= 2x4 = 4.2003 = 4.22 = 5x 10x ) .1 x 440.02656 *.0141 ; whence x = 8.4 + .0131 = 3.4141, 3103.55781 nearly. Taking 8.414 and 8.415 for the numbers, we have 42170.867905346409824 = 206 42195.935727527009375 10023.976207593632 = 10028.74241890125 1787.017385832 1787.654620125 283. 181584 283.2489 42.07 42.075 54307.113082772041824 54337.654666553259375 54337.65465655325 8.415 54337.65466655325 53307.11308277204 8.414 54321 30.54158378121 : *.001 :: 16.65466655325 : 000545 and X=r-z=8.415—.00054558.414455, very near. 5. Given (7xo+4x) +(20x4_10x)=28, to find an approx. imate value of x. Ans. 4.510661. By trial x is found to be between 4 and 5. Let these two numbe taken for the first value; then Ist Sup. 8 ( 733 + 4x) = 9.916 2d Sup. 21.213 31.129 24.73 4 24.73 Therefore 3.27 6.399 : 1 :: 3.27 : =.51; whence 6.399 4.51, nearly. Assuming x=4.51 and 4.52, and repeating the operation, we get x=4.5106, the root, nearly. 6. Given „{144x?—(22—20)} tw{196c?—(+*+24)} = 114, to find a near value of x. Here the root is found, by a few trials, to be rather more than 7; let, therefore, 7 and 7.1 be assumed for the numbers ; then 1st Sup. 7056 144m2 = 7259.04 2d Sup. 4761 (22+20) 4957.5681 ♡ 2295 47.906 12302.4719=47.977 9604 196x2 9880.36 5536.8481 V4343.5119=65.905 47.906 47.977 113.882 7.1 114. 65.383 65.905 113.289 7 113.882 113.289 results 113.882 .593 .118 : .1 x .118 .0118 =.2; whence x=7.12, nearly. By assuming .593 .593 or : .1 84. 7.12 and 7.13 for the numbers, we get 2 7.12209461, the answer, nearly. 1. Given 29+*+x==100, to find an approximate value of x. Here it will be found, by a few trials, that the value of x lies between 4 and 5. Hence, by taking these as the two assumed numbers, the opera. tion will stand thus : First Sup. Second Sup. = 4.264 + .0004299 4.... 5 4.2644299, very nearly, 16....22 25 2. Given (72 15)+XNI 64.... 23 125 =90, to find an approximate 84 Results 155 value of x. (155....5....100 Mere, by a few trials, it will .4 84 be found that the value of x lies between 10 and 11 ; which let, 71 : 1 :: 16 : .225 therefore, be the two assumed And consequently =4+.225 4.225, nearly numbers, agreeably to the di rections given in the rule. Again, if 4.2 and 4.3 be now Then taken as the two assumed num First Sup. Second Sup. bers, we shall have 25 84.64 First Sup. (7x?—15) Second Sup. 31.622 XNX 36.482 4.2 4.3 17.64 18.49 56.622 Results 121.122 Hence 74.088........79.507 121.122 11 .. 121.122 56.622 10 90 64.5 : 1:: 31.122:.482 And consequently 11. 6.369 : .1 :: 2.297 .582-10.518. : 036. Again, let 10.5 and 10.6 be And consequently t=4.3.036=4.264, nearly. the two assumed numbers; Then Again, let 4.261 and 4.265 First Sup. Second Sup. be the two assumed numbers. 49.7025 (2--15) 55.830784 First Sup. Second Sup: 34.0239....XNX.. 34.511099 4.264 4.265 83.7264.. Results .. 90.341883 18.181696 x2 18.190225 Hence 77.526752 77.581310 90.341883.. 10.6 .. 90.341883 83.7264 .. 99.972448 ...: 10.5 ..90. 6.615483.. .1 :: .341883 100.036535 4.265 100 : .0051679, and consequently 99.972448 4.264 .99.972448 10.6 .0051679 = 10.5948321, very nearly. .064087: .001 :: .027552 :.0004299, and consequently, ....4.2..100 Then . 2m-1 2m-2 2m-1 Of Reciprocal Equations. 104. Although no general method has hitherto been discov. ered for the resolution of equations higher than those of the fourth power, there are, notwithstanding, some particular equations, of all orders under the 10th, which, on account of the relations subsisting between their coefficients, may be solved by the rules that have been already given for the first four orders. This is particularly the case with what have been usually called reciprocal equations, which are such that the coefficients of their terms, taken from the beginning of the equation, are the same as those of the corresponding terms, taken from the end, with the same signs; or which remain of the same form when the recipro 1 cal of the unknown quantity, or, is substituted for x ; except , that the terms are then reversed. Thus the equations dat pa2n+qzem?...+qx?+px+1=0, and 22n+2 +pam tazko ....tqz*+px+1=0,may always be transformed into others of a degree which is denoted by half the exponent of the highest pow. er of the unknown quantity, if it be an even number, or by half that exponent when it is diminished by 1, if it be an odd number, the method of resolving them, as far as to the 9th order, inclu. sively, being as follows. To this we may add, that the nature of these equations consists, as abovementioned, in their not being changed by substituting for x; from which it follows, that if a be any one of the roots, its reciprocal å will also be a root; and as +1, or -1, is always a root of the equation, when the number of its dimensions is odd, it may be readily shown from these circumstances, that every equation of the 2mth or 2mth+lth order, can be reduced to another of the mth order. CASE I. When the index of the highest term is an even number. RULE I. If the equation be of the fourth power, as x+px9+ qx+px+1=0, find the two values of z in the equation, z+pz+ 2-2=, and let them be denoted by r and r' ; then the roots of the two quadratic equations, x-r1+1=0, and 22-r'x+1=, — —1 will be the four roots of the proposed equation. 