October 29, 1908J 



NA TURE 



663 



with respect to the last paragraph of Prof. Zeeman's 

 article on Prof. Hale's discovery. The magnetic forces 

 indicated by the splitting up of the lines are not sufficient 

 to produce any direct observable magnetic effect at the 

 distance of the' earth. Arthur Schuster. 



Simla, October 6. 



The Magnetic Disturbances of September 29 and 

 Aurora Borealis. 



Some details of an unusually bright aurora, seen at 

 Omaha, U.S..\., on the night of September 28, local 

 time, may be of interest to the readers of Nature in 

 connection with the three-hour magnetic disturbance re- 

 corded on our magnetograms between 4 a.m. and 7 a.m. 

 of September 29, Greenwich time. 



The details come from Father Rigge, S.J., director of 

 the Creighton University Observatory, Omaha. The sky 

 was perfectly clear throughout the night. The aurora 

 " seemed to commence suddenly at 9.50 p.m.," Sep- 

 tember 28, local time, i.e. at 4.15 a.m., September 29, 

 Greenwich time, when the unifilar magnet at Stonyhurst 

 commenced a rapid westward movement up to 62' of arc 

 at 4.40 a.m.,- returning more slowly in three sudden steps 

 backward at 5.5 a.m., 5.35 a.m., and 6 a.m., accompanied 

 by minor rapid oscillations. 



The aurora was watched for tw-o hours, up to the local 

 midnight, and during this time alternations of the scene 

 were observed between brilliant streamers of various 

 lengths and breadths from a well-defined arch, and a 

 broken-up arch accompanied by drifting luminous patches 

 as of fiery clouds. It would have been interesting to 

 compare the times of these changes with the halting move- 

 ments of the magnetic needle, but the time was recorded 

 onlv of the first appearance of the streamers, the smaller 

 lengths of which " seemed to come directly out of the 

 ground," and the noted time agrees closely with that of 

 a single break in the first long and rapid deflection of the 

 needle — a short step-back followed by a rush forward to 

 its greatest elongation. The aurora was again looked for 

 at 5 a.m. of the following morning, when nothing was 

 seen in the still unclouded sky. 



It is therefore probable that the auroral display began 

 and ended synchronously with this greater deflection of the 

 needle. 



The three-hour wave was, then, followed by the usual 

 rapid oscillations consequent upon a magnetic storm until 

 2.50 p.m., September 29, G.M.T., when another and a 

 greater storm broke out and lasted until 4.30 of the 

 following morning. ;\t Omaha aurora was again seen at 

 7.15 p.m., September 29. local time, but in a less favour- 

 able sky, which clouded over at 9.15, and showed only 

 by the brightened clouds that the aurora was still active 

 at 10 p.m., when the greater oscillations of the magnets 

 were ending. W.iLTER Sidgreaves, S.J. 



Stonyhurst College Observatory, October 21. 



A Method of Solving Algebraic Equations. 



So far as I can ascertain, the method referred lo is not 

 known, at least in its complete form. It is a development 

 of a method described by me in a previous paper (" Verb 

 Functions, with Notes on the Solution of Equations by 

 Operative Division," Proceedings of the Royal Irish 

 Academy, vol. xxv., Sec. A, No. 3, April, 1905), which was 

 reviewed in Nature of April 25, 1905. I give it here as 

 briefly as possible. 



Take, for example, the equation used by Newton to 

 illustrate his method of approximation, namely, 



which has one real root, 2094.;5. . . . Write the equation 

 in the form .r' = 2.v-l-5. Select any real number, .r, ; sub- 

 stitute it for X in the right-hand member of the equation, 

 and then find .v, from », = v'a-v, -1-5. Next substitute x, 

 for X, in the right-hand side of the equation, and find the 

 value of X.J, and so on. We thus have a series of numbers 

 connected by the equation Ji'„+, =2jr„-t-5, and it will be 

 found that whatever number we start with for r,, .-v,, con- 

 stantly approaches the value of the root. Thus, if we 



begin with 11, we have .\j=ii, X2 = 3, a-3 = 2-2240, 



A-, = 2ii4o, A,. = 2097S, •'^■6 = 2-0949 Or, commencing 



with —too, we obtain .x', = — 100, .v,= —5-7989, .\-3= — 1-8756, 

 .v,, = i-0758, .\:s=i-92b8, .\-5 = 2-o688, .v, = 2-0907, .v, = 2-0940, ... 



