a?^.p^^:l?iM^^^i^$'^ QA £18 V4- UC-NRLF $C Ibb Elfi ^^ NEW, SIMPLE, AND GENERAL METHOD OF SOLVING NUMERICAL EQUATIONS -BY THOMAS WEDDLE. ^ NEW, SIMPLE, AND GENERAL METHOD OF SOLVING NUMERICAL EQUATIONS OF ALL ORDERS. BY THOMAS WEDDLE, MATHEMATICAL MASTER, MR BRUCE'S ACADEMY, NEWCASTLE-UPON-TYNE. Sontron : PUBLISHED FOR THE AUTHOR, BY HAMILTON, ADAMS, AND CO., PATERNOSTER ROW; AND J. PHILIPSON, BOROUGH OF TYNEMOUTH. 1842. •.':;••:.: "•;..! The method of solving Numerical Equations, developed in the following pages, occurred to the Author early in 1839. He subsequently drew^ up the Paper which now issues from the press, and sent it as a communication to the Royal Society. It was read before that learned body in June, 1841, and they were pleased to transmit their thanks to its Author. The encouragement which he thus I'eceived induces him to lay the result of his enquiries in this important branch of Mathematics before the Public. Newcastle-upon- Tyne, March, 1842. LIST OF SUBSCRIBERS. Mr. Wm. Armstrong, Liverpool. Peter Barlow, Esq., F. R. S., Royal Military Academy, Woolwich. TLos. H. Bates, Esq., Woolsingham. Wm. Bennett, Esq., London. James Beman, Esq., Chipping-Norton. Wm. Bolton, Esq., Birmingham. David Burn, Esq., Busy Cottage, near Newcastle. Rev. J. C. Bruce, A. M., Newcastle. Two copies. Mr. George Bruce, Do. Messrs. E. & T. Bruce, Do. Two copies. Mr. C. S. Booth, Barnard Castle. E. Chamley, Esq., Newcastle. Wm. Crawhall, Esq., Allenheads. Isaac Crawhall, Esq., White House, Stanhope. Mr. Craigy, Crawcrook, near Wylam. Mr. J. J. Crawford, North Shields. Two copies. B. Donkin, Esq., Engineer, New Kent Road, London. Mr. Jas. P. Dodd, A. M., Trin. Col., Dublin, North Shields. Two copies. Mr. Ed. Dobson, West Matfen Delight. Mr. Wm. Donkin, Newcastle. Mr. Robt. Elliott, Wall, near Hexham. Mr. Peter Elliott, Lamesley. John Fenwick, Esq., Newcastle. S. Fenwick, Esq., Royal Military Academy, Woolwich. Two copies. Mr. Fairbairn, Horsley High Barns. Mr. Jas. 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W. Robinson, Newcastle. Miss Surtees, Stamfordham. Wm. Swan, Esq., Walker, Mr. Wm. Temple, Liverpool Mechanics Institution. Tioo copies. Mr. Ed. Temple, Corbridge. Mr. Thos. Temple, Beaufront. Mr. Thos. Thompson, Newcastle. Mr. Wm. Towns, North Shields. Mr. Tweddell, Wylam. W. S. B. Woolhouse, Esq., F. R. A. S., London. Two copies. Rev. Wm. Walton, F. R. S., AUenheads. Mr. Whinham, Ovingham. Mr. N. Whitfield, Cawn Wood. Mr. Thos. Winship, Springwell Colliery. J. R. Young, Esq., Professor of Mathematics, Belfast. NEW METHOD OE SOLVING NUMERICAL EQUATIONS. 1. The object of the present paper is to develope a new method of solving numerical equations. It is a process of great simplicity and elegance. And although, in comparison with existing methods, it is not without defects, yet I trust it will be found to possess some advantages of considerable importance. 2. We shall, in the first place, exhibit the methods of transformation which it will be necessary to employ. Let Aa;" + Ba;«-' + C.c"-2 +va^ + l3x + » = (X) be an equation of the «.th degree. If the roots of this equation be divided by R, the resulting equation or the transfonn• ^1 + A, n — 2' + A„_xr B„_2 B n — 1 K"' •"'+B„_i«^" Whence the transformed equation (Y) becomes •'"~' + C„_2 ^'""^ +y, a!'2 + ,3, *' + »=0. 4. When the significant part of r consists of a single digit, the addends in the preceding transformation need not be written, as they may be incorporated mentally with great ease. For the operation consists merely in multiplying the successive digits of a certain number by a digit, and adding to each product two other digits, viz., that which was carried forward, and that figure of the same number which stands immediately above the place where the unit's figure of the result is to be set down. 5. Since Aj =(!+»•) A A^ = {\+r)A^ &c. (See Art. S.) .•. Ajr=(l +r) . Ar A^r={l +r) . Ajj* A„_i»-=(l+r) . A„_, Bir=(l+r) . Br B^r=(l+r) . Bj/- &c. Now Ar, Ajr, A^r \-.i^'> Br, Bjr, B^r g _ ^ ; &c., which we shall denote by A', A\, A\ A'^ ij B', B'j, B'2 B'jj 2 J ^c., axe the addends employed in article 3, and it ia plain from the above values that they may be derived from each other by the process in article 4. It is also obvious (article 3) that A^=A + Ar + Ajr +A^_ir =A + A'+A', + A'„_i B„_,=B + Br + B,r +B„_2r =B + B' + B\ + B'„_2 &c. Hence instead of employing the process in article 4 we may employ the following : ABC y fi a. + A' +B' +C' +./ ^^ + A\ +B\ +C\ +y\ + A'„_3 + A'„_2 + B + B «— 3 B— 2 + C'„_3 Cjj 2 V2 + A'„_, B„-x K 6. We shall now proceed to to develope our method. Let us represent the given equation by F(a;), and the first of the succeeding transformed equations by F^{x^), the second by F^i^z)^ the third by F3(a;3), &c. ; also the limiting equations of F(x) by F'(x), F"(x), &c. ; those of F,(.. J by F',(;r,), F'\{x,), &c., &c. Let R be the value of the first significant figure of a root of F(ic), transform the equation by dividing its roots by R, or, which amounts to the same thing, by substituting Ux^ for «, one value of Xj will obviously be between 1 and 2, put »'j for the value of the first significant figure of the decimal part of x^, transform the equation in (1 + /-j) j;^, assume r^ for the value of the first significant figure of the decimal part of x^, and transform the equation in (l+rj) «3, &c. 7. In effecting the preceding transformations we shall not be obliged to extend each column to an indefinite num- ber of decimals, but each must be restrained to that number (5') of places, which is requisite to give the root to the extent required. Suppose that in continuing the transformations we arrive at an equation F^ (.j;^ ) (wliieli denote by Aic" +Bx"~^+Cx"~^ +yx^ +3 ;?;_ + «= 0) whose coefficients are so related to each other, that , Fm (1) . when it is transformed in |l — — — -— | «m + 1 ^'^ addends (see article 3), when restricted to q places of deci- m \ ' Fm (1) mals, are constant in each of the columns, then the root of F^ {x^)\% 1 — ^, — -;— r-. In order to prove this it will obviously be sufiicient to show that the root of the transformed equation F^ ^ xif'm +1) F^(l) is unity. Put — -= r-=«, then since the addends are constant, and as there are n of them in the first column, («— 1) in the second, (m— 2) in the third, &c., we shall have + {y + 2y«}a-'^ + i+ {/3 + ^^^} ■r^+i+«. = A<^,+B4-;+C^»-; +y^^^^+;3^.