in the two equations. For example, the produd ffiew “ how the fecond (or other intermediate) term may of four, viz. abed, having the fame fign in both, and “ be taken away out of an equation.” one equation in the fifth term having"a^X-l-f, and It is plain, that in the equation (d) whofe fecond the other hbcdX—e, it follows, that their produd abede and confequently mud have contrary figns in the two equations : thefe two term is je—pXy', if you fuppofe equations therefore that have the fame roots, but with 3«'—p=o, then the fecond term will vaniffi. contrary figns, have nothing different but the figns of the In the equation (C) whofe fecond term is —Ze+pXy*, alternate terms, beginning with the fecond. From fuppofing the fecond term alfovaniffies. which it follows* “ that if any equation is given, and Now the equation (yZ) was deduced from x3—px*+ u you change the figns of the alternate terms, begin- qx—r—o, by fuppofing and the .equation (C) 2 “ ning with the fecond, the new equation will have roots was deduced from x3-j-pxy—x—e: +qx-{-r=o; by fuppofing “ of the fame value, but with contrary figns.” From whidh this rule may eafily be deduced for It is often very ufeful “ to transform an equation into exterminating the fecond term out of any cubic equation. Rule.