: 



IXVOU'TIOJf AND EVOLUTION. 



INVOLUTION' AND EVOLUTION. 



til* equation of UM evolute of the parabola, which evolute therefore 

 ppn to be whet u called a temi-fMeml parabola. 



For considerations similar to those which precede aee CAUSTICS. 



INVOLUTION and EVOLUTION (Arithmetic). Taking th< 

 words in their etymological mow, they might stand fcir the greater 

 part of mathematical analysis. In their technical algebraical sense, 

 they mean only the raining of powers, and the inverse operation, the 

 extraction of root*. The revival however of a general process, accom- 

 pannHi by an improvement which make* it comparatively eaay, renders 

 it necs*ary to make a more extenaive definition of the terms. We 

 hall not relinquish any characteristic of the old meaning!, and ahall 

 bring all corresponding processes together, by laying down the fol- 

 lowing definition : Involution i the performance of any number of 

 successive multiplication* with the aame multiplier, interrupted or 

 not by additions or lubtmctions ; and evolution is any method of 

 finding out, from the result of an involution, what multiplier was 

 <-tupl<>yl. provided that the said method proceed by involutions. 

 Thus to determine ir 1 + 4jr Si + 10 by involution, we multiply -2 l.y 

 x and add 4 ; then multiply by x and subtract 3, then multiply by j: 

 andadd 10. If this give 1000, then any method of determining x which 

 proceed* by successive involutions is evolution. 



Forty years ago our only instances of evolution would hare been 

 common division, and the extraction of the square and cube roota, 

 with references to Vieta, Harriot, Oughtred, and the older algebraists 

 in general, for evolutionary methods of solving equations, bearing a 

 strong likeness to such extractions. But since the publication of Mr 

 Homer's New Method of Solving Equations of all Orders,' ' PhiL 

 Trans.,' 1818, the process which has rendered it worth while to propose 

 the preceding extension of terms has been in the hands of mathe- 

 maticians. For a more detailed account than we can here give, the 

 reader is referred to the paper just cited, which is reprinted in the 

 Ladies 1 Diary ' for 1888, or to The Theory and Solution of Algebraical 

 Equations,' by Professor Young, of Belfast (London, J. Souter, 1835). 



W e should begin with simple division, and the extraction of the 

 square and cube roots, if we were writing an elementary treatise. But 

 taking it for granted that the reader is familiar with the first two, at 

 least, we shall proceed to describe the general process. This consists 

 of three distinct parts, the first two of which have been long known, 

 while the third, which contains the peculiar distinction of this method 

 is due to Mr. Homer.* 



1. In the article APPROXIMATION it is shown that if a be a value of 

 x which makes fx very small, then a-(fa-f.f'a) is a value of x which 

 makes fx much smaller; so that a continued succession of approxi- 

 mations may be made to a value of x which makes f x absolutely = 0. 

 Here fr means the differential coefficient or derived function, and if 



fx=AX" + BX -' + ex" -' + 



; then 



L Meaning by a root of fx, any value of x which makes fx=0, it is 

 obvious that f(y + ) is a function which has for its roots the roots of 

 fsr, each diminished by a. And the substitution of x + a, instead of x 

 in the preceding value of fx gives a well known development, of 

 which an instance will be more to our present purpose. Let the 

 function be 



A* + BJT-rCr > -rl>x* + Ez + F... .(1). 



Write x + a for x, and this becomes 



xi + (SAO + B) x* + (lOao'-r <na + c)x + (10Aa + 6na + 3ca + D) a* 

 (5AO* + 4B0 1 8ca + 2oo + ) x + AO + Ba + ca" + Do* + Ea + p ; 



which we may represent by 



8. The quantities fa, fa, f,a, Ac., may be determined by a sue- 



ion of involutions, each one making use of the result* of the 



Ung. rind fa by involution, of which the following are the 



***I -"~~ 



A 



AO +B 



-f 



+D 

 Aa f + Ba 1 co* + Da 



Bpsat th. process using the preceding quantities, except the last 

 and we have f'a by the following stop* : 



A 

 2AO 



SAO* + 4ea> 8oa + 2oa + K = f'a. 



rfcht to U. lm,,lon, of which w. abaU pm^Uj 





A repetition of the process, leaving out the last, gives f.o, as 

 follows : 



A 



SAO + B 



o + 6Ba* + Sea + D = f,a, 

 Repetition gives f ,a, and finally f ,a, as follows : 



A 



= f. 



A 

 4AO + B 



1 OAO* + 4Ba + c = f,a 



In numerical application the operations may be made to stand thus, 

 where a new letter below a line stands for the sum of the tw 

 ceding ; and fa, f'a, Ac., are introduced when found. 



B 



Art 



C 



ra 



B 



Ra 



F 



sa 



p 



Aa 



Q. 



Ta 



R 



ca 



I 



va 



u 

 wa 



v 

 xa 



f'a 



w 



Ad 



I 

 TO 



T 

 AO 



f 4 



<t>,a 



<t> t a 



<t>a 



If a be of only one significant figure (as 200, 6, -03), all the operations 

 necessary to fill up this process can be performed in the head, ami w.- 

 have thus (for the method is general, though our example be only f 

 the fifth degree) a working method of answering the following 

 question: Given a certain equation <fx=0; required the equation 

 <l/x=Q, the roots of which are each less by a than those of fx=0. 



If fa came out=0, we should then know that a is a root of the 

 equation : and the method of approximating to a root is as follows : 

 Suppose we have an equation of which the root (unknown to us) is 

 2678. By trial, or otherwise, suppose we find that '20 is the highest 

 denomination of the root, and we thereupon find another equation, 

 each of whose roots is less by 20 than a root of the given equation : 

 this is done by the preceding process, and one of the new roots (but 

 unknown) is 673. If we can find that the highest di-iK.minai. 

 this root is 6, we make another reduction of all the rooU, an.l liixl :i 

 new equation, one of whose roots is 73. If we can then find 7 to be 

 the highest denomination, we repeat the process and tind an equation 

 one of whose roots is -03. In finding the highest denomination of tins 

 root we find the root itself, evidenced by the *a of this final process 

 being =0. 



The first denomination of the root must be found by trial, or by 

 some of the methods referred to in THKOKY OF EQUATIONS. But the 

 second and the remaining ones are found by comparing the results fa 

 and f 'a. If a be nearly a root, 



fa 



fa 



is still nearer. Consequently, by dividing -fa by f'a, we may, after 

 the second process, be sure of finding one figure of the remaining root 

 correct. But after the first process we may be liable to an error of 

 a unit (to be corrected by a new trial), as in extraction of the square 

 root. 



In order to obtain fa and not fa, let the last coefficient, p, have 

 its sign changed, and let the process in the column wliich "utains it 

 be always subtraction, and not addition. In the preceding typo of 

 calculation, we should then have 



F F 



Ka instead of . sa 



Subtr.-fa 



fa Add. 



In carrying on the process, the results fa, f'a, &c., come in a diagonal 

 line ; before taking the next step, the beginner may bring them down 

 into one line, as in the type preceding. In our examples, asterisks or 

 other symbols will mark results of a process. 



We now apply this method to the solution of the equation 



x* + 2x- x*~x- 631 064798 = 0. 

 It will be found that a root lies between 100 an.l 200. 