^_^^^^. + ax^+,{,^.U.;-;+,73r.B..:-%«^=2"C<-^ +2y^^+,+/3}. F^(l) Let^^+, = 1, ...F^^,(l) = F^(l)-j,^^F'^(l) = F^(l)-F^(l) = 0. F^(l) Hence a root of F^+ i (^^+ j) is unity, and consequently a root of F^ (x^^ ) ia i_— ^^-^-. We shall, there- fore, obviously obtain the following development of the root of the original equation F (x), viz. : .=E{1..,} . {1...} . {1..3} {!...-.} . {l-^'^,y}. 8. It is obvious that it would be a great acquisition could we discover an easy method by which r^, r^, r^, &c., may be suggested, nor is this desirable object unattainable, ForF^(^^) = F^ (1+^^_1)=F^(1) + F'^(1).^^-1 + ie^—l 2 P Now whatever be the value of p, {x — 1) is a decimal, and after the first or second transformation, a very small one too, it is therefore obvious that — —, . . - will generally be sufficient to suggest at least the first significant figure of {x — 1), which figure is the value of r . 9. In the preceding investigation, we have supposed the significant part of rj, r„, r^, &c., to consist of only one digit, but this restriction is not necessary as they may evidently consist of any number of digits whatever. f ^P (^) > 10. If any of the transformed equations, F / ), bo such that were it transformed in |1 — —, . | x ^ j, and each column supposed to be restricted to a certain number (X;) of places of decimals, the addends would be con- Fp 0) stant, then it follows from what has been said in article 7, that x =1 = — ~-^, where F (1), F' (1), must be ^ ^'p (1; >' -^ restrained to k places, and the quotient resulting from the (contracted) division of the former, by the latter carried to that number (/) of decimals, which can be found without annexing cyphers to the dividend F„ (1), hence as F« (1) 1— -jT, — — — gives the value of x„ true to about/ places of decimals, it may sometimes be convenient to make r = V F;, ( 1) Fv (1) to /places. We may here remark that should we find that we have taken any of the numbers rj, r^, r^, &c., as r ,a. little too large, we need not erase the transformation, for it will merely have the efiect of making the next r + j negative. 9 11. The calculation of Fi(l), FjCl), FgCl), &c., and of F'i(l), F'^Cl), F'3(l), &c., which must be computed for the purpose of discovering r^, r^, ^3, &c., is easily effected, thus if ThenF^ (1)=A + B + C +y + ^ + <,. AndFp (l)=w. A+«— l.B. +»— 2.C +2y + /3. By referring to the formulas of transformation in Articles 2, 3, and 5, it will be seen that the absolute term re- mains unaltered, and consequently no column will be formed under it, this vacant space we propose to appropriate to the calculation of F;i(l), F2(l), &c. ; also an additional column a little to the right, to the computation of F'i(l), F',(l), &c. In order to elucidate our method, we shall now proceed to apply it to the solution of a few equations. Example 1. Given ;»*— 30 ^^ + 700 «2 + i5i32 «— 1804827 = 0, to find x. Here a; 7 30 -; 40, hence R=30, and the operation is as follows : 10 1 30 900 2700 8100 1620 9720 0^ 1944 11664 2332 8 1399e~8~0 2799 3 6 16796 1 6 167 9 6 16 16964 1 2 169 6 4 16 12 17133 7 6 171 3 3 28 76 17305 1 173 5 04 10 17478 1 69 9 17548 70 1 5 14 1 26 6 40 9 23 17618 2 70 4 5 63 7 30 17688 7 70 7 2 93 5 49 17759 10 4 8 42 6 5 57 17770 10 1 3 99 6 6 21 17780 10 8 20 6 6 85 17791 10 4 6 7 7 05 49 17802 1 4 54 — 30 900 27000 8100 1620 9720 1944 8 11664 2332 8 13996 8 139 9 6 14136 7 6 141 3 6 8 77 14278 142 14420 57 ( 1 3 rs 3 1 3 8 57 14 Tl 37 14478 ( 57 i 3 08 ) 1 44 14536 . 58 ] 5 1 52 I 4 61 14594 ( 8 ' ? 6 13 r 5 68 14603 ^ 8 ' t 1 81 ' 6 20 14612 ] 8 7 8 '6 01 73 14620 9 4 74 700 2100 6300 1260 7560 1512 9072 90 7 2 9162 7 2 91 6 272 9254 3 37 9291 3 37 1 472 174 646 655 9328 5 9334 5 5 301 5 971 1 272 6 005 9339 7 277 1513 2 4539 6 907 9 2 5447 5 2 54 4 752 5501 9 22 952 080 5524 032 3I3 144 5527 3 176 —180 4827 45 X 1 = 63 X 2 = — 81 X 3 = 81 X 4 = -72 =F,(1) -180|5 54 5 90 7 — I40I0 168|0 -7|3 = F,(1) -180 55 92 — 144 174 — 2 48 02 54 21 78_ 35 = FsCl) -180 55 93 -145 177 483 240 285 947 595 310 = F4(1) — 180 55 93 — 146 178 = •00000052) 4827 273176 397277 209474 021454 (30 = R 45 (1-2 = 1 -i-r, 126 —243 324 252 = F'i(l) 55 (1-01 = 1 -1-^2 181 —420 672 488 = F',(1) 55 (1-004 = 1 +r3 185 —438 699 506 = F'3(1) 55 (1-0006 = 1 +r^ 187 —438 710 514 = F'^(1) 55 187 —439 712 Hence we have 30 x 1-2 x 1-01 x 1-004 x lM)00e X 1-00000052 for the development of the root, which may be computed as under : 0006 -267=F^(1) 515 = F'„(1) 258 I I 10 1-0000|0 5 2 6 00 00 1-0006 010 5 2 40 2 4 1-0046 2i9 2 100 4 6 3 1-0146 4 8 9|5 2029 2 9 7 9 004 01 1-2175 7 8 7 4 3 36-527 3 6 2 =« In this solution the process unfolded in Article 3 haa been employed ; if we suppress the addends, as suggested in Art. 4, the work wiU take the following form : 11 1 30 900 2700 8100 9720 11664 13996 8 16796 1 6 16964 1 2 17J33 7 6 17305 1 17478 1|5 16 28 04 14 17548 17618 2 17688 7 17759 4 17770 17780 17791 17802 6 40 5 63 2 93 8 42 1 3 99 8 20 4 7 05 1 4 54 - 30 900 2700 8 100 ~9720 11664 13996 8 ITl36 7 6 14278 1 3 14420 911 14478 6j0 08 14536 5 1 52 14594 6^6 13 57 71 14603 14612 14620 4 1 81 1 8 01 9 4 74 700 2100 6300 7560 9072 9162 7^ 9254 31472 9291 3 6T6 932S|5'301 9334 1 272 93397 277 