2. If it be of the sixth power, as zo+pa+qx+rx+qx+px +1=1, find the three values of z in the equation, 28 +pz? +(2-3)z tr2p=0, and let them be denoted by r. r',p'; then the roots of the three quadratics 2-rx+1=0, 22-r'x+1=0, and a?—-"x+1=0, will be the six roots of the proposed equation. 3. And if it be of the eighth power, as 28+px?+q2m+rasts #rx+qxo+px+1=0, find the four values of z in the equation, z z+pzi+(9-4)2+(r—3p)z+s4219–1)=, and let them be denoted by r, r', r", ". Then the roots of the four quadratic equations. -rx+1=1, 2mpx+1=0,2~q"x+=1, and 22—p"'x+1 =1, will be the eight roots of the proposed equation. CASE II. When the index of the highest term is an odd number. Rule I. If the equation be of the third power, as x+px' +px +1=1, where one of its roots is evidently —1, find the value of z in the simple equation z+p-=0, and let it be denoted by r; then the roots of the quadratic equation 22—r&+1=1, will be the other two roots of the proposed equation. 2. If it be of the fifth power, as 20+px*+quotquopx+1=1, where one of its roots is also —1, find the two values of z' in the equation +61)2+2---=0, and then let them be denoted z? by r, and r?; then the roots of the two quadratics —r&+1=1, and 2—r'3+1 = 0 will be the other four roots of the proposed equation. 3. If it be of the seventh power, as x+px®+226 +ræ*+rast qu'+px+1=1, where one of the roots is -1, as before ; find the the three values of z in the equation 2+(-1)2+(9-7-2) + rpma+1=1, and let them be denoted by r, r', zo"; then the roots of the 3 quadratics 22-rx+1=0, x_r'x+1=0, and 22_7"x+1=1, will be the other 6 roots of the proposed equation. 4. And if it be of the 9th power, as 29+px8+qx?+r2@+sx-+sx"+r2i+qx+px+1=1, where one of the roots is 1, as in the former cases, find the four values of z in the biquadratic equation 2+(p-1)2 + (9-2–3)22 + (r-9-2p+2)z+s~r-a+p+1 '; then the roots of the four quadratics *—rx+1=1, 2—r'x+1=1, 2—7"x+1=1, and 22—p!"x+1 =1, will be the other eight roots of the proposed equation. Note. If an equation of this kind be of an odd number of dimensions, or if the middle term of one of an even number of dimensions be wanting, the same rules will hold when the signs of the terms, taken from the beginning, are + and — alternately. Thus 7?-px?+px-1=0, and 24-px+px-1=0, are reciprocal equations, like those above given, except that + 1 is now one of the roots, instead of -1. 1. Given x +4.x2—192? +4.x+=0, to find the four roots of the equation. Here p=4 and q=-19; hence, by Case 1, =, -9 , be , E S Therefore the four roots of the proposed equation are *+^5, 15, - tin 5, and 115. 2. Given 11x4 +172+1724-11x+1=1," to find the five roots of the equation. Here p=-11, and q=17; hence, by case 2, 2+1)=l+pma becomes z_122=-27, where the two values of z are 9 and 3; x-9x=-1, and x_3x=-1, from the first of which equations x=+IN77, and from the second ==*EIN5. Therefore the five roots of the proposed equation are –1, +177, 177, +1N5, and 25. 105. If the coefficients of either of the orders of, equations, mentioned in the rule, be in part literal, and so constituted as to render the different terms homogeneous, its roots may be determined by means of the simple substitution of a new unknown quantity, as if they were entirely numeral. Thus, let there be taken, as an equation of this kind, x4 + 4ax® -19a*x+4dor + a=0, where the indices of the given and unknown quantities, when added together, are the same as those of the first and last terms. Then, by putting x - az, we shall have afz4+ 4a'z3—19a*z? +4a'z+a* = 0, or, dividing by at, z* +428 – ** --19z++4z+1=0; which equation, like the first of those given above, will have for its roots +115, 45,- + 5, and -N5; and the four roots of the proposed equation are ali + 5), (5), al + 5), and al-1-5). An homogeneous equation, containing only two letters, a and 2, is of the general form com tpaxm-+qaʻzmatraemmit....+kam-ixtla"=0, where p, q, r, k, l, are the numeral coefficients. 1. Given x*—15x+38x2—15x+1=0, to find the four roots of the equation. Ans. +5, and 6£35. Here p=-15, q 38, we have z_152=_36, by case 2; Z=+15=12 or -3. Hence we have 2–12x=-1, and is + —3 2 3.x=-1; .. in the first we have x= -6£135, and in the second case we have x =ŁİN 5. 2. Given x*—4axo+5aʻx_4a xt-a=0, to find the four roots of the equation. Ans. a +15), and a(LEJN-3). Let p=44, q=5; then z2_4z=2–5=-3; 2=3 or 1, and we have xé—3y=-1, and a=iN5. Again, 2 - x=-1;.: x=iİN-3; consequently we have a +i5), and ali+N-3). 3. Given 20–214+372+37x—21x+1=0, to find the five roots of the equation. Ans. -1, +5, and 19+1357. Here p=421, and q=37; then by case 2, 22-222 '. z=11+8 = 19 or 3, and .. y:—19x = -1; then 193357. Again 2–3x=-1, we find x=i+15. 106. Equations of this kind, in which the given and the un X = -57 ; |