M A THEMATICS. ALGEBRA. 



[THEOBY OF INDICES. 



iiilitraction ; and it remains for us to ascertain what this 

 more extended meaning is. The science which concerns 

 i with this second generalisation is called Symbolical 

 Algebra. Thus, then, we hare, in all three sciences 



(1). Arithmetic, in which the symbols employed are 

 particular in form, and particular in ralu. 



Arithmetical Algebra, in which the symbols em- 

 ployed are general in farm, but particular in value. 



(3). Symbolical Algebra, in which the symbols em- 

 ployed are grneral in form, and also general in ra/. 



Thu, as the second of these sciences is a generalisation 

 of the first, so the third is a generalisation of the second. 

 The principle in accordance with which this second 

 generalisation is conducted is called that of "The per- 

 manence of equivalent forms." The principle may be 

 stated as follows : 



" Whatever algebraical forms are equivalent, when 

 the symbols are general in form, but specific in value, 

 will be equivalent likewise when the symbols are general 

 in value, as well as in form." 



For the full exposition of relations between these two 

 sciences, the advanced reader is referred to a Treatise 

 on Algebra, by George Peacock, D.D., to whom is due 

 the detection of the co-existence of these two sciences in 

 tli.it which is generally treated as one science Algebra. 

 We shall have several occasions to make use of the 

 principles above enunciated in the course of the following 

 pages. As an example of their application, we will re- 

 consider the Theory of Indices already treated in p. 465. 



2. On the Theory of Indices. 



We have already seen that o" signifies o X o X a X a, 

 ic. , to m factors, when, of course, m must be a whole 

 u umber. 



In like manner a" signifies a X a X , <tc., to n factors. 

 Hence 



o" X a oX a X a to (m + n) factors, 

 and therefore a" X a" = a" + " 



This is a result in A rithmetical A Igebra. It is perfectly 

 general in form, but it is particular in value ; for m and 

 n arc, by the definition of a, limited to being positive 

 whole numbers. If we suppose m and n to have negative 

 or fractional values, this involves a generalisation of our 

 original definition ; and the question arises, what meaning 



j> 



wo must assign to such expressions as a-" o'. To answer 

 it, wo proceed in the following manner : By assu 

 m and n general in value as well as in form, we enter 

 the domains of Symbolical Algebra ; hence, by the prin- 

 ciple of the permanence of equivalent forms, under all 

 circumstances 



a" X a" = a" + , 



.'. a" X a" X a r = a" + " X ar= a + " + ' ; 

 and so on for any number of terms. Hence 



X I, X a X . . . to q terms = +7 + f + *A" 



of. 



i.e., o must (in accordance with our general principle) 

 tignify the ' root of the ra> power of a. 

 Again, a" X a" X a-" a~ + " = a", 

 .'. a- X a-" = 1. 



. 



Heuco, we see that in assigning the meaning *J a to 

 ai, we are doing so not arbitrarily, but in accordance 

 with a principle which lies at the foundation of Algebra. 



3. On Impossible Expressions. 



Again, we know that V * In like manner if 

 we werejto liave a j , this is = a 2 X ( 1), and .-. J a" 

 a V 1 . The expression ,/ ^1 is frequently spoken 

 of an an " impossible quantity," an "imaginary ex- 

 preion," and so on, since 1 cannot bo produced by 

 multiplying cither + 1 by + 1, or 1 by 1. In 



reality, however, J 1 is as possible or as impossible 

 M 1 ; for in arithmetical Algebra a and V are "'y 

 admissible on the supposition that a is positive. In 

 symbolical Algebra this restriction is removed, and there- 

 fore in that both a and J a are admissible. Hence 

 in future investigations wo shall make use of J 1 just 

 as freely as J a, whenever it may suit our purpose, quite 

 undeterred by the circumstance of its so-called impos- 

 sibility. Of course there are many ilill'ereuces between 

 the symbols land J 1; for instance, the interpre- 

 tation of the former is a much simpler matter than the 

 interpretation of the latter, and in some cases a 6 

 belongs to arithmetical Algebra ; but a + 6 ,/ 1 never 

 does. We cannot enter further into the matter now ; 

 what we have said will be enough to explain that we are 

 justified in introducing into our calculations expressions 

 which are called "imaginary," or "impossible." 

 4. To prove that if A + A,x+A 2 x 2 H ----- -f Atvx- 



, for all values of x. 



o 2 , A n 



.+%*+..., 



Then A = a , A l - o 1 , A a 



For since A -f- A,x -j- . . . + A n x" = n + a, a; -J- . . . 

 + <inX" are equivalent for ALL values of x; they are equi- 

 valent when x = 0. . . Ao = o , and .'. A,x -f- A 2 x" 

 + . . . + A n x"= a t x +o 2 x" -f . . . + a"x" ; . . Aj -f- A .. 

 x + + A..X"- 1 = o, + a 2 x + . . . + az"-' for all 

 values of x : and hence when x = o, .'. A t = o lf and so 

 on. Hence A 2 = a 2 . . . and A, = a n . 



This is called the principle of Indeterminate Coefficients. 



It will be seen, that in the case supposed, where the 

 number of terms in each series is finite, the proof is quite 

 rigid. If each series were infinite the proof would not 

 then be conclusive ; and, accordingly, we shall refrain 

 from using this principle except in cases where no objec- 

 tion can bo raised to its use. Such as the following : 



ra - 



(1). To resolve (x _ l) (x _ 2) (x _ 3 ii 



tions. 



l_2x + 3x* A t , 



Assume ; 



UD (x l)(x 2)(x 3) x 1'x 2^x o 

 .-. 3x* 2x + l = A,(x 2)(x 3) + A.,(x 3) 

 (x 1) + A 3 (x 1) (x 2) = x a (A l + A a + A 3 ) 

 x(5A, + 4A 2 -f 3A 3 ) + GA, 3A a + 2A 3 . 

 This being tme for all values of x, we havo 

 A, -4- A. +A 3 =3. 

 r.A, +4 A, +3A 3 = 2 



..2 2 A t + Ar=-r. 



' ' ' 5 



2A, 



> i A 1 



* . Aj = 1. 



.-. A 2 = 9. 

 A a =- 11. 



= 1 9 , 11 



" '(x l)(x 2)(x 3) x 1 2^ x 3' 



N. B. A fraction written in the above form is said to 

 bo resolved into its partial fractions. 

 x 1 



_ 2x -f 



(2). Resolve 



Assume 



, . 



~ 



into its partial fractions. 



A M "< + N 



x 1 = x*(\ + M) + .''(M + N) + 3A + N 

 . A + M ~ 



M + N = 1 



3A + N = 1 

 . A N = 1 

 . 4A 2 



A -- J- 



M-J- 

 g 1 



_ 

 ' (x + 1) (x + 8) 



The student can prove the following : 



1 

 Sfx + 1) " 



i- 1 



54(-c + : ') ' 



(1). 



' 





. - 



(x + 1 ) (f. 1; (x - 1') - .",- + 1 T" X 1 , 