1513 2 4539 6 5447 5 2 5501 9 952 5524 032 5527 3 176 1-0000|0 5 2 1-0006 0|0 5 2 1-0046 2|9 2 1-0146 4 8 9|5 1-2175 7 8 7 4 30 36-527 3 6 2=a; 0006 004 •01 •2 -180 4827 45 63 - 81 81 ■£72 -180 54 90 -140 168 -7 — 180 55 92 — 144 174 — 180 55 93 — 145 177 — 180 55 93 — 146 178 •00000052) 5 5 7 0_ 3~ 48' 02 54 21 78_ 35 483 240 285 947 595 310 4827 273176 397277 209474 021454 -267 258 ~9 10 (30 45(1-2 126 -243 324 252 55(1-01 181 -420 67_2 iis 55(1-004 185 -433 699 506 55(1-0006 187 -438 TIQ 514 55 187 -439 712 5r5 But perhaps the best way of conducting the solution is to perform the first transformation according to Article 2, the second and third according to Article 4, and the remaining two according to Article 5, aa follows : 12 1 30 900 2700 8100 9720 11664 13996 8 16796 1 6 16964 1 17133 7 17305 1 17478 1 69 9 70 1 70 4 70 7 2 16 6 28 0104 5 14 1 26 9 23 7 31 5 50 17759 4 8 44 10 6 5 57 10 6 6 21 10 6 6 85 10 6 7 49 17802 1 4 56 - 30 900 2700 8100 ^720 To 11664 13996 8 14136 7 6 8 14278 1 3 57 14420 9 1 71 57 6 8 37 57 9 1 44 58 1 4 60 14594 6 6 12 8 7 5 68 8 7 6 20 8 7 6 73 14620 9 4 73 700 2100 6300 7560 9072 9r62 7 2 9254 3 37 37 1 1513 2 4539 6 472 174 655 9328 5 5 5 301 5 971 6 005 9339 7 277 5447 5 2 5501 9 952 5524 032 5527 3 176 1-0000|0 5 1 1-0006 0\0 5 1 1-0046 2|9 1 1-0146 4 8 914 1-2175 7 8 7 3 30 36-527~3 6 2=x •0006 •004 •01 •2 -180 4827 (30 45 45(1-2 63 126 -81 -243 81 324 -72 252 —180 5 54 5 55(1-01 90 7 181 — 140 —420 168 672 ~7 3~ 488 -180 48" 55 02 55(1-004 92 54 185 —144 21 —433 174 78 699 —2 35 506 —180 483 55 240 55(1-0006 93 285 187 —145 947 -438 177 595 710 — 310 514 —180 4827 55 273176 55 93 397277 187 —146 209473 —439 178 021456 712 )0051) —264 515 258 II 6 5 13 o C4 CO CO M U) I U) I « o © »-( 1-f ^^ ^^-^ (N (N i-H CO ■* r-< CO 0> CO ■* IN rH t- rH CD (N CO CO •* CO 05 (N CO i- «:i CO I-t r-( 1-1 00 >o O r-H CO CO IN (N CO (N eo ' I I I (N — I I o >« t^ o CD >0 CO 00 >C >0 CO •* O OS CN -* T-H ""^ Tt< 00 CO rH ■* rt r— CO CO r-H rt (N rH O i>- CO 00 >0 CD ■* IN I-H CO t- i- CO 171 CO TO I- CO e ■* l—t CO >C 10 CO 7-t CO CO CO IN 00 © IN IN (M (N CO CO CO CO CO CO CO © © o 00 © © «5 © >o (N »0 (N (N IN 10 ■« >o © »0 00 rH f-l r-i T-t IN rH CO IN IN t- CO CO 00 CO r-< CO (N ■* ■* CO CO ^^ CO l~ 00 I- CO r-i «! 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H IN •* IN II t- t> IN >« •* ■* t~ eo o o CO o ■*■*■<(< o OS OS OS OS OS "^ >o o CJS OS T-l f— ( OS OS OS OS OS OS OS lO OS OS OS o CO "C O IN --1 t- t~ O tH 00 CO 00 CO t~ 00 IN CO rH lO OS lO »0 t- 00 O rH COCDMSOOSOOOOt- rH IN rH O OS t- CO «S ISO rH 00 CO CO rH rH rH O O (N CO CO CO CO CO rH CO eo eo CO CO rHt-INt-CDCOCDCD CDO-*lNOOOOO COrHlNOOrHrHrHr-rH CD IN OS CO CO © rH rH (N CO 1 1 1 1 1 00 MS >o «S "O ws oo "Mill rH 17 Example 3. Given a;* + 17 a?— 9, «— 630 = 0, to find x. Here x l —3 7 —4, hence R= — 3. 1 —3 9 -27 81 —243 + 729 787-3 850-3 918-3 9917 1071 1 1156 8 46 46 46 46 47 47 2 5 30 9 6 40 6 05 45 17 38 7 33 5 84 4 42 3 08 1 81 61 1184 8 7 4 47 9 4 7 90 9 4 8 66 9 4 9 42 9 5 17 9 5 93 ' 9 5 1 69 17 —51 153 —459 495-7 2 535-3 7 716 578-2 017 81 2 3 12 83 2 3 2 2 08 2 3 3I1 37 —8 + 24 25-9 2 585 1 4 4 4 7 4 09 6 8 14 6 8 51 6 8 89 586 5 7 9 63 26-0 2"60 2368 4450 —•0000 7 04 •9999|9 2 96 1-0007 9|2 95 1-0047 9 6|12 1-0851 7 9 81 —3 0008 004 08 - 63 2 4 — 45 9 72 9 — 33 6 — 63 2 6 - 57 8 115 7 —2 5 - 63 2 6-02 — 58 5-17 118 4-87 4-28 1190 5 7 3 24 -3-2555 3 9 4=ar. — 630 26-04450 — 586-57963 1190-57324 •00000704) 3811 3786 25 (-3 2(1-08 -138 437 301 3(1-004 -173 694 524 3(1-0008 -176 711 538 26 -1760 7143 5409 II 12. The preceding belongs to a class of equations, to which it appears to me, that the method of solution here pre- posed applies very happily. I allude to those which are incomplete, or which want several of their terms. For if any term be absent in the given equation, the corresponding term will also be absent in each of the transformed equa^ tions, consequently if an equation have m terms less than a complete equation of the same dimensions, we shall get rid of m columns. The preceding example with those following, will, it is thought, demonstrate the superiority of our method in the solution of equations of this kind. E 18 Example 4. Given ':[x^^+n(ia? + Q5'2!>^+Ui2a? + 5Z28x—29\%5{M=0, to find x. 7 14 28 56 112 224 448 896 1792 3584 7168 14336 28672 57344 114688 229376 458752 504627^ 555089- 610598- 671658- 738824 812707 893977 983375 1081713 1189884 1308872 1439760 1583736 1742109 1916320 2107953 2 92 91 80 68 15 86 65 21 54 99 29 32 95 6 95 2 04|7 2171191 2236327 2303417 2372519 2443695 2517006 2592516 2670291 2750400 2832912 2917900 3005437 3095600 3188468 3284122 3382645 63 8 9 38 8 1 97 6 78 66 1 37 8 6 170 340 680 1360 2720 5440 1088 2176 4352 4787 2 5265 9- 5792 5- 6371 7- 7008 9- 7709 8 8480 8 9328 8 9 8|5 2 3 2 9 5 3 4 6 8 9608 9897 10193 10499 10814 11139 11473 11817 70 71 71 72 72 73 73 73 2 3 5 5 9 6 6 90 3 3 7 5 1 8 6 2 5 4 9 9 3 4 4 2 4 1 6 4 9 5 5 4 08 8 8 9 3 8 2 5 2 8 2 6 6 6 1 759 12396 8 6 1 6 2 62 6 2 62 62 6 2 6 2 6 8 9 7 9 8 4 3 15 3 4 6 3 0773 10 8 8 13 9 4 1705 2 16 12446 5 4327 9 9 5 7 2 9 9 5 8 9 9 5 8 8 9 9 5 9 6 9 9 6 4 9 9 6 12 9 9 6 2 9 9 6 2 8 652 1304 2608 5216 1043 2 1147 5-2 1262 2-7 2 1388 4-9 9 2 1527 3 -4 9|12_ 1573 1 6 9)59" 1620 3 6 4 68 1668 9 7 5I62 1719 4 4 89 10 3 1 4 27 10 3 7 6 16 10 4 3 8 42 10 5 1 05 1760 6 7 4 79 8 8 034 8 8 078 8 8 1 22 8 8 1 66 1764 1 9 8 79 1 4 1 14 1 4 1 15 1 4 1 16 1 4 1 17 1764 7 6 3 41 134 2 268 4 536 8 590 4-8 649 5-2 8 669 0-1 3|84 689 8 4 13 4 15 4 26 4 51 9 31 5326 1065 2 1171 7-2 1206 8-7|16 1214 111 283 1214|7 1 989 1214 8 1 707 697 3 3 3 7 8 08 4 8 69 4 8 86 698 7 5 63 5 5 85 5 5 85 698 1 8 7 33 l-OOOOOll 1 1-00008 111 1-00058 1 1 1-00658 4 6 1-03678 2 2 1-14046 4 00008 0005 •006 03 1 2-28092 8 6 8=ar. 12454 5 112 7 19 —39 18500 1 1 1 4 j4_6x f 339 t —33 9 10 = 6 = —39 _21 -39 1 19 12 7 17 18 83 82" -39 185 121 70 J 76 1 240 37 224 — 354 39 1850 1215 698 1764 1 2447 37 5230 -496 -39185 00 121 481707 69 818733 176 476341 1245 451127 37571 071579 0000011445)- 700513 612068 88445 61207 27238 24483 2755 2448 307 306 (2 0(1-1 3 46 _28 _77 0(1-03 1 7 211 127 346 0(1-006 1 9 338 203 551 0(1-0005 1 10 872 223 606 0(1-00008 1 10 375 225 en 121 140 706 9964 375711 225426 Continuation of the first column. 