POWKK. 



POWER. 



and |o m j that the fourth power of x is the result of four multi- 

 plication* by f, unity being understood M the commencement Tliiw 

 the successive power* of x, firet, Moond, third, Ac., are 1 x x, 1 x x x x, 

 1 x t x x x i, *c. : denoted by jr 1 , x 3 , x", Ac. And the term roof in the 

 UTCTM of powrr, a follow* : If A be the nth power of B, B in the mth 

 root of A. denoted by V*- The peculiar algebraical character of the 

 root* a explained in ROOT. 



It i thiu easily proved that when m and * are any two integers, 



that when M is greater than , 



Also that 



and that V-*"=*", 



whenever i divisible by without remainder. These rules, if 

 applied in defiance of the restrictions first mentioned, lead to such 

 results 



which are unintelligible so far as the definitions have yet been stated- 

 Their proper interpretations [INTERPRETATION] are as follows : First, 

 j* must be allowed to stand for unity, whatever x may be ; secondly, 



3C~* must be understood to be IH-X"; thirdly, x*, m and being 

 positive integers, must stand for V-e". When these new definitions 

 are added, all the rules remain true, whether m and n be positive or 

 negative, integral or fractional : and the system of algebraic powers is 

 complete. 



An algebraic expression is said to be arranged in powers of a letter, 

 say x, when the powers of that letter which enter are made to enter 

 in ascending or descending order of algebraic magnitude. Thus 

 ojp + bx-J-x* x~ l is not at present arranged at all. To arrange it in 

 ascending or descending powers of x, we must write it thus 



bx~* x~* +ax* x 4 , ascending ; 

 x 4 + oz* *-' + bx-*, descending. 



But even yet it is incomplete for many algebraical purposes, having no 

 written indication of the fact that the ascent or descent is interrupted. 

 Completely written in ascending powers, it should be 



* *- 1 + .0* + Ox. 1 



Written in this form, which may remind us of the use of a cipher in 

 writing ordinary numbers, it is clear that we hardly read the expression 

 less easily, and write it much more briefly, if we omit .' and its powers 

 altogether, and make some distinctive mark, analogous to the decimal 

 point, between the parts which belong to the positive and negative 

 powers. Thus the above might be written 



4 + 0-1+ | O + + o + Ol, 

 or -l+0 + o+O + O | 1+0 + i; 



the mark | being on that side of the adjacent + or which belongs to 

 the positive powers. This mark however is not necessary in what 

 follows. 



The late Mr. Homer [INVOLUTION, Ac.] was the first who suggested 

 the systematic rejection of the ascending or descending powers. An 

 example of multiplication and division will sufficiently explain its use 

 Suppose it required to multiply 7-r 1 -2*-8 and 2*<+*-4.j:-5 : 



2+0+1-4-5 



7-2+0-8 



14 + + 7-28-85 

 -4 + 0- 2+ 8 + 10 



- 6+ 0- 8 + 12 + 15 



14-4 + 7-88-27+ 7 + 12 + 16 



Accordingly the answer is 1 4x? - 4x* + 7x - 86x- 27** + 7x> + 1 2* + 1 5 ; 

 and every stroke of the pen which the usual method contains, more 

 than is in the preceding, is mere waste, and risk of error into the 

 bargain. Now let it be proposed to divide 4a-8o.^ + 2a 

 by x' + 2oje > -o': 



4- 8+ 2+ 0+ 0-11-1(1 + 2 + 0-1 

 *+ + 0- 4 4-11 + 24-44 



-11+ S+ 4+ 

 -11-2J+ 0+11 



24+ 4-11-11 

 24 + 48+ 0-24 



-44-11 + 13- 1 

 -44-88+ + 44 



77 + 18-45 



Accordingly the quotient u 4x > -ll<w* + 2l<i 1 .r 44<r*, and the re- 



,.'*+ 18<ix 45a : . 



Mr. Horni-r himself .li.l not live to print this suggestion, which, 

 simple as it is, seems never to have been made before him. Tin- 

 possessor of his ]npers, Mr. T. S. Davies, of Woolwich, published some 

 extracts from those papers in an appendix to a reprint ot the paper on 

 the solution of equations, which reprint appeared in the ' Ladies' I 

 for 1888 ; having previously introduced the simplification ii 

 llth edition of Mutton's Course. Since that time a paper on 

 ' Algebraical Transformation,' sent by Mr. Homer to the Royal Society, 



i * printed in the ' Philosophical Transactions,' has been 

 lished in the first and second volume of the ' Mathematician.' Details 

 and examples are given in Mr. Davies' ' Solutions of Questions con- 

 tained in Mutton's Course,' 1840, and in the 12th edition of that 

 course, 1841. 



