impossible; except e", ", "", &c. which are each equal to unity.* For instance; I. Let a be one of the impossible roots of the equation y3-1=0, then 1, a, a'; or 1, a1, a3; or 1, a2, a"; or 1, a1o, a"; &c. &c. ad infinitum, are so many different forms under which the three roots of that equation may be represented. A II. Let be one of the impossible roots of the equation ys-1=0; then 1, 6, 8', s3, 8; or 1, 86, 87, 88, 39; or 1, 6, 8, 63, 615; &c.&c. ad infin. are different forms under which the five roots of the equation 35-10 may be represented; the several powers of in the former case, and of ẞ in the latter, being all impossible quantities. These sets of quantities are indeed only the same roots under different forms. For a13«=1× « =« } .'. 1, a1, as are of the same value as «5=«3 × «2 = 1 × «2=a') 1,«, a2; &c. &c. &c. value as 1, ß, ß3, ß3, ß*; ̃ ̄&c. &c. 60. Next, let 1, ç, ç2, &c. be the roots of the equation y'-10, and 1, σ, o3, &c. ... the roots of the equation y"-1=0, 7 and m being each of them prime numbers; and = let • If n be not a prime number, then other possible quantities besides e", ea", e3", &c. will be found among the terms of the foregoing series. For let n=7m, then g"=gl=1; and em√ √11 or +1, according as is an odd or an even number. In the same manner it may be proved that e'±1; and since I and m are each of them less than n gande must lie between 1 and g"; for the same reason, quantities of the form 1 will lie between g" and g2; &c. &c. H let the roots of the former équation be multiplied into each Now since the equations y'-10, and y-10, are each of them divisors of the equation y-10, the several noots of the equations y'-1-0 and y-10-must each of them be roots of the equation' yl-1=0; and since the remaining quantitiès noiteɔilqiclum ad; mom grieins Jouborg adt gnied giltig gimp&cit în reimbazd go2, tę"o",\&citewpgboto';p&eo&cat are also roots of that equation, it follows that the foregoing I'm quantities are the Im roots of the equation y-1=0.‡ From hence we infer, conversely, that "if n be a number arising from the product of two prime numbers, bánālm, "the n roots of the equation y"—10 will be the quantities " arising from the multiplication of the roots of the equation y'-1=0 by each of the m roots of the equation y′′-1=0." epson x moitsups sit to etoor 9.3 digim 66 This of course goes upon the supposition that "every equation has as many roots as it has dimensions" and no more; (see the end of this Chapter). being the product arising from the multiplication of each of 2 T In the same manner we might exhibit the roots of the equation xa—1=0, by means of the roots of the two equations x3-10, and x-1=0; and the roots of the equation 35-10 by means of the roots of the two equations x3-1=0,3 and x-1=0. By continuing the process, we might exhibit the roots of the equation 10510, by means of the roots of the three equations, x3-1=0, -1=0, and x2-10; and from what was shewn at the end of Art. 58, all this might be effected by the solution of equations not exceeding three dimensions. 10^ v noi!" If `n be "the" square of a prime number (mm), or y"—1=yTM”—1=0, then let y", in which case yTM"=x*; སྒྱུ་ ས 21mm hence hence yTM-x=0, and x-1=0, are two equations for determining the values of y and z. Let the m values of z in the equation on bed, Xite'&c,lon dogy and det them be substituted successively for z in the equation y"-x=0; then Jet 191qd broɔɔ2 bit ni mzode nood ybsolle end i &c. T =0 are m equations of m dimensions, for = determining १६ mm) values of y; ai ¿noir determining the n (or mm) y ງ mitsup by in the equation y a nois q i fred i fe + 9d lliw i For instance, if 1,,, be the three roots of the equation yil 10, then y➡0, y2+0, y→«2=0, will be three cubic equations for determining the nine values of goin the equation yil 0 Now (by Art. 56) the three roots of the equation: yi→ɑ#Ordforos, A, vein that Article correspond respectively to 1, a, a in this Article, and n=3,) ate Ja, aja, à Va, and the three roots of the equation 30, are a2, «Ya2, à «2; hence the nine roots of y the equation y=0, are vitnetp toonday of elon an£' doidw 1,«, « Ja, a Sha, a Tag Valga Sh **Va̸'odt of For the same reason, if 1, 8, 82, ß3, 8a be the five roots of the equation y-10, the roots of the equation 5 →→ 250 o will be T + 1, ß, In the same manner, having found the m roots of the equation y1=0 (m being any prime number whatever), we might write down the roots of the equation ya—1=0, where n is equal to the square of m. B, od modt On the general solution of Cubic Equations, oitsups nortups et at a hot vigne-boue bosusitede 62. It has already been shewn in the Second Chapter, that the cubic equation 3 − px2+qx¬r±ō, whose roots are 101 ora , 8, 7, may be transformed into another deficient in its second term, by substituting y++p y+p for x in the given equation; in which case the roots of the transformed equation 58 will be, a+p, ß+šp1v+ý p. Let apa; ß+}p=ß'; 44/thent÷p; p='p; p=p;} if therefore the roots of the transformed equation be known, the footstof the given equation will be known alsooit Hence the solution of a cubic équation complete in all its terms wilkube effected oif we can arrive at the solution of it in the form £9ærỏ breidt i 1 o ylovitogen noitsup bd to 21001 990 gt bus to 69. In the solution of the equation +qx-7=0 by Cardan's rule, the unknown quantity (2) is supposed to be equal to the sum of two other, unknown quantities (~+x); which being substituted for x in the given equation, we have 101 @2±(u+z)2=u3μ3u2z+3ux}+x}=w+3uz(w+12)(p231⁄2) (3 +q(u+z) nd I=0 -rS +9x= -T= plovito9q297 شاهه اعلام شماهه Now as another unknown quantity has been introduced into the equation, another condition may be annexed to its solu tion. Let this condition be, that 3uz+q=0, or z= in which case the transformed equation becomes 34 or q3 27 u |