612068 338264 5 2029 5 2041 7 2054 2066 3 2078 7 2091 2 2103 7 2116 3 2129 2141 8 2154 7 2167 6 2180 6 2193 7 2206 8 2220 1 286 750 502 561 970 774 017 743 997 825 272 384 206 785 168 401 531 372241 186 186 186 186 186 186 186 186 186 186 187 187 187 187 187 187 1 172 2 051 1 357 668 9 983 9 303 8 628 7 957 291 630 973 321 674 031 393 760 131 375230 1 3 323 30 1 841 30 2 081 30 2 321 30 2 561 30 2 801 30 3 041 30 3 281 30 3 521 30 3 761 30 4 001 30 4 241 30 4 481 30 4 721 30 4 961 30 5 201 30 5 441 375710 7 1 579 20 Example 5. Given x^^ = 100000, to find x. 1 —100 000 (2 2 2 22(1-4 4 8 16 32 64 128 256 512 1024 2048 -100 000 2 —98 -10 83 -117 —10 00 9 25 2867-2 4014-08 5619-71 2 7867-59 6 8 11014-63 5 5 15420-48 9 7 21588 68 5 6 30224 15 9 8 42313 82 3 8 59239 35 3 3 82935 09 416 83764 84602 85448 86302 87165 88037 88917 89806 90704 91611 92528 647 652 656 661 666 670 675 680 684 689 694 44 5 09 11 1 59 2 61 8 27 4 64 6 82 3 89 1 94 05 69 23 79 39 02 68 38 10 2 3 2 7 2 5 9 6 4 1 00 1 1 3 1 5 2 5 5 7 9 9 8 4 2 2 9 4 2 3 1 5 66,2 6 4910 3 99907 39 4 2 5 7 99 2 5 9 7 99 3 2 3 7 99 3 8 7 7 99 4 5 1 7 99 5 1 5 7 99 579 7 99 6 4 3 7 99 707 7 99 7 7 1 7 99 8 3 5 7 99 8 9 9 -\75 —1010000 9 1 9907 "P93 —100000 999 95-34794 42293)— 4-65206 4 39980 91(1-01 102(1-007 110(1-00008 1099949 25226 21999 3227 2200 1027 990 37 83 1-00000|42 2 9 3 1-00008 42l2 9 6 1-00708 48 1 19 2 1-01715 56 6 7|4 1-42401 79 3 4 4 2 -00008 -007 -01 -4 2-84803 58 6 9=«. 99995 34 7 9 4 21 13. It is to be observed that there is anotlier method of obtaining the transformed equations, which may often be employed with great advantage, particularly in the solution of very high equations. If the roots of the equation Aa;" +B;b"~ +Cx" +ya^ + /8ar + »=0 be divided by any number P, the resulting equation is AP"a;^+BP"-V"~'+C-P"-V»-^ +YV^a/^ + fil?-x' + x=0, (Z). Now if we compute P^, P^ P" in succession, or if the equation be incomplete, such of these powers as we have occasion for, and then take the product of A and P", of B and P" , of C and P" of y and P^, and of /3 and P, we shall obviously obtain the coefficients of the transformed equation (Z). The two examples which follow, will, it is hoped, fully elucidate this method. Example 6. Let the equation 102 a;2o_iii a;'9 + 72 2;'8— 85 a;"— 31 a;'** + 67 a;" + 201 x^* + 6 x^^—SS x^^ — .5.'?' + 123ari«— 234a:9— 22a^ + 93a;7— 7-6 a^ + -932 a^ + 82-2 ar*— 73-86 ar'-94 a^— 2-638 a;— 290.565 = 0, be proposed for solution. The solution of this equation extends over the six sucpeeding pages, and the three last columns of the work are devoted to the computation of 1-5^ l-5=» 15^", of 1-02^ l-02« l-022", &c. F 22 102 3 3 25-25 673015 -1 11 2 216-83 782010 3 3 25-25 67302 66 50 51346 3 3 91 76-18648 1-4 85 94 7 39597 3 3 91 76 18648 1 3 56 70 47459 2 71 34 09492 16 95 88093 3 05 25857 13 56705 2 37423 10175 3052 170 31 2_ 5 39 97-97107 1-1 27 09 264152 5 39 97-97107 5 03 99 79711 1 00 79 95942 35 27 98580 45 35982 1 00800 30240 2016 50 25 1 5 6 80 52-40454 1-0 08 03 047305 5 6 80 52 40454 45 44 41924 17 04157 22722 3976 170 3_ 5 7 26 14-13406 1-0 01 60 121658 5 7 26 14 13406 5 72 61413 3 43 56848 57261 11452 573 344 29 5_. 5 7 85 31-01331 1-0 00 04 000076 5 7 35 31 01331 22 94124 40 3_ 5 7 35 53-95498 2 216 83 78201 221 68 37820 22 16 83782 2 460 68-99803" 1-456 81 117252 2 460 68 99803 984 27 59921 123 03 44990 14 76 41399 1 96 85520 2 46069 24607 2461 1722 49 12 3 584 76-06553" 1-120 37 041901 3 584 76 06553 358 47 60655 71 69 52131 1 07 54282 25 09332 14339 358 ^323 4016 25-979"73 1-007 62 742208 4 016 25 97973" 2 811 38186 240 97559 8 03252 2 81138 16065 803 80 3 4 046 89-35059 1-001 52 109489 4 046 89 35059 4 04 68935 2 02 34468 8 09379 40469 3642 162 32 4 4 053 04-92150' 1-000 03 800068 4 053 04 92150 12 15915 3 24244 24 3 4 053 20-32336 72 1 477-89 18801 1 034 52 43161 29 55 78376 1 064 08-21537 1-428 24 624757 1-064 08 21537 425 63 28615 21 28 16431 8 51 26572 21 28164 4 25633 63845 2128 426 74 5 1 1 519 77-13431 1-113 68 828927 1 519 77 13431 151 97 71343 15 19 77134 4 55 93140 91 18628 12 15817 1 21582 3040 1216 187 3 1 1 692 55-15472 1-007 22 453227 1 692 55 15472 11 84 78608 33 85103 3 38510 67702 8463 508 34 3 1 1 704 77-94404 1-001 44 097961 1 704 77 94404 1 70 47794 68 19118 6 81912 15343 1193 153 10 1 707 23-59927 1-000 03 600061 1 707 23 59927 5 12171 1 02434 JO 1 707 ^9'^542 85 985-2 612534 'r882 90027 492 6 30627 8 874 7-20654 1-400 2 414192 8 874 7 20654 3 349 8 88262 1 6 74944 3 34989 8875 3350 84 75 2 1 1 726 6-80785 1-1 070 4 601319 1 1 726 6 30785 1 172 6 63074 82 86415 4 69065 70360 117 35 1 1 1 2 98 19-19803 " 1-0 06 82 180355 1 2 98 19 19803 7 78 91519 1 03 85536 2 59638 12982 10386 39 6 1 1 3 07 04-79910 1-0 01 36 087074 1 3 07 04 79910 1 30 70480 39 21144 7 84229 10456 915 9 1 1 3 08 82-67144 1-0 00 03 400054 1 3 08 82 67144 3 92648 52353 7 1 1 3 08 87-12153 - 8 1 6 56-8 408356 1 9 70 5 22507 65 6 84084 2 36 2-06591 1-3 72 7 857051 36 2 06591 6 10 8 61977 1 42 5 34461 4 72413 1 4 25344 1 62896 10181 1425 10 2 7 95 2-75298 1-1 00 4 438531 27 2 2 75298 5 27530 1 18110 1 11811 8386 839 140 8 3 76 0-42122 1-0 06 4 192359 3 76 42122 18 1 56253 30417 30760 27684 615 92 15 3 3 95 7-87961 1-0 01 2 807683 95 3 7 87961 95788 6 19158 2 47663 2167 186 25 1 3 99 7-52949 1-0 00 320005 3 99 7 52949 92993 