But the greatest improvement in the operation of division, and 0110 

 which contains the principle of a class of improvements, is one which 

 Homer called the iythrtic net/iwl, which amounts to deferring the 

 actual steps of subtraction until they are wanted. If we were to 

 proceed one step farther with the preceding division, 44 in the 

 quotient would be followed by + 77. This + 77, if we look at all iu 

 components from the beginning, arises from +011-0 + 88. In like 

 manner 44 arise* from +0+4048. How arrange the process aa 

 follows : 



14-8+2+0 + 0-11- 1 

 -2 -8+0+4 -11 + 24-44 

 + +22+ + 0+0 



+ 1 -48 +88 



4-11 + 24-441 + 77 + 18-45 



Write the coefficients of the dividend horizontally o, 6, c, ic., and of 

 the divisor vertically p, q, r, 4c., taking care to change the sign of every 

 term of the divisor except the fnt. 



pa + b + c + d + e + / + g + k 

 +g +g+r + ut +i(t + vt + irt +rt+yt 

 + r + r q + vr + m + vn + xt + yt 



u+ v + w + x 



Divide o by;>, giving a, and then write 117, ur, ta, and tit in the suc- 

 cessive columns which follow that of u. Make up + h + ut), the second 

 column, and divide by p, giving : write n/, IT, r.<. r/. in the successive 

 columns which follow that of v. Make up c + ur + rq and divide by p, 

 giving w : write wq, wr, ws, wt, in the columns which follow that of to, 

 and so on. Then u + t' + ir + , &c ., will give the coefficients of tin- 

 quotient, and u' + v' + , Ac., made from the columns which ha\ 

 been used to find quotient terms, will give the coefficients of tlie 

 remainder. For example, we want to find some terms of the quotient 

 of a* + l divided by x t + x*-3x: 



11+0+0+0+ 1 



1+8+0+ 0+0+ 0+0 



+ 1-8+12 -21+60 



-4+ 7 -20 



-1 



+ 3 

 + 



ll+4-7 + 20|-41+60 + 

 Hence the quotient is x l + 4x~' 7x~* + 20x~ 5 , and the remainder 



When the firet coefficient is anything; but unity, fractions are intro- 

 duced into the quotient. To avoid this, proceed as follows : L. 

 the coefficient of the first term of the divisor. Multiply the sue. 

 coefficients of the dividend by 1, a, a*,a s , &c. : turn the first coefficient 

 of the divisor into 1, and multiply the second, third, fourth, &c., by 

 1, a, a", &c. Proceed as above with the coefficients thus altered, and 

 suppose that in the last line the quotient terms become u + r + w + , &c., 

 and those for the remainder ' + r'+, 4c. To find the true quotient 

 coefficients write 



* + + ^ + & c ., 

 a a o 8 



The Royl Society hu bwn wry unfortunate In lt decisions about paprrs 

 involving Improvement of calculation. Ilorncr'a paper on equations was nearly 

 rejected, and was only saved by the earnest remonstrance of Mr. I > 

 Barrett's Improvement of life contingency calculations was rejected; o waa 

 Wecldle's remarkable paper (afterwards published by himself) on the solution of 

 equations by successive factors. Homer's paper on transformation was vil>,. 

 drawn, not rejected. The report on it act forth that Mr. Homer had : 

 himself the moat able and the most dexterous of all who had recently investigated 

 the wlution of equations ; but there was a doubt as to the methods of the paper 

 being so much better than others, and possessing to much of noveltv, as to flt 

 them for the Transactions. The council were embarrassed : they could neither 

 reject nor accept the paper; they suggested that it should be withdrawn. 

 It seems then that Mtrr methods than any in use were not flt for the Phil. 

 Trans., unleu the superiority was very decided. Now, flrst, this was a vrry 

 unsound principle, and very unworthy of a scientific body; secondly, Mr. 

 Horner'n methods da posaew a Tory decided superiority, as all who have used 

 them know. 