Again, take the equation ^" — i5.v-4 = o, which has three 

 real roots, 4 and —2 + .4/3, that is, 4, —0-2678, and 

 — 3-7321. Write it in the form .\°=i5.v + 4, and begin 

 with any number above the limits of the positive roots, 

 say 16. Substitute this for x in the right side of the equa- 

 tion, and proceed as before. Then a;j=i6, .1:, = 6-2488, 



.Vj = 4-6o62, Xj = 4.oi24, X5 = 4.oo39 , which is nearly 



the first root. 



In order to obtain the next lower root take for x^ a 

 number which is a little less than the first root, say 3-9, 

 and substitute it for x, not in the right side of the equa- 



, ,:)_., 



tion, but in the /('/( side, so that now 



15 

 Thus we obtain -v, = 3-9, .v, = 3-6880, -'>:3 = 30558, .Vj = 1-6356, 



A-5 = o-035i, A.'5= —0-2666, .v,= — 0-2679, which is 



nearly the second root. 



For the third root take a number, say —0-3, which is a 

 little less (algebraically) than the second root, and substi- 

 tute it for -V in the right side of the equation, as done for 

 the first root. We thus obtain a;[=— 0-3, Xn=— 0-7937, 

 .V3=-2-o, .v^= -2-9625, .->--i = -3-4313. a;J= -3-6203, 



Aj=— 3-6910, -■>-'s = ""3'7ib9, .Vg=— 3-7267 , which is 



nearly the third root. 



We can solve the equation in the same manner by 

 beginning with any number, say —5, which is below the 

 limit of the negative roots, and substituting it for x in 

 the right side of the equation ; then after finding the lowest 

 root, substitute a greater number for it in the left side of 

 the equation, and so on. We may thus either descend from 

 the highest to the lowest root, or ascend from the lowest 

 to the highest. It is evident that a root is obtained when 

 ;r,(+j=A-„, because the equation is then satisfied. 



We took the original equations in the forms x' = 2X + ^ 

 and .v^ = 15.V-I-4, but we may take them also in the forms 

 A'" = 2-f-5/.\ and .v' = 15-1-4 .v, ^^ ^'^ other forms obtained 

 by ordinary algebraic or operative transformations ; and 

 the method of solution is the same. 



The rule is most easily explained geometrically. Let 

 f{x) = o be the original equation. Write it in the form 

 /.,(a) = /|(.v), as may usually be done in many ways. 

 Draw the curves /,(.i.') = y and /,(.t) = y. Then the roots of 

 /,(a) = /,(.v) are evidently the abscissse of the points of inter- 

 section of the two curves. The procedure adopted above is 

 really as follows. Select any point, .v,, on the axis of x, 

 and draw a straight line from it parallel to the axis of y, 

 either in the positive or in the negative direction, until it 

 meets the nearer of the two curve."; — let us say f^(x) = y. 

 From this second point draw a line parallel to y until it 

 meets f„(x) = y. From the third point draw a line parallel 

 to X until it meets /i(-v) = y again, and from the fourth 

 point one parallel to y until it meets /2(.v) = y again, and 

 so on. Then the abscissa of the first and second points 

 is x^, of the third and fourth points is x,, of the fifth and 

 sixth points is x.^, and so on, and «„ must generallv 

 approach nearer and nearer to the point of intersection of 

 the two curves — that is, to a root of the original equation. 



Fig. I represents an intersection where the lines drawn 

 according to the rule all lie within the angles formed by 

 the converging curves. In this case, analytically, x,, x„, 

 .\',, . . . ., are all either greater or all less than x, the 

 abscissa of the point of intersection, although they con- 

 stantly approach it. Fig. 2 illustrates the case where the 

 lines ultimately approach the intersection spirally. Here, 

 analytically, .v,, .v,. .v, . . . alternately oscillate above and 

 below X, although they constantly approach it. The 

 former, or " staircase " procession, occurs while the differ- 

 ential coefficients of the two curves have the same sign : 

 the latter, or alternating " spiral " procession, while they 

 are of opposite signs. 



The staircase procession trends in the same direction as 

 the tangent vectors of the curves if a.', — -v, is positive, and 

 in the opposite direction if x.. — x, is negative. A similar 

 law holds for the direction of rotation of the spiral pro- 

 cession. Thus x,, x„, x, . . . will increase or decrease, 

 either continuously or alternately, according to whether we 

 have taken x, on one or the other of the two curves 



NO. 20.35, ■^'OL. 78] 