6199 2 3 99 8-52143 23 67 437- 8 938904 2 627 3~63342 306 5 257 23 2 933 8-89065 1-345 8 683383 2 933 8~89065 880 1 66720 117 3 55563 14 6 69445 2 3 47111 1 76033 23471 880 88 28 1_ 3 948 6-28400 1-093 8 800726 3 948 6 28400" 355 3 76556 11 8 45885 3 1 58903 3 15890 276 8 2 4 319 ¥25920 1-006 168291 319 25 3 25920 9 15956 43193 25916 3455 86 39 4 345 3-14565 1-001 2 006722 4 345 4 3 14565 3 45315 8 69063 2607 304 9 1 4 350 5-31864 1-000 300004 4 350 5 31864 1 30516 2_ 4 350 ¥62382 2 01 2 91-92 926026 5 83 85~85205 2 91 92926 5 86 77^8181 1-31 94 787631 JlerrmsT 1 76 03 33439 5 86 77781 5 28 10003 23 47111 4 10744 46942 4107 352 18 1_ 7 74 24-08629 1-08 73 559370 7 74 2ir08629 61 93 92690 5 41 96860 23 22723 3 87120 38712 6968 232 54^ 8 41 87-53988 1-00 56 145833 8 41 8753988 4 20 93770 50 51252 84188 33675 4209 673 25 3^ 8 46 60-21783 1-00 11 205826 8 46 60 21783 84 66022 8 46602 1 69320 4233 677 17 5^ 8 47 55-08659 1-00 00 280004 8 47 5Tb8659 1 69510 67804 3^ 8 47 57-45976 19 4 -619507 riiT"- 71704 1-293 606630 1 167 71704~ 233 54341 105 09453 3 50315 70063 701 70 4 1 510-56651 1-080 870713 1 510 56651 120 84532 1 20845 10574 106 2 1 632-72710 1-005 212498 1 632 72710~ 8 16364 32655 1633 327 65 15 1 1 641-23770 1-001 040499 1 641 23770~ 1 64124 6565 66 15 1 1 642-94541 1-000 026000 1 642 94541 3286 986 1 642-98813 - 83 129-7 463379 1 037 9 70703 38 9 23901 1 076 8-94604 1-268 2 417946 1 076 8 94604 215 3 78921 64 6 13676 8 6 15157 2 15379 43076 1077 754 97 4 1 1 365 7^62746 1-074 4 241677 1 365 7 62746 95 6 03392 5 4 63051 5 46305 27315 5463 137 82 9 1 1 467 ?08501 1-004 8 105 741 1 467 4 0850r 5 8 69634 1 1 73927 14674 734 103 6_ 1 474 4-67579 1-000 9 604225 474 1 4 67579 3 27021 88468 590 29 3 1 1 475 8-83691 1-000 240003 1 475T83691 29518 5904 1 475¥~19113 — 5 86-497558 4 32-48779 1-24 337431 4 32 48779 86 49756 17 29951 1 29746 12975 3027 173 13^ 5 37-74420 1-06 801607 ¥3774420 32 26465 4 30195 538 323 4 5'74-31945 1-00 440881 5 74 31945 2 29728 22973 459 46 1 5 76-85152 1-00 088035 5 76 85152 46148 4615 17 3^ 5 77-35935 1-00 002200 5 77 35935 1155 115 5 77-37205 24 123 5 7-66 5 03906 23 4 3 8-44 3 35938 5 7 66 5 0391 1 1 53 3 0078 1 72 9 9512 7 92-7 9981 1-2 18 9 94420 18 70 56 6 92 7 9981 5 5996 9 2800 7 4240 3 8352 6 3835 2837 284 14 8 6 46-0 8339" 1-0 61 6 46194 8 6 46 8339 5 18 7 6500 8 6 4608 5 1 3 8765 4584 5188 86 78 3 9 1 79- 1-0 04 8151 07208 9 1 79 36 8151 7 1633 6425 184 7 9 2 15 1-0 00 8 6400 8 00288 9 2 15 7 8 6400 3 7269 184 74 7 9 2 23' 1-0 00 2 3934 20000 9 2 23 2-3934 1 8446 9 2 23-4 2380 7 6 88 6 7188 1 1 53 3 0078 1 53 7 7344 8 9 95-7 4610 1-1 95 92569 8 9 95 7 4610 8 99 5 7461 8 09 6 1715 44 9 7873 8 0962 1799 450 54 8 1 7 50^7 4932 1-0 5 53 1 43083 10 7 501^4932 5 87 5 3747 53 7 5375 3 2 2522 1 0751 4300 322 8 113 45-4 1957 1 36 57654 1 1 3 45 4 1957 34 3626 6 8 0725 5673 794 68 6 113 86-3 2849 1-0 07 2 02304 1 1 3 86 3 2849 7 9 7043 2 2773 228 34 1 1 8 94-5 2927 1-0 00 1 80001 1 1 3 94 5 2927 1 1395 9116 1 1 3 94-7 3438 -2 2 2 5-6 289068 5 12 57818 5 1 25781 93 17-0 859875 5 6 3-83594 1-1 7 165938 6 6 0-62366 1-0 4 902019 6 6 62366 2 6 42495 5 94561 1321 7 6 6 9 3-00756 1-0 320448 6 9 3 00756 2 07902 13860 277 28 6 6 9 5-22829 1-0 064018 6 9 5 22829 41714 2781 7 6^ 6 9 5-67387 1-0 001600 6 9 5 67387 •696 417 6 9 5-68450 1 53 7 73438 5 1 25781 1 58 8MJ9219~ 1-14 8 685668 992T9~ 89922 55969 71194 95340 12712 794 95 10 1 1 82 5-252^6" 1-04 2 763606 58 15 6 1 82 7 25256 01010 65051 27768 10951 548 110 1 1 90 8-30695 1-00 2 803362 1 90 3 30695" 3 80661 1 52265 571 57 11 1 90 8-64260 1-00 560134 1 90 8 64260 95482 11452 19 6 1 1 90 9-71170 1-00 014000 1 90 9 71170 1910 764 7-6 1 1-3 90625 7 9 73438 6 83437 8 6-56875 1-1 261624 8 6 56875 8 65688 1 73137 51941 866 519 17 3 9" 7- 49046 1-0 865443 9 7 49046 2 92471 58494 4875 390 39 3 1 1-05318 1-0 240240 10 1 05318 20211 . 4042 20 4 1 1-29595 1-0 048010 10 1 29595~ 4052 810 1 1 1-34458 1-0 001200 1 1 34458" 101 20 1 1-84579 1 90 9-73844 25 •9 32 7-59375 6 8 3438 2 2781 15^9 7-0 7738 1-1 04081 7^0 7738 7 0774 2831 57 1 7-8 1401 1-0 30362 7 8 1401 2 3442 234 47 2 8^0Tl26" 1-0 02002 8'(r5i26~ 1610 2 8-0 6738 1-0 00400 8 6738 323 8-0 7061 1-0 00010 8 7061 8 8-0 7069 8 2-2 5-0 625 4 5 000 1 125 1 0125 4 1 1 6-1375 -0 8 243216 4 1 6 1375 3 3 29100 83228 16645 1248 83 4 2 4 5 0-44060 1-0 2 421687 4 5 44060~ 9 00881 1 80176 9009 450 270 36 3 4 6 1-34885 1-0 160096 4 6 1 3488'5~ 46135 27681 41 3 4 6 2-08745 1-0 032004 4 6 2 08745~ 13863 924 2 4 6 2-23534 1-0 000800 4 6 2 23534 370 4 6 2-23904 - 7 3-86 3-3 75 2 2 1 58 2 2 158 5 1702 3693 2 4 9-2775 1-0 6 1208 2 4 9 2775 1 4 95665 24928 4986 199 2 6 4-53528 1-0 1 810822 2 6 4 53528 2 64535 2 11628 2645 212 5 1 2 6 9-32554 1-0 120048 2 6 9 32554 26933 5387 11 2 2 6 9-64887 1.0 024002 2 6 9 64887 5393 1079 1 2 6 9-71360 1-0 000600 2 6 9 71360 162 2 6 9-71522 9 4 2-2 5 —2-638 1-5 18 8 18 8 47 2 1 1-5 1-0 4 04 2~nr'5~ 8 46 846 2 2 0-0446 1-0 1 2036 2 2 0446 2 20045 44009 660 132 2 2 2-69306 1-0 080016 2 2 2 69306" 17815 2 1 2 2 2-87124 1-0 016001 2 2 2 87124 2229 1337 2 2 2-90690 1-0 000400 2 2 2 90690 89 2 2 2-90779 2 638 1 319 3-957 1-02 3 957 7914 4-03614 1-006 4 036fr 2422 4-06036" 1-0004 4 06036~ 162 F06l98" 1-00008 F06198 32 ?06230 1-000002 4 06230" 1 4-06231 26 -29 0565 (1-5 -29 057 — 29056 5 Ox 1 = ... ...0(1-02 0... 0(1-0004 — 406231 Ox 2 = — 22 — 22 290779 — 45 Ox 3 = — 27 — 26 971522 - 81 Ox 4 = 46 46 223904 185 Ox 5 = 1 807069 4 Ox 6 = — 10 — 10 134579 - 61 2x 7 = 1 190 1 190 973844 1337 — Ix 8 = - 1 — 69 — 1 — 69 568450 — 557 — 9x 9 = - 8 - 1 135 — 10 — 1139 473438 — 10255 7x 10 = 7 918 9 922 342380 9223 Ox 11 = — 57 — 1 - 57 737205 — 635 — 1 Ix 12 = — 13 — 1 467 -18 — 1475 919113 - 17711 1 X 13 = 1 163 2 164 298813 1643 5 9x /10 = 4 = 59 493 24 8 419 84 8475 745976 84757 2 9x 10 = i 5 = 29 34 33903 15 4 319 43 4350 662382 43507 — 2 Ox ;io=- 6 = - 20 22 21753 12 - 3 076 - 31 - 3099 852143 — 30999 — 8 4x /10=- 7 = _ 84 — 18 — 18599 59 -12 982 — 130 — 13088 712153 — 130887 10 6x 10 = 8 = 106 — 91 — 91621 85 16 926 169 17072 974542 170730 —24 6x {'?:= 246 135 136584 221 -40 163 — 402 — 40532 032336 — 405320 33 9x 20 = 678 — 361 — 364788 —11 9~ 341 56 805 278 1136 57355 395498 114 7108 9668 572 29938)— 173541 57 -29 06 '115934 1 0. 0( 1-006 -290 565 57607 — 2 — 41. 0(1-00008 52170 — 3 — 2229 0(1-000002 5437 5 — 2696 — 1 5217 4621 2 1 81 220 18 1 1013 — i 174 — 7 — 1 1 9086 13 46 — 1 08 - 10 — 6952 - 6 46 86 9 — 11 3863 - 102 — — 5 — 1 9 2159 92 - 1 37 — 16 — 5769 — 6 15 2 - 14 7447 - 177 7 74 77 31 1 6412 16 5 1-00000029938 1-00000229938 -000002 -00008 3 95 40 20 84 6602 847 339 1-00008229956 1-00048233248 -0004 -006 — 2 80 - 28 - 17 43 4531 435 217 1-00648522647 1-02661493100 -02 -5 — 11 73 - 117 - 82 - 30 9579 — 310 — 186 l-5399223965=a;. 15 20 152 122 —130 7048 — 1307 — 915 -35 85 - 359 — 323 170 4779 1705 1364 50 40 1008 -404 6894 — 4047 — 3 54 508 - 3642 572 6141 11452 4769 5787 4630 1 139 116 27 1 2 3 5' 7' 11 17' 25' 38' 57' 86' 129' 194' 291' 437- 656' 985' 1477' 2216' 3325' 5 25 375 0625 59375 390625 0859375 62890625 44335937 66503906 49755859 74633789 61950684 92926026 89389039 84083558 5 26125337 89188006 83782010 25673015 1-02 1-0404 1-061208 1-08243216 1-104080803 1-126162419 1-148685667 1-171659381 1-195092568 1-218994419 1-243374308 1-268241794 1-293606630 1-319478763 1-345868338 1-372785705 1-400241419 1-428246247 1-456811172 1-485947395 2 26 65 00 62 99 39 56 45 06 32 09 19 57 52 97 1-006 1-012036 1-01810821 1-02421686 1-03036216 1-03654433 1-04276360 1-04902018 1-05531430 1-06164619 1-06801607 1-07442416 1-08087071 1-08735593 1-09388007 1-10044335 1-10704601 1-11368828 1-12037041 1-12709264 6 530 649 949 553 716 828 413 129 772 273 701 263 307 319 927 901 152 1-0004 1-0008001 1-0012004 1-0016009 1-0020016 10024024 1-0028033 1-0032044 1-0036057 1-0040072 1-0044088 1-0048105 1-0052124 1-0056145 1-0060168 1-0064192 1-0068218 1-0072245 1-0076274 1-0080304 6 8006 6025 0063 0127 6223 8357 6536 0767 1055 7407 9830 8330 2913 3586 0355 3227 2208 7305 00008 000160 000240 000320 000400 000480 000560 000640 000720 000800 000880 000960 001040 001120 001200 001280 001360 001440 001521 001601 0064 01920 03840 06400 09601 13442 17923 23044 28806 35208 42251 49934 58258 67223 76828 87074 97961 09489 21658 00000 00000 00000 00000 00001 00001 00001 00001 00001 00002 00002 00002 00002 00002 00003 00003 00003 00003 00003 00004 2 400000 600001 800002 000004 200006 400008 6000 U 800014 000018 200022 400026 600031 800036 000042 200048 400054 600061 800068 000076 Example 7. Given 1379664 afi^^ + 2686034-10*'* a;i*3--17290224-105»8 g^ ^ 252415e-10=7*=0, to find x. The solution of this equation also extends over the six following pages, of which four are occupied by the computa- tion of 8«22, 8i«3, and 880. of i.04«22, l-04'53, and 1-04'"', &c. ; these powers are all calculated in the same manner, except that 1-0000024" is obtained by multiplying 1-0000024^ and 1-00000248 together. We have not thought it necessary to compute Fj(l), F\(l), F2(l), and V ^{l), because it is evident at a, glance that- FxO) F'i(l) will give no approximation to the value of r^, nor- F (l^ ^'^ -^ to that of »-, F'.(l) 28 o o o o o o eo«ot-t-((Nsoi-i|TO (M 05 »H 00 rt \<^ (N TO 'IM , II II II II II O O O CO O © M CO O "O O !M to I-H 04 •* o CO CO ■* (N !0 ■* ra- «5 >C t- OS ce (N 1^ o I—* 33 X O 1^ 1—t 00 y o ■* ■* •o (N >o (N 05 o ■* o> (N C © 1-H t- i-H 05 t^ t^ Ci (N 00 T— ( <3i (M Oi lO O "<*< 00 «5 I-H rH CO i-H 00 ■* 00 >o t^ C5 t- -* CO TO « 05 rH CI (M IM (N ■* W5 ■* TO CO © CD rl n o © to ■^ 00 t^ 00 =0 TjH t^i^-*rocoTOt^'*cD 'O>C'-ICO'*©00O5«5 ooeOTO-*(Nb-t^TO t--t~o TO CO © "O © 00 00 00 CO ■* © © (N (N 00 00 IN C X © "5 TO © © r-i TO TO © t— rH © r-l (^ X © >o t— 1 X © (N © ■* tH 1-1 •* © T-4 rH *iiNro-* ^HXTO"^'^i-l*Or-t ©»OMli— iX©TtC »o -^ »C "^ "^ X rH •* © 1-1 IN © © © l-t X ■* t- © (N © X e^ ■* i- TO t- t- © X © © X •C IN o X tH CD © © © X ■Oi t- O IN © © 1-1 X © © lo IN TO l^ 1^ ■<»< © © © TO © >0 TO (N © 1- X •O © T-l — 1 «> t- © t- © TO IN X o 1—1 n TO t- TO (N © I- TO 1 (NTOro©ro'.TXTfTO'Oi,— -*©©©'orox©t>T-i TO-HC5>OTOi— i-iTO ©t-i^Ot-fNX-* © t^ © IN !N t- in »C 1-1 !N X l^ © © X 1-1 (N -fl" 1-1 (N X >0 (N rH © oTOi^©©-*-H©'*©ro ■*TOro(N©X'C©Xi-< © f— 1 TOf-iTO©iO'<*i©C^ 1-1 TO »n TO IN t- © TO "O 1-1 ■* © >0 IN >0 1-1 0 f-(C^TtHi— l"^"^OC0„ CDOOCir-lCOOfNOS CO CO i-H — t (N *o (M 50 !•* " •0< i-i >o (N (N 00 «5 C5 CO 00 o oa C O N CO T-1 03 o ■* «o o o CO O 00 «o CO O (N »0 r-l t~ 1-c t- O ■* >0 C3 (N CO 00 W3 02 CO (N CO OJ CD t- l-t (M 00 (N Ci 1-1 00 C3 03 ^~ 1 •* 1 (M C5 Oi CO "O I© >o 00 t- 5 o (N «5 C3 >o I— t >o f- 1-H (N OS 03 l~ t- 00 ! 05 (N •0 ■* t-o 1 CO -1 O:iO3«5'*lO00CO00O3N — lOOOCOOOCOl^OOrt t~t~»coor-iooeoi-< !M .-I o> t- CO CO 03 oi I— t o CO CO I- CO lO O CT 03 O t- CO t- >-l CO X •* •o 03 "O f^ •* CO 03 CO CO I-H CO irq 03 CO T-< «500COCOCO'(fl N O IN t- t~ CO ■* o CO >0 i-H 0 (N O i> CO rH -I o ■* l-l rH >o CO 03 CO I-H 1 (M WJ CO rH 03 11 "«< i-H ■* CO i-H in ■*! t^ 00 CO t- "5 CD »0 03 "O 1-1 T-H -H CO »H r- •o ■o CO CO 03 03 O IH iH in 00 00 •* CO .00 o ■* o IN CO CO -H ■rtl .— CO -H CD i-c TH O CO IN t- CO "O •o CD li IN O CO © O l> © ® © >c ©©©©■* © © o © © iH 00 VI fi 03 t- b- 00 ■* 00 ■* CO t^ rH Tfl 00 ■* t^ © IN 00 •* (N 00 rH ^ (N CD CO o IN tr- IN ! CO 03 1 >f> © 00 © so 1-4 (Neo>ooo>ot^>-ico C3 00 t^ © 00 -* eo 00 CO © i^ eo eo •o 00 t- t^ l~ CO t^ CO ^ "C CO 00 CO ■* CO IN IN 00 «^ 03 CD IM CO 03 © 00 © 00 © CO w* ■* 03 CO CO 00 eo eo IN t- CO (N 1> 0 t~ 00 I-H * ffl © ^ © IN IN IN CO CO ira © © CD CD CD 03 t- © © l^ t~ © © © © © "O W3 © © © © ■* © © © © © © CO CO 00 OOe00^0300INCD00C0 iNco©rHeo'Ooo'*eo eOr-ITtieO©^rH<^ (N rH 00 IN © W3 CO CD CO "n 00 © (N ■* IN ^ rH 1— CD 00 "O CO CO rH IN CD CO X ■* 1^ f- «3 © CO CO tf IN >« © Wi 00 CO CO 00 © © rH „rH10—0 CO CO I- •'J 00 CO CO CD ^ CD © W •* CD 00 03 IN «3 © CO rH "O W3 00 CO © 1> 00 IN © I- CO © rH © © IN CO •n © © ■* r^ IN © © © t-C0rHCDC0©©?5 © >0 00 00 CD rH rH © © CO 00 © © >« © 00 r- CO >0 IN © © © © ■* © <-! © 30 8 64 512 4096 32768 262144 2097152 16777216 134217728 1073741824 8589934592 68719476736 549755813888 4398046511104 8>«= 35184372088832 35184372088832 105553116266496 17592186044416 351843720888 281474976711 14073748836 1055531163 246290605 7036874 281475 28147 2815 106 7 8-'»= m794003928539 x 10" 123794003928539 "123794003928539 24758800785708 3713820117856 866558027500 111414603536 4951760157 3713820 1114146 24759 9904 619 37 11 8"* = 539198933343026 x 10'' 876 ^ 431359146674421 x 10" 8" = 345087317339537 X 10" 1-0 4 129407744002326 17254365866977 2156795733372 34508731734 3019514027 129407744 4313591 3019514 129408 12941 3882 216 13 3 8>«= 148856570735748 x 10'^* 8'=^= 119085256588598 X lO'^s 8'5»= 95268205270878 x lO'^s 8»»«= 76214564216702 x 10'^ 83" = 6668774368961 571609231625 19053641054 952682053 381072821 47634103 5716092 381073 19054 953 572 67 7260824748428 x lO^ss 7260824748428 860= 153249554086592 x 10*» 8'«= 35184372088832 459748662259776 76624777043296 1532495540866 1225996432693 61299821635 4597486623 1072746879 30649911 1225996 122600 12260 460 31 87»= 539198933343026 X 10«' 508257732390 14521649497 4356494849 58086598 1452165 290433 50826 2904 581 29 1 1 86"''= 527195760274 x lO^^" 1-0 816 1-1 24864 1-1 6985856 1-2 166529024 1-2 65319018496 1-3 15931779235 84 1-3 68569050405 27 1-4 23311812421 48 1-4 80244284918 34 1-5 39454056315 07 1-6 01032218567 67 1-6 65073507310 38 1-7 31676447602 80 1-8 00943505506 91 1-8 00943505506 91 1 8 00943505506 91 1 4 40754804405 53 1620849154 96 72037740 22 5402830 52 900471 75 9004 72 900 47 10 80 1 62 2 3-2 43397510027 52 3-2 43397510027 52 9 7 30192530082 56 6 48679502005 50 1 29735900401 10 9730192530 08 973019253 01 291905775 90 22703782 57 1621698 76 32433 98 64 87 22 70 1 62 6 1 0-51962740805 271 1-8 00943505506 91 1 51962740805 27 8 41570192644 22 946766466 72 42078509 63 3155888 22 525981 37 5259 81 525 98 6 31 95 1 1 8-94525466088 49 31 18-9 452 54660 8 8149 19-7 030 64847 3 2l03 20-4 911 87441 2 1 31 1-005 1-010025 1-015075J25 39 4 061 29694 6 4 1 7 881 22593 8 9 3 1 773 27583 6 2 6 19 70306 4 8 5 1 97030 6 4 8 1 57624 5 19 13792 14 5 788 1 2 3 78 8 1 2 19 7 3 9 4 2 40 3-739 19495 218 2 41 9-888 76275 9 3 43 6-684 31326 0^9 7 45 4-151 68579 14 1 17 4 673 72530 4 4 2 1 834 21566 3 1 746 73725 3 43 66843 1 3 21 83421 5 7 43668 4 3 26201 6 3493 4 7 218 3 4 30 5 7 3 9 3 4 2 19 8 320-91702 6 19 8 320-91702 6 19 8 320 91702 6 17 8 488 82532 3 1 5 865 67336 2 594 96275 1 39 66418 3 1 78488 8 1983 2 1388 2 40 1 2 39 3 311 86129-9 1-02015050062 5 1-02525125312 813 1-03037750939 377 1-03552939694 074 04070704392 544 04591057914 507 05114013204 080 05639583270 100 06167781186 450 06698620092 382 07232113192 844 07768273758 808 g 1-07768273758 808 1 07768273758 808 7543779163 117 754377916 312 64660964 255 8621461 901 215536 548 75437 792 3233 048 754 378 53 884 8 621 862 9 1-4 536325249 91547 1-4 609006876 2|o45 1-4 682051910 5 855 1 4 609006876 2 05 5 843602750 4 82 876540412 5 72 116872055 10 2921801 3 75 73045 34 1460 9 01 1314 8 11 14 6 09 7 30 1 17 7 1 1-16140008289 535 1-161 40008289 535 1 16140008289 535 11614000828 954 6968400497 372 116140008 290 46456003 316 9291 200 232 280 92 912 10 453 581 35 6 1-34885015254 934 1-07768273758 808 r3488"50l5254 934 9441951067 845 944195106 784 80931009 153 10790801 220 269770 031 94419 511 4046 550 944 195 67 443 10 791 1 079 11 r45363252T99 547 2-1 449019731 2-1 556264830 2-1 664046154 8 54 5 13 6 66 2-1 772366385 4 39 4 3 328092309 3 3 2 166404615 4 7 1 516483230 8 3 151648323 8 4332809 2 3 649921 3 8 129984 2 8 12998 4 3 649 9 2 173 3 1 10 8 3 87 6 4-7 167755027 4 4-7 1677550 27 4 1 8 867102010 8 3 301742851 9 47167755 28300653 3301742 9 330174 3 23583 9 2358 4 9 4 3 3 2 2- 247971 U29~ 82 1-0007 1-00140049 1-002101470 1-002802941 1-003504903 1-004207356 1-004910302 1-005613739 1-006317668 1-007022091 1-007727006 1-008432415 1-009138318 1-009844715 1-010551606 1-010551606 343 3722 4312 8636 0134 2248 8423 2105 6743 5790 2699 0927 3933 3933 1-05388320 5|8295 1-05462092 410736 1-05535915 8 720 4 1 05462092 4 074 5273104 6 204 527310 4 620 31638 6 277 5273 1 046 949 1 588 10 5 462 5 2 731 8 437 738 21 1-010551606 3933 10105516 0639 505275 8032 50527 5803 1010 5516 606 3310 6 0633 3032 909 30 3 1-021214549 1840 1-02121454 9 1840 1 02I2T4549 1840 20424290 9837 1021214 5492 204242 9098 10212 1455 4084 8582 510 6073 40 8486 9 1909 1021 817 41 1-042879155 4651 1-01055160 6 3933 1-042879155 4651 10428791 5547 521439 5777 52143 9578 1042 8792 625 7275 6 2573 3129 939 31 3 1-053883205 8295 11300385 •11378295 ■11456260 1 198 3 894 1 962 11534279 5 783 11456260 1 96 11145626 20 1114562 6 02 557281 3 01 33436 8 78 4458 2 50 222 9 13 78 19 10 31 5 57 78 9_ 1-24311936 8 54 1-24311936 8 54 1 24311936 8 24862387 3 4972477 4 372935 8 12431 1 1243 1 1118 8 37 2 74 9 1-54534576 4 2 00006 0001200 0001800 0002400 0003000 0003600 0004200 0004801 0005401 0006001 0006601 0007202 0007802 0008403 036 10800 21600 36001 54003 75606 00811 29617 62025 98035 37647 80861 27678 0009003 78098 0009003 78098 1 0009003 78098 9008 10340 3 00270 70063 8007 90 8_ 1-0018015 66876 1-0018015 66876 1 0018015 66876 10018 01567 8014 41254 10 01802 5 00901 60108 6011 801 70 6_ 1-0036063 79396 1-0009003 78098 1 0036063 79396 9032 45741 3 01082 70252 8029 90 8 1-0045100104598 I-OO45702I75198 1-0046305 49415 1 0045702 7520 40182 8110 6027 4217 301 3711 5 0229 4018 904 40 1 1 1-0092219 8751 33 1009221 1-009282 1-009343 1-009403 98751 54083 09778 65837 1-00000 1 009343 0978 9084 0879 403 7372 3 0280 6056 505 81 3 _1 1-018834 6155 1-018 834 61 55 1 018834 616~~ 10188 346 8150 677 815 068 30 565 4 075 611 10 5 1_ 1-038023 974 000005 400001 1-00000 4 800006 2 000010 400002 1-00000 7 200018 2 000014 400003 1-00000 9 600035 2 000019 400004 1-00001 2(000058 2 000024 400005 1-00001 4 400087 2 000029 ' 400006 1-00001 6 2 800122 000034 400007 1-00001 9 200163 1-00001 6 800122 1 00001 9 200163 1 000192 6 000115 800015 122 1-00003 6 000607 1-00003 6 000607 1 00003 6 000607 3 001080 6 000216 607 1-00007 2 002510 1-00007 2 002510 1 00007 2 002510 7 005040 2 000144 2510 1-00014 4 010204 1-00003 6 000607 1 00014 4 010204 3 004320 6 000864 607 1-00018 015995 2 000360 '400072 1-00018 2 2 416427 000365 400073 1-0001 8 4816865 1-0001 8 2416427 1 0001 8 481687 1 001848 8 001479 200037 40007 1643 1-0003 6 726701 200073 40015 1-0003 6 966789 200074 40015 1-0003 7 206878 200074 40015 1-0003 7 446967 1-0003 7 206878 1 0003 3 7 44697 01123 7 00262 20007 688 1-0007 1-0007 robo7 7 4 66777 4 66777 0523 4 0030 6004 678 1-0014 9 3913 1-00018 4 816865 34 14. We shall, in conclusion, advert to a method of computing the powers of 1+r.^, 1 H-r^, &c., although it may not, perhaps, be of much importance. By the binomial theorem, ifYl \ Now when r is very small a few of the leading terms of this expansion will generally give the value of (1 +r )*" to that number of decimals which we require, and the computation of these will be very easy, for sup- pose we can forsee that a certain number of terms, say four, will be sufficient, then we shall have sm , m— 1 „ m— 1 m — 2 „ (1 +r ) =1 +m . r +m 1" +'m . . . /•„ , which put= 1 +A»-+Br ^ + C r ^, which we compute thus, multiply C by r , and add B to the product, multiply this sum by r , and add A to the pro- duct, multiply once more by r , and add unity to the product, v. e tl nil tl lis evidently obtainl + A r + B r^ + C r * or (1 +r )"*. Should it take a greater number of terms to give the value of (1+r^)'" to the extent which we require, we must proceed in precisely the same way. To exemplify this method, let Example V., that is «" = 100000, be re-proposed, the root to be found to about twenty places of decimals. We shall effect the first transformation according to Art. 2, the four next according to Art. 4, and the remaining two according to Art. 13. Inthisexample wehave (1+r )'" = (1+/- )" = 1 + 11 r +55 . r^^^ies /•^3 + 33o^^4 + &(., We shall hence easily obtain 1-000004" = 1 + -00011 x -4 + -0000000055 x -4^ + -000000000000165 x -4^ + -0000000000000000033 X -4*, and 1-00000022916" = 1 + -000011 x -22916 + -000000000055 x -22916^ + -000000000000000165 x -22916^. 35 1 2 i 8 16 32 64 128 256 51 2 1024 2048 2867- 4014- 5619- 7867- 11014- 15420- 21588- 30224- 42313- 59239- 82935' 9 9907-894356320750180966 2 08 712 5968 63552 489728 6856192 15986688 823813632 3533390848 09467471872 83764-4456214659072 84602-09007768056627 2 85448-11097845737193 4 86302-59208824194565 4 87165-61800912436511 88037-27418921560876 1 88917-,64693110776484 9 89806-82340041884249 7 90704-89163442303092 2 91611-94055076726123 2 92528-0599562749338414 9 9915- 9 9923- 9 9931- 9 9939- 9 9947- 9 9955- 9 9963- 9 9971' 9 9979' 9 9987' 9 9995 1-0000 72 07 61 72 34 83 81 04 36 93175 93827' 94484 95146 95812 96482 97158- 97838' 98523 99212 99907 75637596885838 98667060064038 78257729484487 17605533590878 19928772326014 88468273732296 26487551648423 37272964509961 24133875261531 90402812388362 1 27 9 94 2 67 6 78 8 29 9 33 12 9 73 7 07 4 29 386947869256 380178825086 374049239392 368559163331 363708648064 359497744756 355926504575 •352994978696 ■350703218294 •349051274551 ■348039198653 440008800105 46972 01024 01712 16848 23497 08011 66060 02665 32233 78588 75002 600845 9 9995 348039198653 75002 3 999813921567 94615 399981392156 79462 79996278 43136 7999627 84314 999 95348 49 99767 5 99972 800 \ 40 5 9 9999-747922509340 72461 1-0000 025207628882 887937 9 9999 747922509340 72461 199999495845 01868 49999873961 25467 1999994958 45019 69999823 54576 5999984 87535 199999 49585 79999 79834 7999 97983 799 99798 19 99995 7 99998 80000 7000 900 30 7 3943563207508019 66 » 9999-999998162742 02056 36 10 0000 2 (2 22(1-4 9 8 10,0 8 3 91(1-01 - 1|7 -10 00 9 25 102(1-007 \'± -10 0000 9 9907 110(1-00008 1—93 -10 00000 9 99953 110(1-000004 - 00000000000000000330 -00000000000016500132 - 000000005500066000528 • 000110002200026400211 12 1-000004"= 1-000044000880010560084 5 -000000000000000165 33 33 1485 165 99 550000000000 - 47 10 0000 9 99 99-747 923 --252077 219999 - 0000000000550000378114 ~ 110000075623 11000007562 4950003403 55000038 33000023 110000000000000000 •0000110000126038086649 1099997(1-00000022916 111 32078 2916 — 100000- 99999-99999816274202056 16702345268)— 183725797944 109999999998 22000025207617330 2200002520761733 990001134342780 11000012603809 6600007562285 1 - 0000002291 6» = 1 - 0000025207628882887937 1099999-99998 73725797946 65999999999 111 7725797947 702345268 1-0000000000016702345268 2000000000003340 200000000000334 90000000000150 1000000000002 600000000001 1-0000002291616702349095 1-0000042291625868815904 1-0000842295009198885409 1-0070848191074263277607 1-0171556672985005910383 1-4240179342179008274536 00000022916 •000004 ■00008 ■007 •01 •4 2-848035868435801654907=a;. BOROUGH OF TrNEMOUTH: FEINTED BY J. PHILIPSON. THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $1.00 ON THE SEVENTH DAY OVERDUE. Ara 17 1934 OCT 1 1955 [. u iDM60VP -H6^ 9rt \933 necD L^ MAY 3 1938 JUL8 ^ PHQTOGOR Y . OCTai'88 NOV ii4: 1938 MAR 13 1943 APR 29 1 946 ?947 ^"^^ -\m ^^ L^ LD21-100m-7,'33 ^ * r? 5 9 u i '^ '■••* UNIVERSITY OF CAUFORNIA LIBRARY • ■»-»J3M( •»-'"i ■i'J-Y'f sr^wi •"•^